# Smith–Treumann theory and the linkage principle

Simon Riche  Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-
Ferrand, France<EMAIL_ADDRESS>and  Geordie Williamson  School of
Mathematics and Statistics F07, University of Sydney NSW 2006, Australia.
<EMAIL_ADDRESS>Dedicated to Roman Bezrukavnikov,
in admiration.

###### Abstract.

In this paper we apply Treumann’s “Smith theory for sheaves” in the context of
the Iwahori–Whittaker model of the Satake category. We deduce two results in
the representation theory of reductive algebraic groups over fields of
positive characteristic: (a) a geometric proof of the linkage principle; (b) a
character formula for tilting modules in terms of the $p$-canonical basis,
valid in all blocks and in all characteristics.

##  1\. Introduction

###  1.1. Geometric representation theory of reductive algebraic groups

Let $\mathbf{G}$ be a connected reductive algebraic group over an
algebraically closed field $\Bbbk$ of characteristic $\ell>0$, and consider
the category $\operatorname{Rep}(\mathbf{G})$ of finite-dimensional algebraic
representations of $\mathbf{G}$. The study of this category has led to
significant progress in modular representation theory over the last fifty
years; however several fundamental questions (e.g. dimensions and characters
of simple and tilting modules) remain only partially understood.

One tempting avenue of pursuit is to find relationships to
$\mathscr{D}$-modules or constructible sheaves, and hence bring sheaf theory
into play. The archetypal example of the success of such an approach is the
Beĭlinson–Bernstein localization theorem, which establishes such an link for
modules over complex semi-simple Lie algebras. Beĭlinson–Bernstein
localization is an indispensable tool in modern Geometric Representation
Theory, leading to proofs of the Kazhdan–Lusztig conjectures, character
formulas for real reductive groups, etc.

Back in the setting of $\operatorname{Rep}(\mathbf{G})$, the _geometric Satake
equivalence_ provides such a connection to constructible sheaves. To
$\mathbf{G}$ we can associate the affine Grassmannian ${\mathcal{G}r}$ of its
complex Langlands dual group (an infinite-dimensional algebraic variety), and
one has an equivalence of tensor categories between
$\operatorname{Rep}(\mathbf{G})$ and a certain category of perverse sheaves
with $\Bbbk$-coefficients on ${\mathcal{G}r}$. The geometric Satake
equivalence is central to modern approaches to the Langlands program, and has
become a cornerstone of Geometric Representation Theory.

However, in contrast to Beĭlinson–Bernstein localization, the geometric Satake
equivalence has been surprisingly ineffective at answering questions about
$\operatorname{Rep}(\mathbf{G})$. For example, several basic statements and
constructions involving $\operatorname{Rep}(\mathbf{G})$ (e.g. the linkage
principle, or Frobenius twist) have no geometric explanation. This is the more
surprising, as several known or conjectured formulas (e.g. Lusztig’s character
formula) involve Kazhdan–Lusztig polynomials or their $\ell$-counterparts,
which encode dimensions of stalks of sheaves on the affine Grassmannian and
flag variety. Nowadays we have several proofs of Lusztig’s character formula
for large $\ell$, however none of them pass through the geometric Satake
equivalence!

###  1.2. Overview

The main result of the present paper is a proof of the linkage principle via
the geometric Satake equivalence. Our proof also explains that each “block” in
the linkage principle is controlled by a partial affine flag variety for the
Langlands dual group, which gives us new proofs of Lusztig’s conjecture on
simple characters (for large $\ell$) and of a conjecture of the authors on
tilting characters (for all $\ell$). The techniques of this paper provide a
powerful new tool in the study of representations of $\mathbf{G}$.

Our argument is a simple application of two new tools in Geometric
Representation Theory. The first one is Smith–Treumann theory, which is a
variant of equivariant localization for tori. In this theory the circle action
is replaced by the action of a cyclic group of order $\ell$, and the
coefficients must be of characterstic $\ell$. We apply this theory to the loop
rotation action on the affine Grassmannian. Whilst the fixed points under the
full loop rotation action are rather boring (infinitely many finite partial
flag varieties), the fixed points under the subgroup of $\ell$-th roots of
unity are rich (finitely many partial affine flag varieties).

The second ingredient is the Iwahori–Whittaker realisation of the Satake
category. This replaces the category of perverse sheaves in the Satake
equivalence with an equivalent category satisfying a certain equivariance
condition with respect to the pro-unipotent radical of the Iwahori subgroup.
(This condition is inspired by “Whittaker conditions” in the representation
theory of $p$-adic groups, hence the name.) It turns out that in the
Iwahori–Whittaker realisation, the components of the fixed points discussed
above match precisely the decomposition of $\operatorname{Rep}(\mathbf{G})$
given by the linkage principle. Our main theorem asserts that the Smith
restriction functor gives an equivalence between tilting sheaves in the
Iwahori–Whittaker realisation and a certain category of parity complexes on
the fixed points. It is then straightforward to deduce the linkage principle.
The character formulas for simple and tilting modules alluded to above are
also an immediate consequence.

In the rest of the introduction, we give a more detailed overview of the
techniques and results of this paper.

###  1.3. The linkage principle

As above, let $\mathbf{G}$ be a connected reductive algebraic group over an
algebraically closed field $\Bbbk$ of characteristic $\ell>0$, and let
$\operatorname{Rep}(\mathbf{G})$ be its category of finite-dimensional
algebraic representations. Fix a maximal torus and Borel subgroup
$\mathbf{T}\subset\mathbf{B}\subset\mathbf{G}$, and let
$\mathfrak{R}^{+}\subset\mathfrak{R}\subset\mathbf{X}$ denote the (positive)
roots inside the lattice of characters of $\mathbf{T}$.111We warn the reader
that in the body of the paper we switch to a Langlands dual notation. The
simple objects in $\operatorname{Rep}(\mathbf{G})$ are classified by dominant
weights $\mathbf{X}^{+}\subset\mathbf{X}$; given $\lambda\in\mathbf{X}^{+}$ we
denote by $\nabla(\lambda)$ the induced $\mathbf{G}$-module of highest weight
$\lambda$, and by $\mathsf{L}(\lambda)$ its simple socle.

Let $W_{\mathrm{f}}$ denote the Weyl group of $(\mathbf{G},\mathbf{T})$, and
consider the affine Weyl group

$W_{\mathrm{aff}}:=W_{\mathrm{f}}\rtimes\mathbb{Z}\mathfrak{R}$

which acts naturally on $\mathbf{X}$. The _linkage principle_ [Ve, Hu, J1, A1]
states that we have a decomposition

(1.1)
$\operatorname{Rep}(\mathbf{G})=\bigoplus_{\gamma\in\mathbf{X}/(W_{\mathrm{aff}},\bullet_{\ell})}\operatorname{Rep}_{\gamma}(\mathbf{G}),$

where each summand is the Serre subcategory

$\operatorname{Rep}_{\gamma}\mathbf{G}=\langle\mathsf{L}(\lambda):\lambda\in\gamma\cap\mathbf{X}^{+}\rangle.$

Notice that we do not consider the standard action of $W_{\mathrm{aff}}$ on
$\mathbf{X}$, but rather the “dot” action (denoted $\bullet_{\ell}$); that is,
if $\rho:=\frac{1}{2}\sum_{\alpha\in\mathfrak{R}^{+}}\alpha$, then

$(w\mathsf{t}_{\mu})\bullet_{\ell}\lambda:=w(\lambda+\ell\mu+\rho)-\rho$

for $w\in W_{\mathrm{f}}$, $\mu\in\mathbb{Z}\mathfrak{R}$ and
$\lambda\in\mathbf{X}$.

######  Remark 1.1.

The subcategory $\operatorname{Rep}_{\gamma}(\mathbf{G})$ will be called the
_block_ of $\gamma$. This is an abuse since this subcategory might be
decomposable, hence is not a “block” in the strict sense, but is convenient.
In fact the blocks of $\operatorname{Rep}(\mathbf{G})$ in the strict sense
have been described by Donkin [Do]. We believe our methods should allow to
shed some light on this description, and hope to come back to this question in
a future publication.

###  1.4. The geometric Satake equivalence

Let $H$ be the complex222In a few paragraphs we will instead assume that $H$
is defined over a field of characteristic $p$ where $p\neq 0,\ell$. reductive
group which is Langlands dual to $\mathbf{G}$, and denote its maximal torus by
$T$ (so that $X_{*}(T)=\mathbf{X}=X^{*}(\mathbf{T})$). Let $LH$ and $L^{+}H$
denote the “loop” group (ind-)schemes whose $R$ points are $H(R((z)))$ and
$H(R[[z]])$ respectively, for any $\mathbb{C}$-algebra $R$; and let
${\mathcal{G}r}_{H}:=LH/L^{+}H$ denote the affine Grassmannian. The affine
Grassmannian is an ind-projective ind-scheme whose $T$-fixed points (resp.
$L^{+}H$-orbits) are in bijection with $\mathbf{X}$ (resp. $\mathbf{X}^{+}$).

The _geometric Satake equivalence_ [L3, Gi, BD, MV1] gives an equivalence of
Tannakian categories

(1.2)
$(\mathsf{Perv}_{L^{+}H}({\mathcal{G}r}_{H},\Bbbk),\star)\cong(\operatorname{Rep}(\mathbf{G}),\otimes)$

where $\mathsf{Perv}_{L^{+}H}({\mathcal{G}r}_{H},\Bbbk)$ denotes the category
of $L^{+}H$-equivariant perverse sheaves on ${\mathcal{G}r}_{H}$ with
coefficients in $\Bbbk$, with its natural convolution product $\star$.

###  1.5. Smith–Treumann theory

A fundamental role in our proof is played by Treumann’s “Smith theory for
sheaves” [Tr]. The basic idea is that, when dealing with coefficients of
characteristic $\ell$, one should be able localize to fixed points of actions
of cyclic groups of order $\ell$. (This theory should be compared with Borel’s
“localization theorem” for manifolds equipped with an action of the circle
group; in this analogy, finite cyclic groups become “discrete circles”. See
[W2] for more comments on this analogy)

More precisely, let $X$ be a variety endowed with an action of the group
${\boldsymbol{\mu}}_{\ell}$ of $\ell$-th roots of unity. One has two (Verdier
dual) restriction functors

$D^{\mathrm{b}}_{{\boldsymbol{\mu}}_{\ell}}(X,\Bbbk)$$D^{\mathrm{b}}_{{\boldsymbol{\mu}}_{\ell}}(X^{{\boldsymbol{\mu}}_{\ell}},\Bbbk)$$i^{!}$$i^{*}$

between the ${\boldsymbol{\mu}}_{\ell}$-equivariant derived categories of
constructible $\Bbbk$-sheaves on $X$ and on $X^{{\boldsymbol{\mu}}_{\ell}}$.

A fundamental observation of Treumann is that the compositions of these
functors with the quotient functor to the _Smith category_

$\mathsf{Sm}_{\mathrm{Treu}}(X^{{\boldsymbol{\mu}}_{\ell}},\Bbbk):=D^{\mathrm{b}}_{{\boldsymbol{\mu}}_{\ell}}(X^{{\boldsymbol{\mu}}_{\ell}},\Bbbk)/\left\langle\text{${\boldsymbol{\mu}}_{\ell}$-perfect
complexes}\right\rangle$

become canonically isomorphic. Here an object in
$D^{\mathrm{b}}_{{\boldsymbol{\mu}}_{\ell}}(X^{{\boldsymbol{\mu}}_{\ell}},\Bbbk)$
is _${\boldsymbol{\mu}}_{\ell}$ -perfect_ if its stalks (naturally complexes
of ${\boldsymbol{\mu}}_{\ell}$-modules) may be represented by a bounded
complex of free $\Bbbk[{\boldsymbol{\mu}}_{\ell}]$-modules. The resulting
_Smith restriction_ functor

$D^{\mathrm{b}}_{{\boldsymbol{\mu}}_{\ell}}(X,\Bbbk)$$\mathsf{Sm}_{\mathrm{Treu}}(X^{{\boldsymbol{\mu}}_{\ell}},\Bbbk)$$i^{!*}$

has remarkable properties. For examples, it commutes with essentially all
sheaf theoretic functors [Tr]. It can be thought of as an analogue of
hyperbolic localization for $\mu_{\ell}$-actions.

For technical reasons (namely, to ensure that the Smith category of a point
satisfies appropriate parity vanishing properties), we use a variant of
Treumann’s construction, proposed by the second author in [W2]. Namely, we
assume that the action of ${\boldsymbol{\mu}}_{\ell}$ can be extended to an
action of the multiplicative group $\mathbb{G}_{\mathrm{m}}$ on $X$ and
consider the _equivariant Smith category_

$\mathsf{Sm}(X^{{\boldsymbol{\mu}}_{\ell}},\Bbbk):=D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}}}(X^{{\boldsymbol{\mu}}_{\ell}},\Bbbk)/\left\langle\begin{array}[]{c}\text{complexes
whose restriction}\\\ \text{to
${\boldsymbol{\mu}}_{\ell}\subset\mathbb{G}_{\mathrm{m}}$ are
${\boldsymbol{\mu}}_{\ell}$-perfect}\end{array}\right\rangle.$

With this definition, the theory of parity complexes from [JMW] applies in the
Smith quotient, which will be crucial for our arguments.333The fact that
Smith–Treumann theory can be made to accommodate the theory of parity sheaves
was first pointed out by Leslie–Lonergan [LL]. The version they use is however
different, and—from our point of view—technically more involved.

###  1.6. Fixed points

To apply this idea in our setting, note that ${\mathcal{G}r}_{H}$ has a
natural action of $\mathbb{G}_{\mathrm{m}}$ via “loop rotation,” induced by
the action of $\mathbb{G}_{\mathrm{m}}$ on $\mathbb{C}((z))$ which “rescales”
$z$. A beautiful fact (that we first learned from R. Bezrukavnikov) is that,
if ${\boldsymbol{\mu}}_{\ell}\subset\mathbb{G}_{\mathrm{m}}$ denotes (as
above) the subgroup of $\ell$-th roots of unity, we have a decomposition

(1.3)
$({\mathcal{G}r}_{H})^{\mu_{\ell}}=\bigsqcup_{\gamma\in\mathbf{X}/(W_{\mathrm{aff}},\square_{\ell})}{\mathcal{G}r}_{H,\gamma},$

where the action $\square_{\ell}$ is defined by
$(w\mathsf{t}_{\mu})\square_{\ell}\lambda=w(\lambda+\ell\mu)$ for $w\in
W_{\mathrm{f}}$, $\mu\in\mathbb{Z}\mathfrak{R}$ and $\lambda\in\mathbf{X}$.
Moreover, each component on the right-hand side is a partial affine flag
variety for the loop group $L_{\ell}H$ representing $R\mapsto
H(R((z^{\ell})))$, whose “partiality” is governed by the stabilizer of an
element in $\gamma$. For example, for $\gamma=W_{\mathrm{aff}}\square_{\ell}0$
we obtain the “thin affine Grassmannian” (defined as above for
${\mathcal{G}r}_{H}$, but now with $z$ replaced by $z^{\ell}$); and if
$\gamma$ has trivial stabiliser under $W_{\mathrm{aff}}$ then
${\mathcal{G}r}_{H,\gamma}$ is the full affine flag variety for $H$.

The similarity between (1.3) and (1.1) is rather striking; for example there
are as many components in the right-hand side of (1.3) as summands in the
decomposition (1.1). However there is a fundamental difference: (1.1) involves
the dot action (with $W_{\mathrm{f}}$ fixing $-\rho$); whereas (1.3) involves
the unshifted action (with $W_{\mathrm{f}}$ fixing $0$). Thus we do not expect
the Smith restriction functor to realise the linkage principle in this
setting.444The effect of Smith restriction in this setting is investigated in
[LL]. The authors show that it realises the “Frobenius contraction” functor of
Gros–Kaneda [GK]..

###  1.7. The Iwahori-Whittaker model

To get around this issue, we replace the “traditional” Satake category
$\mathsf{Perv}_{L^{+}H}({\mathcal{G}r}_{H},\Bbbk)$ with the “Iwahori–Whittaker
model” considered in [BGMRR]. There it is proved that (under a mild
assumption, satisfied e.g. if $H$ is of adjoint type, which we assume from now
on for simplicity) one has an equivalence of abelian categories

(1.4)
$\mathsf{Perv}_{L^{+}H}({\mathcal{G}r}_{H},\Bbbk)\xrightarrow{\sim}\mathsf{Perv}_{\mathcal{IW}}({\mathcal{G}r}_{H},\Bbbk)$

where the right-hand side denotes a category of perverse sheaves of
${\mathcal{G}r}_{H}$ which satisfy a certain equivariance condition with
respect to the pro-unipotent radical $\mathrm{Iw}_{\mathrm{u}}^{+}$ of an
Iwahori subgroup; such perverse sheaves are called “Iwahori–Whittaker.”555One
can make sense of this condition in various ways. In this work (following
Bezrukavnikov) we use étale sheaves and the Artin–Schreier covering, which
necessitates passing to ${\mathcal{G}r}_{H}$ defined over a field of
characteristic $p>0$ (with $p\neq\ell$). The geometric Satake equivalence is
also available in this setting.

A crucial point for us is that on simple objects the equivalence (1.4) sends
the intersection cohomology complex associated with the $L^{+}H$-orbit
parametrized by $\lambda\in\mathbf{X}^{+}$ (which corresponds to
$\mathsf{L}(\lambda)$ under (1.2)) to the Iwahori–Whittaker intersection
cohomology complex associated with the $\mathrm{Iw}_{\mathrm{u}}^{+}$-orbit
parametrized by $\lambda+\rho$. Thus, after passage to the Iwahori–Whittaker
model, our issue with the two distinct actions goes away, and the linkage
principle is reflected perfectly in geometry of the
${\boldsymbol{\mu}}_{\ell}$-fixed points. In particular, two simple
Iwahori–Whittaker perverse sheaves, parametrized by some weights $\lambda$ and
$\mu$, lie in the same summand in the linkage principle if and only if the
corresponding fixed points $L_{\lambda}$ and $L_{\mu}$ lie in the same
component of the fixed points!

Another favorable property of the $\mathrm{Iw}_{\mathrm{u}}^{+}$-action on
${\mathcal{G}r}_{H}$ is that each orbit is isomorphic to an affine space. This
setting is known to imply nice properties for categories of perverse sheaves
(see e.g. [BGS]), and in particular that this category admits a transparent
structure of highest weight category. The situation is even more favorable
here in that the “relevant” orbits (i.e. those which support a nonzero
Iwahori–Whittaker local system) have dimensions of constant parity in each
connected component of ${\mathcal{G}r}_{H}$. This implies that the tilting
objects in $\mathsf{Perv}_{\mathcal{IW}}({\mathcal{G}r}_{H},\Bbbk)$ are parity
in the sense of [JMW]; in particular the indecomposable tilting perverse
sheaves coincide with the self-dual indecomposable parity objects.

###  1.8. Main theorems

Recall that we let $\mathbb{G}_{\mathrm{m}}$ act on ${\mathcal{G}r}_{H}$ via
loop rotation. The Iwahori–Whittaker condition and the loop rotation
equivariance are compatible; we thus obtain a Smith restriction functor

$i^{!*}:D^{\mathrm{b}}_{\mathcal{IW},\mathbb{G}_{\mathrm{m}}}({\mathcal{G}r}_{H},\Bbbk)\to\mathsf{Sm}_{\mathcal{IW}}(({\mathcal{G}r}_{H})^{{\boldsymbol{\mu}}_{\ell}},\Bbbk).$

We write
$\mathsf{Parity}_{\mathcal{IW},\mathbb{G}_{\mathrm{m}}}({\mathcal{G}r}_{H},\Bbbk)$
(resp.
$\mathsf{SmParity}_{\mathcal{IW}}(({\mathcal{G}r}_{H})^{{\boldsymbol{\mu}}_{\ell}},\Bbbk)$)
for the additive category of parity sheaves in the source (resp. target) of
this functor. Our first main result is the following.

######  Theorem 1.2.

Smith restriction yields a fully faithful functor

$i^{!*}:\mathsf{Parity}_{\mathcal{IW},\mathbb{G}_{\mathrm{m}}}({\mathcal{G}r}_{H},\Bbbk)\xrightarrow{\sim}\mathsf{SmParity}_{\mathcal{IW}}(({\mathcal{G}r}_{H})^{{\boldsymbol{\mu}}_{\ell}},\Bbbk).$

Remarkably, the proof of this theorem is a few lines once one has the
appropriate technology in place. It is an easy consequence of Beĭlinson’s
lemma, once one knows that $i^{!*}$ preserves standard and costandard objects;
this in turn follows because $i^{!*}$ commutes with $*$\- and $!$-extension.

Recall from §1.7 that the self-dual indecomposable Iwahori–Whittaker parity
complexes on ${\mathcal{G}r}_{H}$ coincide with the indecomposable tilting
perverse sheaves in $\mathsf{Perv}_{\mathcal{IW}}({\mathcal{G}r}_{G})$ (which
in turn correspond to tilting modules under the geometric Satake equivalence).
Given $\lambda\in\rho+\mathbf{X}^{+}$, we denote by
$\mathscr{E}_{\lambda}^{\mathcal{IW}}$ the corresponding indecomposable parity
complex.

By Theorem 1.2, the image of any $\mathscr{E}^{\mathcal{IW}}_{\lambda}$ under
$i^{!*}$ has to be supported on a single component. This has the following
consequence, from which one easily obtains the promised proof of the linkage
principle.

######  Corollary 1.3.

If
$\operatorname{Hom}(\mathscr{E}^{\mathcal{IW}}_{\lambda},\mathscr{E}^{\mathcal{IW}}_{\mu})\neq
0$ then
$W_{\mathrm{aff}}\square_{\ell}\lambda=W_{\mathrm{aff}}\square_{\ell}\mu$.

Theorem 1.2 implies that many questions about tilting Iwahori–Whittaker
perverse sheaves on ${\mathcal{G}r}_{G}$ (and hence about tilting
$\mathbf{G}$-modules) may be answered after applying Smith restriction.
However, in order for this to be useful, one needs another way of
understanding the Smith parity complexes on
$({\mathcal{G}r}_{H})^{{\boldsymbol{\mu}}_{\ell}}$, which is the subject of
our second main result.

Consider the following diagram of quotient and forgetful functors:

$\mathsf{Sm}_{\mathcal{IW}}(({\mathcal{G}r}_{H})^{{\boldsymbol{\mu}}_{\ell}},\Bbbk)$$D^{\mathrm{b}}_{\mathcal{IW}_{\ell},\mathbb{G}_{\mathrm{m}}}(({\mathcal{G}r}_{H})^{{\boldsymbol{\mu}}_{\ell}},\Bbbk)$$D^{\mathrm{b}}_{\mathcal{IW}_{\ell}}(({\mathcal{G}r}_{H})^{{\boldsymbol{\mu}}_{\ell}},\Bbbk)$.$Q$$\mathsf{For}$

(Here the subscript in $\mathcal{IW}_{\ell}$ indicates that the
Iwahori–Whittaker condition is imposed with respect to the action of an
Iwahori subgroup in $L_{\ell}H$ now.) Our second main theorem is the
following.

######  Theorem 1.4.

The functors $Q$ and $\mathsf{For}$ preserve indecomposable parity complexes.

The first step towards the proof of Theorem 1.4 is the observation that given
parity complexes $\mathscr{E},\mathscr{F}\in
D^{\mathrm{b}}_{\mathcal{IW}_{\ell},\mathbb{G}_{\mathrm{m}}}(({\mathcal{G}r}_{H})^{{\boldsymbol{\mu}}_{\ell}},\Bbbk)$
of the same parity we have canonical isomorphisms:

(1.5)
$\displaystyle\operatorname{Hom}^{\bullet}(\mathsf{For}(\mathscr{E}),\mathsf{For}(\mathscr{F}))$
$\displaystyle=\operatorname{Hom}^{\bullet}(\mathscr{E},\mathscr{F})\otimes_{x\mapsto
0}\Bbbk.$ (1.6)
$\displaystyle\operatorname{Hom}(Q(\mathscr{E}),Q(\mathscr{F}))$
$\displaystyle=\operatorname{Hom}^{\bullet}(\mathscr{E},\mathscr{F})\otimes_{x\mapsto
1}\Bbbk.$

(Here, $x\in\mathsf{H}^{2}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\Bbbk)$
denotes the equivariant parameter, and
$\operatorname{Hom}^{\bullet}(\mathscr{E},\mathscr{F})$ is viewed as an
$\mathsf{H}^{\bullet}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\Bbbk)$-module in
the natural way. The tensor products are taken over
$\mathsf{H}^{\bullet}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\Bbbk)$, with the
indicated module structure on $\Bbbk$.) The isomorphism (1.5) is simply the
equivariant formality of homomorphisms between parity complexes, which follows
from a standard parity argument. The isomorphism (1.6) essentially follows
from the analysis of the Smith category of a point.

The statement for $\mathsf{For}$ in Theorem 1.4 is immediate from (1.5), and
is a basic ingredient in the theory of parity complexes. The statement for $Q$
is potentially more surprising, as inverting the equivariant parameter rarely
preserves indecomposability. The key point is that the
$\mathbb{G}_{\mathrm{m}}$-action on
$({\mathcal{G}r}_{H})^{{\boldsymbol{\mu}}_{\ell}}$ factors through an action
of
$\mathbb{G}_{\mathrm{m}}/{\boldsymbol{\mu}}_{\ell}=\mathbb{G}_{\mathrm{m}}$.
As a consequence, this action is indistinguishable from the trivial action
when we take $\Bbbk$-coefficients. Hence we obtain canonical isomorphisms:

(1.7) $\displaystyle\operatorname{Hom}^{\bullet}(\mathscr{E},\mathscr{F})$
$\displaystyle=\mathsf{H}^{\bullet}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\Bbbk)\otimes_{\Bbbk}\operatorname{Hom}^{\bullet}(\mathsf{For}(\mathscr{E}),\mathsf{For}(\mathscr{F})),$
(1.8) $\displaystyle\operatorname{Hom}(Q(\mathscr{E}),Q(\mathscr{F}))$
$\displaystyle=\operatorname{Hom}^{\bullet}(\mathsf{For}(\mathscr{E}),\mathsf{For}(\mathscr{F})).$

Now the statement for $Q$ in Theorem 1.4 follows from the statement for
$\mathsf{For}$, as a finite-dimensional graded algebra is local if and only if
its degree-$0$ part is.

######  Remark 1.5.

The isomorphism (1.8) shows that “$Q\circ\mathsf{For}^{-1}$” behaves like a
degrading functor. Degrading functors are ubiquitous in modern Geometric
Representation Theory. In algebra, they are often realised by forgetting the
grading on a graded module; in geometry they are often associated with
forgetting a mixed structure. The above shows that Smith–Treumann theory
provides another possible geometric realisation of degrading functors.

###  1.9. Tilting characters

A fundamental question in the representation theory of $\mathbf{G}$ is to
determine the characters of the indecomposable tilting modules. In [RW1] we
started advocating the idea that character formulas for $\mathbf{G}$-modules
should be expressed in terms of the $\ell$-canonical basis of
$W_{\mathrm{aff}}$, and illustrated this idea by a conjectural formula for
characters of indecomposable tilting modules in the principal block, under the
assumption that $\ell$ is bigger than the Coxeter number $h$ of $\mathbf{G}$.
This formula was proved in case $\mathbf{G}=\mathrm{GL}_{n}(\Bbbk)$ in [RW1],
and then for a general reductive group in a joint work with P. Achar and S.
Makisumi, see [AMRW]. A simple consequence of the results of §1.8 is a new and
much simpler proof of this character formula, along with an extension to a
formula valid in all blocks of $\operatorname{Rep}(\mathbf{G})$, without any
restriction of $\ell$.

To state this formula, recall that the summands on the right-hand side of
(1.1) can be parametrized by the weights in the intersection $\mathcal{A}$ of
the weight lattice with the closure of the fundamental alcove for the dot
action of $W_{\mathrm{aff}}$. For $\lambda\in\mathcal{A}$ we denote by
$W_{\lambda}\subset W_{\mathrm{aff}}$ the stabilizer of $\lambda$ for
$\bullet_{\ell}$ (a standard parabolic subgroup), and by
$W_{\mathrm{aff}}^{(\lambda)}$ the subset of $W_{\mathrm{aff}}$ consisting of
elements $w$ which are both maximal in $wW_{\lambda}$ and minimal in
$W_{\mathrm{f}}w$. Then the indecomposable tilting $\mathbf{G}$-modules in the
block of $W_{\mathrm{aff}}\bullet_{\ell}\lambda$ are in a natural bijection
with $W_{\mathrm{ext}}^{(\lambda)}$, and we denote by
$\mathsf{T}(w\bullet_{\ell}\lambda)$ the module of highest weight
$w\bullet_{\ell}\lambda$.

The tilting character formula alluded to above can be stated as follows.

######  Theorem 1.6.

For any $\lambda\in\mathcal{A}$ and $w\in W_{\mathrm{aff}}^{\lambda}$ we have

$\mathsf{ch}(\mathsf{T}(w\bullet_{\ell}\lambda))=\sum_{y\in
W_{\mathrm{aff}}^{(\lambda)}}{}^{\ell}n_{y,w}(1)\cdot\chi_{y\bullet_{\ell}0},$

where $\chi_{\mu}$ is the Weyl character formula attached to the dominant
weight $\mu$, and ${}^{\ell}n_{y,w}$ is the antispherical
$\ell$-Kazhdan–Lusztig polynomial attached to $(y,w)$.

See [RW1] for a comparison of this formula with an earlier formula conjectured
by Andersen [A2], which was one of our sources of inspiration.

###  1.10. Simple characters

Using ideas of Andersen [A2] recently refined by Sobaje [Sob], from the
formula in Theorem 1.6 one can in theory deduce a character formula for simple
$\mathbf{G}$-modules, in all blocks and all characteristics. This can be done
in at least two ways. The first possibility is to use a “reciprocity formula”
due to Andersen [A3] (based on earlier work of Jantzen) and which expresses
multiplicities of simple modules in Weyl modules in terms of multiplicities of
induced modules in indecomposable tilting modules. This method has the
advantage of allowing to deduce Lusztig’s conjectural formula [L1] in case the
relevant $\ell$-Kazhdan–Lusztig polynomials coincide with the corresponding
standard Kazhdan–Lusztig polynomials, but it requires the assumption that
$p\geq 2h-2$, and does not produce a very natural formula in general, since it
involves a certain “twist” (denoted $y\mapsto\hat{y}$ below) on indices.

To explain this in more detail, let us assume that $p\geq 2h-2$ and that
$\mathbf{G}$ is quasi-simple, and denote by $\alpha_{0}^{\vee}$ the highest
coroot. We then set

$Y:=\\{w\in W_{\mathrm{aff}}\mid\text{$w$ is minimal in $W_{\mathrm{f}}w$ and
$\langle w\square_{\ell}\rho,\alpha_{0}^{\vee}\rangle<p(h-1)$}\\}.$

This subset does not depend on $p$, and is an ideal in the Bruhat order on the
set of elements $w$ minimal in $W_{\mathrm{f}}w$; in fact, in terms of the
notation of [RW2], it consists of the elements $w$ sending the fundamental
alcove $A_{\mathrm{fund}}$ inside the portion of the dominant region delimited
by the hyperplane orthogonal to $\alpha_{0}^{\vee}$ and passing through
$\rho$. Consider also the operation $y\mapsto\hat{y}$ on $W_{\mathrm{aff}}$
corresponding to the operation $A\mapsto\hat{A}$ on alcoves considered in
[Soe] or [RW2]. Then, in view of [RW1, Proposition 1.8.1], from Theorem 1.6 we
obtain that for any $w\in Y$ we have

$[\nabla(w\bullet_{\ell}0)]=\sum_{y\in
Y}{}^{\ell}n_{w,\hat{y}}(1)\cdot[\mathsf{L}(y\bullet_{\ell}0)]$

in the Grothendieck group of $\operatorname{Rep}(\mathbf{G})$. In order to
compare this formula with that in Lusztig’s conjecture, one needs to invert
these equations. In general we do not know how to do that explicitly. However,
if we assume that each polynomial ${}^{\ell}n_{w,\hat{y}}$ in these formulas
coincides with the corresponding “standard” Kazhdan–Lusztig polynomial
$n_{w,\hat{y}}$ (as considered e.g. in [Soe]), then the inverse matrix is
computed in [Soe, Theorem 5.1]; from this result we obtain that

$[\mathsf{L}(w\bullet_{\ell}0)]=\sum_{y\in
Y}(-1)^{\ell(w)+\ell(y)}h_{y,w}(1)\cdot[\nabla(y\bullet_{\ell}0)]$

for any $w\in Y$, as predicted by Lusztig in [L1]. This property is well known
to hold in large characteristic (without any explicit bound), which explains
why Theorem 1.6 provides a new proof of Lusztig’s conjecture in large
characteristics.

######  Remark 1.7.

The condition on $w$ considered above is not the same as in Lusztig’s
formulation of his conjecture. However, the two versions are known to be
equivalent under the present assumptions, due to results of Kato; see [W1,
§§1.12–1.13] for more details and references.

The other method to obtain a character formula for simple
$\mathbf{G}$-modules, which works for all values of $p$ thanks to the results
of [Sob], is to express multiplicities of the simple $\mathbf{G}$-modules
whose highest weight is restricted in the baby Verma
$\mathbf{G}_{1}\mathbf{T}$-modules. In this way one obtains a formula that may
be compared with the “periodic” formulation of Lusztig’s conjecture, see [L2].
This formula was made explicit in [RW2], under the assumption that $p\geq
2h-1$. The extension of the tilting character formula in Theorem 1.6 makes it
desirable to extend the validity of these results to smaller values of $p$,
and we plan to come back to this question in a future publication.

###  1.11. Acknowledgements

It is a pleasure and an honor to dedicate this paper to Roman Bezrukavnikov,
who taught us so much about our subject. His inspiring influence on our works
should be clear, and cannot be overestimated.

We thank Luc Illusie and Weizhe Zheng for their help with the proof of
Proposition 2.6. We would also like to thank Ofer Gabber who provided an
alternative proof. Finally we thank Timo Richarz for indicating to us the
reference [dCHL], whose constructions play a crucial role in the proof of
Proposition 4.6, and Pramod Achar for pointing some typos.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 677147).

##  2\. Preliminaries on equivariant sheaves

###  2.1. Equivariant sheaves

We start by recalling some generalities on étales sheaves on schemes endowed
with an action of a finite group. We fix a coefficient ring $\Bbbk$ (not
necessarily assumed to be commutative).

Let $X$ be a scheme, and let $A$ be a finite group acting on $X$ (by scheme
automorphisms). For any $g\in A$, we denote by
$\alpha_{g}:X\xrightarrow{\sim}X$ the action on $X$. Recall that an
$A$-equivariant (étale) sheaf of $\Bbbk$-modules is the datum of an étale
sheaf $\mathscr{F}$ of $\Bbbk$-modules on $X$ together with a collection
$(\varphi_{g})_{g\in A}$ where, for any $g\in A$,

$\varphi_{g}:\alpha_{g}^{*}\mathscr{F}\xrightarrow{\sim}\mathscr{F}$

is an isomorphism of sheaves of $\Bbbk$-modules, this collection satisfying
the condition that for $g,h\in A$ we have

(2.1) $\varphi_{h}\circ\alpha_{h}^{*}(\varphi_{g})=\varphi_{gh}$

as morphisms from $\alpha_{gh}^{*}\mathscr{F}$ to $\mathscr{F}$. (We will
often abuse notation, and omit the isomorphisms $(\varphi_{g})_{g\in A}$ from
the notation.) Morphisms of $A$-equivariant sheaves are defined as morphisms
of sheaves compatible (in the natural way) with the isomorphisms
$\varphi_{g}$. The (abelian) category of $A$-equivariant sheaves of
$\Bbbk$-modules will be denoted $\mathsf{Sh}_{A}(X,\Bbbk)$. We have a natural
“forgetful” exact functor

(2.2) $\mathsf{For}_{A}:\mathsf{Sh}_{A}(X,\Bbbk)\to\mathsf{Sh}(X,\Bbbk)$

(which simply forgets the collection of isomorphisms $(\varphi_{g})_{g\in
A}$), where $\mathsf{Sh}(X,\Bbbk)$ denotes the category of sheaves of
$\Bbbk$-modules on $X$. If $A$ acts trivially on $X$, we have a canonical
identification

(2.3) $\mathsf{Sh}_{A}(X,\Bbbk)=\mathsf{Sh}(X,\Bbbk[A]).$

If $X,Y$ are two schemes with actions of $A$ (with actions denoted
$\alpha^{X}_{-}$ and $\alpha^{Y}_{-}$ respectively), and $f:X\to Y$ is an
$A$-equivariant morphism, then for any $g\in A$ we have a canonical
isomorphism

$(\alpha^{X}_{g})^{*}\circ f^{*}\cong f^{*}\circ(\alpha^{Y}_{g})^{*}.$

As a consequence, the functor $f^{*}$ induces an exact functor

$\mathsf{Sh}_{A}(Y,\Bbbk)\to\mathsf{Sh}_{A}(X,\Bbbk),$

which will also be denoted $f^{*}$. Similarly, we have a canonical isomorphism

$(\alpha^{Y}_{g})^{*}\circ f_{*}\cong f_{*}\circ(\alpha^{X}_{g})^{*}.$

(Here we use the fact that
$(\alpha^{Z}_{g})^{*}\cong(\alpha^{Z}_{g^{-1}})_{*}$ for $Z=X$ or $Y$.) As a
consequence, $f_{*}$ induces a functor

$f_{*}:\mathsf{Sh}_{A}(X,\Bbbk)\to\mathsf{Sh}_{A}(Y,\Bbbk).$

###  2.2. Equivariant sheaves and injective resolutions

We consider again a scheme $X$ endowed with an action of the finite group $A$.
For any $\mathscr{F}$ in $\mathsf{Sh}(X,\Bbbk)$ we set

$\mathsf{Av}_{A}(\mathscr{F}):=\bigoplus_{g\in A}\alpha_{g}^{*}\mathscr{F}.$

We endow $\mathsf{Av}_{A}(\mathscr{F})$ with the structure of an
$A$-equivariant sheaf by defining, for any $g\in A$, the isomorphism

$\varphi_{g}:\alpha_{g}^{*}\mathsf{Av}_{A}(\mathscr{F})\xrightarrow{\sim}\mathsf{Av}_{A}(\mathscr{F})$

as the canonical identification

$\alpha_{g}^{*}\left(\bigoplus_{h\in
A}\alpha_{h}^{*}\mathscr{F}\right)=\bigoplus_{h\in
A}\alpha_{hg}^{*}\mathscr{F}=\bigoplus_{a\in A}\alpha_{a}^{*}\mathscr{F}.$

This construction extends in a natural way to an exact functor

$\mathsf{Av}_{A}:\mathsf{Sh}(X,\Bbbk)\to\mathsf{Sh}_{A}(X,\Bbbk).$

######  Lemma 2.1.

The functor $\mathsf{Av}_{A}$ is left and right adjoint to the forgetful
functor (2.2).

###### Proof.

To prove the lemma we have to define morphisms of functors

$\mathsf{For}_{A}\circ\mathsf{Av}_{A}\to\mathrm{id},\quad\mathrm{id}\to\mathsf{For}_{A}\circ\mathsf{Av}_{A},\quad\mathsf{Av}_{A}\circ\mathsf{For}_{A}\to\mathrm{id},\quad\mathrm{id}\to\mathsf{Av}_{A}\circ\mathsf{For}_{A},$

and check the appropriate zigzag relations. Here we have

$\mathsf{For}_{A}\circ\mathsf{Av}_{A}=\bigoplus_{g\in A}\alpha_{g}^{*},$

and the first two morphisms are defined as the projection to and embedding of
the factor $\alpha_{e}^{*}=\mathrm{id}$. On the other hand, for
$(\mathscr{F},(\varphi_{g})_{g\in A})$ in $\mathsf{Sh}_{A}(X)$, the morphisms

$\mathscr{F}\to\mathsf{Av}_{A}\circ\mathsf{For}_{A}(\mathscr{F})\to\mathscr{F}$

are defined as $\bigoplus_{g\in A}(\varphi_{g})^{-1}$ and $\bigoplus_{g\in
A}\varphi_{g}$ respectively. (These morphisms of sheaves are morphisms of
$A$-equivariant sheaves thanks to the cocyle condition (2.1).) The zigzag
relations are all trivial to check. ∎

Lemma 2.1 implies that the functor $\mathsf{Av}_{A}$ sends injective objects
of $\mathsf{Sh}(X,\Bbbk)$ to injective objects of $\mathsf{Sh}_{A}(X,\Bbbk)$.
Since the category $\mathsf{Sh}(X,\Bbbk)$ has enough injectives (see [SP, Tag
01DU]), it follows that the same property holds in $\mathsf{Sh}_{A}(X,\Bbbk)$.
In fact, if $\mathscr{F}$ belongs to $\mathsf{Sh}_{A}(X,\Bbbk)$, and if
$\mathscr{I}$ is an injective object of $\mathsf{Sh}(X,\Bbbk)$ such that we
have an injection $\mathsf{For}_{A}(\mathscr{F})\hookrightarrow\mathscr{I}$,
then the map $\mathscr{F}\to\mathsf{Av}_{A}(\mathscr{I})$ deduced by
adjunction is injective, and $\mathsf{Av}_{A}(\mathscr{I})$ is injective in
$\mathsf{Sh}_{A}(X,\Bbbk)$.

In particular, if $f:X\to Y$ is an $A$-equivariant morphism between schemes
with $A$-actions, recall that we have the (non derived) pushforward functor

$f_{*}:\mathsf{Sh}_{A}(X,\Bbbk)\to\mathsf{Sh}_{A}(Y,\Bbbk).$

From the considerations on injective objects above we deduce that this functor
admits a derived functor

$Rf_{*}:D^{+}\mathsf{Sh}_{A}(X,\Bbbk)\to D^{+}\mathsf{Sh}_{A}(Y,\Bbbk),$

which can be computed by means of injective resolutions.

######  Lemma 2.2.

For any injective object $\mathscr{I}$ in $\mathsf{Sh}_{A}(X,\Bbbk)$, the
object $\mathsf{For}_{A}(\mathscr{I})$ is injective in $\mathsf{Sh}(X,\Bbbk)$.

###### Proof.

If $\mathscr{J}$ is an injective object in $\mathsf{Sh}_{A}(X,\Bbbk)$ and
$\mathsf{For}_{A}(\mathscr{I})\hookrightarrow\mathscr{J}$ is an embedding, the
injection $\mathscr{I}\hookrightarrow\mathsf{Av}_{A}(\mathscr{J})$ must be
split since both objects are injective. Hence $\mathscr{I}$ is a direct
summand in $\mathsf{Av}_{A}(\mathscr{J})$, so that
$\mathsf{For}_{A}(\mathscr{I})$ is a direct summand in the injective sheaf
$\mathsf{For}_{A}\circ\mathsf{Av}_{A}(\mathscr{J})=\bigoplus_{g\in
A}\alpha_{g}^{*}\mathscr{J}$, hence is injective. ∎

From this lemma one obtains that for an equivariant morphism $f:X\to Y$ as
above we have a natural commutative diagram

$\textstyle{D^{+}\mathsf{Sh}_{A}(X,\Bbbk)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Rf_{*}}$$\scriptstyle{\mathsf{For}_{A}}$$\textstyle{D^{+}\mathsf{Sh}_{A}(Y,\Bbbk)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{For}_{A}}$$\textstyle{D^{+}\mathsf{Sh}(X,\Bbbk)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Rf_{*}}$$\textstyle{D^{+}\mathsf{Sh}(Y,\Bbbk),}$

where the lower horizontal arrow is the standard pushforward functor. In
particular, in case $X$ and $Y$ are of finite type over some field
$\mathbb{F}$ of finite cohomological dimension (e.g. algebraically closed),
and $\Bbbk$ is torsion (e.g. a field of positive characteristic), it is known
that the “standard” functor $Rf_{*}$ sends
$D^{\mathrm{b}}\mathsf{Sh}(X,\Bbbk)$ into
$D^{\mathrm{b}}\mathsf{Sh}(Y,\Bbbk)$, see [SP, Tag 0F10]. It follows that the
“equivariant” functor $Rf_{*}$ considered above restricts to a functor

$Rf_{*}:D^{\mathrm{b}}\mathsf{Sh}_{A}(X,\Bbbk)\to
D^{\mathrm{b}}\mathsf{Sh}_{A}(Y,\Bbbk).$

Of course, since the functor

$f^{*}:\mathsf{Sh}_{A}(Y,\Bbbk)\to\mathsf{Sh}_{A}(X,\Bbbk)$

is exact, we have an induced functor

$f^{*}:D\mathsf{Sh}_{A}(Y,\Bbbk)\to D\mathsf{Sh}_{A}(X,\Bbbk)$

which maps $D^{+}\mathsf{Sh}_{A}(Y,\Bbbk)$ into
$D^{+}\mathsf{Sh}_{A}(X,\Bbbk)$ and $D^{\mathrm{b}}\mathsf{Sh}_{A}(Y,\Bbbk)$
into $D^{\mathrm{b}}\mathsf{Sh}_{A}(X,\Bbbk)$ (and is compatible with the
usual pullback functor $f^{*}$ in the obvious way). It is easily checked that
the functor

$f^{*}:D^{+}\mathsf{Sh}_{A}(Y,\Bbbk)\to D^{+}\mathsf{Sh}_{A}(X,\Bbbk)$

is left adjoint to

$Rf_{*}:D^{+}\mathsf{Sh}_{A}(X,\Bbbk)\to D^{+}\mathsf{Sh}_{A}(Y,\Bbbk).$

###  2.3. Equivariant sheaves and quotient map

From now on we assume that $X$ is of finite type over some base scheme $S$,
and that each $\alpha_{g}$ is an automorphism of $S$-schemes. We assume
furthermore that the action is admissible in the sense of [SGA1, Exposé 5,
Définition 1.7]. Then we have a quotient scheme $X/A$, and a finite quotient
morphism $q:X\to X/A$, see [SGA1, Exposé V, Corollaire 1.5]. (Here, by
definition $X/A$ is the scheme which represents the functor
$Z\mapsto\operatorname{Hom}(X,Z)^{A}$, where $A$ acts on
$\operatorname{Hom}(X,Z)$ via its action on $X$. It can be constructed by
gluing affine schemes of the form $\operatorname{Spec}(R^{A})$ where
$\operatorname{Spec}(R)\subset X$ is an $A$-stable affine open subscheme of
$X$.)

By finiteness the functor $q_{*}$ is then exact (see [SP, Tag 03QP]), and we
therefore obtain a functor

$q_{*}:\mathsf{Sh}_{A}(X,\Bbbk)\to\mathsf{Sh}_{A}(X/A,\Bbbk)\overset{\eqref{eqn:equ-
sheaves-triv-action}}{\cong}\mathsf{Sh}(X/A,\Bbbk[A]).$

######  Remark 2.3.

As explained in [SGA1, Exposé V, Proposition 1.8], the action of $A$ on $X$ is
admissible iff each orbit of $A$ is included in an affine open subset of $X$.
This condition is automatic if $S=\operatorname{Spec}(\mathbb{F})$ for some
field $\mathbb{F}$ and $X$ is quasi-projective over $S$, see e.g. [Se, p. 59,
Exemple 1]. (This setting is the only one we will consider in practice.)

Recall that a complex $\mathscr{G}\in D\mathsf{Sh}(X/A,\Bbbk[A])$ is said to
be _of finite tor dimension_ if there exist $a,b\in\mathbb{Z}$ such that for
any $\mathscr{G}^{\prime}\in\mathsf{Sh}(X/A,\Bbbk[A]^{\mathrm{op}})$ we have

$\mathscr{H}^{n}(\mathscr{G}^{\prime}\mathchoice{\mathbin{\mathop{\otimes}^{L}}_{\Bbbk[A]}}{\otimes^{L}_{\Bbbk[A]}}{\otimes^{L}_{\Bbbk[A]}}{\otimes^{L}_{\Bbbk[A]}}\mathscr{G})=0\quad\text{unless
$n\in[a,b]$,}$

see [SP, Tag 08FZ]. Recall also that $\mathscr{G}$ has finite tor dimension
iff for any geometric point $\overline{x}$ of $X$ the complex of
$\Bbbk[A]$-modules $\mathscr{G}_{\overline{x}}$ has finite tor dimension, see
[SP, Tag 0DJJ].

The action of $A$ on $X$ induces an action on geometric points. Namely, if

$\overline{x}:\operatorname{Spec}(K)\to X$

is a geometric point and $g\in A$, then the geometric point
$g\cdot\overline{x}$ is the composition

$\operatorname{Spec}(K)\xrightarrow{\overline{x}}X\xrightarrow{\alpha_{g}}X.$

We will say that the $A$-action on $X$ is _free_ if each geometric point of
$X$ has trivial stabilizer for this action.

######  Lemma 2.4.

Assume that $\Bbbk$ is a field, and that the $A$-action on $X$ is free. Then
for any $\mathscr{G}$ in $\mathsf{Sh}_{A}(X,\Bbbk)$, the sheaf
$q_{*}\mathscr{G}\in\mathsf{Sh}(X,\Bbbk[A])$ has finite tor dimension.

###### Proof.

By the comments above, to prove the lemma it suffices to prove that for any
geometric point $\overline{y}:\operatorname{Spec}(K)\to X/A$ the
$\Bbbk[A]$-module $\mathscr{G}_{\overline{y}}$ has finite tor dimension. Now
by [SP, Tag 03QP] we have

$(q_{*}\mathscr{G})_{\overline{y}}=\bigoplus_{\overline{x}}\mathscr{G}_{\overline{x}},$

where $\overline{x}$ runs over the set $X_{\overline{y}}$ of maps
$\overline{x}:\operatorname{Spec}(K)\to X$ such that
$q\circ\overline{x}=\overline{y}$. However $q$ is surjective, and its fibers
are the $A$-orbits (see [SGA1, Exposé V, §1]). Hence $X_{\overline{y}}$ is
nonempty, and $A$ acts transitively on this set. Our assumption ensures on the
other hand that this action has trivial stabilizers. Therefore, if we fix some
$\overline{x}\in X_{\overline{y}}$, we deduce a bijection
$A\xrightarrow{\sim}X_{\overline{y}}$ determined by $g\mapsto
g\cdot\overline{x}$. The $A$-equivariant structure on $\mathscr{G}$ provides a
canonical isomorphism

$\mathscr{G}_{\overline{x}}\xrightarrow{\sim}\mathscr{G}_{g\cdot\overline{x}}$

for each $g\in A$. Using these data we obtain an isomorphism

$(q_{*}\mathscr{G})_{\overline{y}}=\Bbbk[A]\otimes_{\Bbbk}\mathscr{G}_{\overline{x}},$

which is easily seen to be $A$-equivariant. Hence
$(q_{*}\mathscr{G})_{\overline{y}}$ is free as a $\Bbbk[A]$-module, hence of
finite tor dimension. ∎

###  2.4. Stalks at fixed points

We continue with the assumptions of §2.3. The closed subscheme $X^{A}\subset
X$ is the scheme which represents the functor
$Z\mapsto\operatorname{Hom}(Z,X)^{A}$, where $A$ acts on
$\operatorname{Hom}(Z,X)$ via its action on $X$. (The representability of this
functor is easy in our setting: since our action is admissible it suffices to
treat the case $X=\operatorname{Spec}(R)$ is affine; then $X^{A}$ is the
spectrum of the maximal quotient of $R$ on which $A$ acts trivially, i.e. the
quotient of $R$ by the ideal generated by the elements $x-g\cdot x$ for $x\in
R$ and $g\in A$.) As a set, the closed subscheme $X^{A}\subset X$ consists of
the points $x\in X$ which are fixed by $A$ and such that the induced action on
the residue field $k(x)$ is trivial.

By definition, any geometric point of $X$ which is stable under the $A$-action
considered in §2.3 factors through a geometric point of $X^{A}$. In
particular, if $A$ is a simple group then the $A$-action on the open subset
$U:=X\smallsetminus X^{A}\subset X$ is free.

######  Lemma 2.5.

The $A$-action on $U$ is admissible.

###### Proof.

Consider the closed subset $X^{A}\subset X$. Since the quotient morphism $q$
is finite, hence closed, the subset $q(X^{A})\subset X/A$ is closed. Now from
the definition of the subset $X^{A}\subset X$ given above and the fact that
the fibers of $q$ are the $A$-orbits in $X$, one sees that

$U=q^{-1}(X/A\smallsetminus q(X^{A})).$

Hence the claim follows from [SGA1, Exposé V, Corollaire 1.4]. ∎

From this lemma we obtain in particular that the quotient $U/A$ exists as a
scheme. In fact, in the proof of this lemma we have seen that the open
embedding $j:U\hookrightarrow X$ induces an open embedding
$\overline{\jmath}:U/A\to X/A$, with complement $q(X^{A})$, and that the
quotient morphism $q_{U}:U\to U/A$ is the restriction of $q$ to $U$.

We now denote by $i:X^{A}\to X$ the closed embedding. Note that $i$ is
$A$-equivariant for the trivial $A$-action on $X^{A}$; we therefore have a
functor

$i^{*}:D\mathsf{Sh}_{A}(X,\Bbbk)\to
D\mathsf{Sh}_{A}(X^{A},\Bbbk)\overset{\eqref{eqn:equ-sheaves-triv-
action}}{\cong}D\mathsf{Sh}(X^{A},\Bbbk[A]).$

In other words, if $\mathscr{F}$ belongs to $D\mathsf{Sh}_{A}(X,\Bbbk)$, then
$i^{*}(\mathsf{For}_{A}(\mathscr{F}))$ admits a canonical “lift” as a complex
of sheaves of $\Bbbk[A]$-modules. In particular, for any geometric point
$\overline{x}$ of $X^{A}$ the complex

$\mathscr{F}_{i(\overline{x})}=(i^{*}\mathscr{F})_{\overline{x}}$

is in a natural way an object of $D(\Bbbk[A]\mathsf{-Mod})$.

From now on we assume that $S=\operatorname{Spec}(\mathbb{F})$ for some field
$\mathbb{F}$ of finite cohomological dimension, and that $X$ is of finite type
over $\mathbb{F}$. (This assumption implies that $U$ is also of finite type
over $\mathbb{F}$. By [SGA1, Corollaire 1.5] we deduce that $X/A$ and $U/A$
also are of finite type.) We also assume that $\Bbbk$ is a field of
characteristic $\ell>0$ which is nonzero in $\mathbb{F}$. The proof of the
following proposition was explained to us by L. Illusie and W. Zheng. (A
different, longer and slightly less easy proof can also be deduced from [DL,
Proposition 3.7].)

######  Proposition 2.6.

Assume that $A$ is a simple group. If $\mathscr{F}\in
D^{\mathrm{b}}\mathsf{Sh}_{A}(U,\Bbbk)$ is such that
$\mathsf{For}_{A}(\mathscr{F})$ has constructible cohomology sheaves (see [SP,
Tag 03RW]), then for any geometric point $\overline{x}$ of $X^{A}$ the complex
of $\Bbbk[A]$-modules

$(Rj_{*}\mathscr{F})_{\overline{x}}$

is perfect.

###### Proof.

Recall that the morphism $j$ has finite cohomological dimension by [SP, Tag
0F10]. As a consequence the complex $Rj_{*}\mathscr{F}$ is bounded, so that
$(Rj_{*}\mathscr{F})_{\overline{x}}$ is bounded. On the other hand, by
[SGA4$\frac{1}{2}$, Th. finitude, Théorème 1.1], the complex
$Rj_{*}\mathscr{F}$ is constructible; hence its stalk
$(Rj_{*}\mathscr{F})_{\overline{x}}$ has finite-dimensional cohomology. By
[SP, Tag 0658], we deduce that to prove that this complex is perfect it
suffices to prove that it has finite tor dimension. But since the image $x$ of
$\underline{x}$ is fixed by $A$ we have $q^{-1}(q(x))=\\{x\\}$, which implies
that

$(Rj_{*}\mathscr{F})_{\overline{x}}=\bigl{(}q_{*}(Rj_{*}\mathscr{F})\bigr{)}_{q(\overline{x})}$

by [SP, Tag 03QP]. Now we have $q\circ j=\overline{\jmath}\circ q_{U}$, so
that

$q_{*}(Rj_{*}\mathscr{F})=R\overline{\jmath}_{*}((q_{U})_{*}\mathscr{F}).$

By Lemma 2.4 the sheaf $(q_{U})_{*}\mathscr{F}$ has finite tor-dimension,
hence by [SGA4, Exposé XVII, Théorème 5.2.11] and [SP, Tag 0F10] the complex
$R\overline{\jmath}_{*}((q_{U})_{*}\mathscr{F})$ also does, so that its stalk

$\bigl{(}R\overline{\jmath}_{*}((q_{U})_{*}\mathscr{F})\bigr{)}_{q(\overline{x})}$

must have finite tor dimension, which finishes the proof. ∎

##  3\. Smith theory for étale sheaves

###  3.1. $\varpi$-equivariant derived categories

The formalism of “Smith theory” that we will build will use the equivariant
derived category of Bernstein–Lunts [BL]. This category is explicitly
constructed only in a topological setting in [BL], but it is well known that
it applies also in the setting of étale sheaves under appropriate assumptions,
see [BL, §4.3]. In this subsection we briefly recall this construction in the
particular case that we require.

So, from now on we fix an algebraically closed field $\mathbb{F}$ of
characteristic $p$, and a finite field $\Bbbk$ of characteristic $\ell\neq p$.
We will consider (admissible) actions of the finite $\mathbb{F}$-group scheme
$\varpi={\boldsymbol{\mu}}_{\ell}$ of $\ell$-th roots of unity on
$\mathbb{F}$-schemes of finite type. Here since $\mathbb{F}$ is algebraically
closed, ${\boldsymbol{\mu}}_{\ell}$ is the constant group scheme associated
with the finite group ${\boldsymbol{\mu}}_{\ell}(\mathbb{F})$, so that the
constructions of Section 2 also apply in this setting. For simplicity, we will
not explicitly distinguish the group scheme ${\boldsymbol{\mu}}_{\ell}$ and
the finite group ${\boldsymbol{\mu}}_{\ell}(\mathbb{F})$.

The construction of [BL] uses some “acyclic resolutions.” In this case these
resolutions can be constructed explicitly as follows: for any $n\geq 0$ we set

$V_{n}:=\mathbb{F}^{n}\smallsetminus\\{0\\},$

with the (admissible) action of $\varpi$ induced by the dilation action of the
multiplicative group $\mathbb{G}_{\mathrm{m}}$. We have

$\mathsf{H}^{m}(V_{n};\Bbbk)=\begin{cases}\Bbbk&\text{if $m=0$;}\\\ 0&\text{if
$1\leq m\leq 2n-2$.}\end{cases}$

From this we see that for any $\mathbb{F}$-scheme $X$ of finite type the
projection

$p_{X,n}:V_{n}\times X\to X$

is $(2n-2)$-acyclic, in the sense that for any $X$-scheme $Y$ of finite type
the morphism $p_{X,n}^{Y}:Y\times_{X}(V_{n}\times X)\to Y$ is such that for
any (étale) $\Bbbk$-sheaf $\mathscr{F}$ the morphism

$\mathscr{F}\to\tau_{\leq
2n-2}((p_{X,n}^{Y})_{*}(p_{X,n}^{Y})^{*}\mathscr{F})$

induced by adjunction is an isomorphism. (In fact, here by the Künneth formula
[SP, Tag 0F1N] we have
$(p_{X,n}^{Y})_{*}(p_{X,n}^{Y})^{*}\mathscr{F}\cong\mathsf{H}^{\bullet}(V_{n},\Bbbk)\otimes_{\Bbbk}\mathscr{F}$.)

We now fix an $\mathbb{F}$-scheme $X$ of finite type endowed with an
admissible action of $\varpi$. For any $n\geq 1$ we set

$P_{n}^{X}:=V_{n}\times X,$

and consider the projection $f_{n}^{X}:P_{n}^{X}\to X$ as above. Since the
actions of $\varpi$ on $V_{n}$ and $X$ are admissible, this property also
holds for the product $V_{n}\times X$, so that we can consider the quotient

$\overline{P}_{n}^{X}:=P_{n}^{X}/\varpi.$

With the notation of §2.4, we have $(P_{n}^{X})^{\varpi}=\varnothing$;
therefore, by [SGA1, Exposé V, Corollaire 2.3] the quotient map
$q_{n}^{X}:P_{n}^{X}\to\overline{P}_{n}^{X}$ is étale. In fact, in view of
[SGA1, Exposé V, Proposition 2.6] this map is an étale locally trivial
principal homogeneous space for $\varpi$ in the sense of [SP, Tag 049A].

For any $n$, we will denote by

$D^{\mathrm{b}}(X,\varpi,n,\Bbbk)$

the category whose

  * •

objects are triples $(\mathscr{F}_{n},\mathscr{F}_{X},\beta)$ where
$\mathscr{F}_{n}$ is an object of
$D^{\mathrm{b}}\mathsf{Sh}(\overline{P}_{n}^{X},\Bbbk)$, $\mathscr{F}_{X}$ is
an object of $D^{\mathrm{b}}\mathsf{Sh}(X,\Bbbk)$, and

$\beta:(q_{n}^{X})^{*}\mathscr{F}_{n}\xrightarrow{\sim}(p_{n}^{X})^{*}\mathscr{F}_{X}$

is an isomorphism;

  * •

morphisms from $(\mathscr{F}_{n},\mathscr{F}_{X},\beta)$ to
$(\mathscr{G}_{n},\mathscr{G}_{X},\gamma)$ are pairs
$(\varphi_{n},\varphi_{X})$ with

$\varphi_{n}:\mathscr{F}_{n}\to\mathscr{G}_{n},\qquad\varphi_{X}:\mathscr{F}_{X}\to\mathscr{G}_{X}$

such that
$\gamma\circ((q_{n}^{X})^{*}\varphi_{n})=((p_{n}^{X})^{*}\varphi_{X})\circ\beta$.

For any bounded interval $I\subset\mathbb{Z}$ we denote by
$D^{I}(X,\varpi,n,\Bbbk)$ the full subcategory of
$D^{\mathrm{b}}(X,\varpi,n,\Bbbk)$ whose objects are the triples
$(\mathscr{F}_{n},\mathscr{F}_{X},\beta)$ where
$\mathscr{H}^{m}(\mathscr{F}_{X})$ vanishes unless $m\in I$. Then the category
$D^{I}(X,\varpi,n,\Bbbk)$ does not depend (up to canonical equivalence) on the
choice of $n$, as long as $2n-2\geq|I|$, where $|I|$ is the length of $I$; in
fact, by the same arguments as in [BL, §2.3.4], if $n,m$ satisfy this
condition then the natural functors from $D^{I}(X,\varpi,n,\Bbbk)$ and
$D^{I}(X,\varpi,m,\Bbbk)$ to the category defined similarly with $P_{n}^{X}$
and $P_{m}^{X}$ replaced by

$V_{n}\times V_{m}\times X=P_{n}^{X}\times_{X}P_{m}^{X}$

(with the diagonal $\varpi$-action) are equivalences of categories.

We can therefore define the $\varpi$-equivariant derived category

$D^{\mathrm{b}}_{\varpi}(X,\Bbbk)$

as the direct limit of the categories $D^{I}(X,\varpi,n,\Bbbk)$ where $I$ runs
over the bounded intervals of $\mathbb{Z}$. This category admits a canonical
structure of triangulated category, see [BL, §§2.5.1–2.5.2]. By construction,
we have a canonical triangulated forgetful functor

(3.1) $\mathsf{For}_{\varpi}:D^{\mathrm{b}}_{\varpi}(X,\Bbbk)\to
D^{\mathrm{b}}\mathsf{Sh}(X,\Bbbk)$

which sends a triple $(\mathscr{F}_{n},\mathscr{F}_{X},\beta)$ to
$\mathscr{F}_{X}$.

######  Remark 3.1.

As explained in [BL, Lemma 2.3.2], if $2n-2\geq|I|$ the functor

$D^{I}(X,\varpi,n,\Bbbk)\to
D^{\mathrm{b}}\mathsf{Sh}(\overline{P}_{n}^{X},\Bbbk)$

sending $(\mathscr{F}_{n},\mathscr{F}_{X},\beta)$ to $\mathscr{F}_{n}$ is
fully faithful. In particular, morphisms between objects in
$D^{\mathrm{b}}_{\varpi}(X,\Bbbk)$ can always be computed as morphisms in
derived categories of quotients of “sufficiently acyclic” resolutions.

This construction is of course functorial in $X$. Namely, consider
$\mathbb{F}$-schemes of finite type $X,Y$ endowed with admissible actions of
$\varpi$, and a $\varpi$-equivariant morphism of $\mathbb{F}$-schemes $f:X\to
Y$. (Note that $f$ is automatically quasi-compact since $X$ is Noetherian. It
is also locally of finite type by [SP, Tag 01T8], hence of finite type.)

  1. (1)

We have a ($*$-)pullback functor

$f^{*}:D^{\mathrm{b}}_{\varpi}(Y,\Bbbk)\to D^{\mathrm{b}}_{\varpi}(X,\Bbbk),$

which can be explicitly described in terms of the pullback functors associated
with $f$ and the induced morphism
$\overline{P}_{n}^{X}\to\overline{P}_{n}^{Y}$.

  2. (2)

If $f$ is quasi-separated we have a ($*$-)pushforward functor

$Rf_{*}:D^{\mathrm{b}}_{\varpi}(X,\Bbbk)\to D^{\mathrm{b}}_{\varpi}(Y,\Bbbk).$

(Here we use the fact that the usual pushforward functors respect the bounded
derived categories, see [SP, Tag 0F10], and also that the morphisms
$p_{n}^{X}$, $p_{n}^{Y}$, $q_{n}^{X}$, $q_{n}^{Y}$ are smooth and that the
induced morphism $\overline{P}_{n}^{X}\to\overline{P}_{n}^{Y}$ is quasi-
compact and quasi-separated, which allows to use the smooth base change
theorem [SP, Tag 0EYU] to “transport” the isomorphism $\beta$.)

  3. (3)

If we assume that $f$ is separated and that $Y$ is quasi-separated, then we
also have a $!$-pushforward functor

$Rf_{!}:D^{\mathrm{b}}_{\varpi}(X,\Bbbk)\to D^{\mathrm{b}}_{\varpi}(Y,\Bbbk),$

see [SGA4, Exposé XVII]. (Here, the fact that the $!$-pushforward functors
respect bounded derived categories follows from [SGA4, Exposé XVII, Corollaire
5.2.8.1], and we use the base change theorem [SGA4, Exposé XVII, Théorème
5.2.6] to transport $\beta$.)

  4. (4)

Under these assumptions we also have a $!$-pullback functor

$f^{!}:D^{\mathrm{b}}_{\varpi}(Y,\Bbbk)\to D^{\mathrm{b}}_{\varpi}(X,\Bbbk),$

see [SGA4, Exposé XVIII]. (The fact that the $!$-pullback functors respect
bounded derived categories is explained in [SGA4$\frac{1}{2}$, Th. finitude,
comments after Corollaire 1.5]. And once again we use the smoothness of
$q_{n}^{X}$, $q_{n}^{Y}$, $p_{n}^{X}$, $p_{n}^{Y}$, and the fact that for
smooth maps the $*$\- and $!$-pullback functors coincide up to shift, see
[SGA4, Exposé XVIII, Théorème 3.2.5], to transport the isomorphisms $\beta$.)

By construction, all of these functors are compatible with the forgetful
functor (3.1) in the obvious way.

######  Remark 3.2.

In practice all the schemes we will consider will be quasi-projective, hence
separated, so that any morphism between them will automatically be separated.

###  3.2. Equivariant derived categories and equivariant sheaves

We continue with the setting of §3.1. In Section 2 we have studied equivariant
sheaves on schemes, and in §3.1 we have considered the equivariant derived
category. It is now time to explain the relation between these two
constructions. This relation is based on the observation that (for any
$\mathbb{F}$-scheme $X$ of finite type with an admissible $\varpi$-action, and
any $n\geq 1$) the natural pullback functor

$\mathsf{Sh}(\overline{P}^{X}_{n},\Bbbk)\to\mathsf{Sh}_{\varpi}(P^{X}_{n},\Bbbk)$

is an equivalence of categories, by the sheaf condition applied to the étale
covering $q_{n}^{X}:P_{n}^{X}\to\overline{P}_{n}^{X}$. Therefore, for any
$\varpi$-equivariant sheaf $\mathscr{F}$ on $X$ the pullback
$(p_{n}^{X})^{*}\mathscr{F}$ admits a natural structure of
$\varpi$-equivariant sheaf on $P_{n}^{X}$, hence descends to a sheaf
$\mathscr{F}_{n}$ on $\overline{P}_{n}^{X}$. Using this construction we define
a canonical triangulated functor

(3.2) $D^{\mathrm{b}}\mathsf{Sh}_{\varpi}(X,\Bbbk)\to
D^{\mathrm{b}}_{\varpi}(X,\Bbbk).$

######  Proposition 3.3.

The functor (3.2) is an equivalence of categories.

###### Proof.

Let us first show that our functor is fully faithful. For this we need to show
that for any $\varpi$-equivariant sheaves $\mathscr{F},\mathscr{F}^{\prime}$
on $X$ and any $m\in\mathbb{Z}_{\geq 0}$, for $n\gg 0$ the natural map

$\operatorname{Hom}_{D^{\mathrm{b}}\mathsf{Sh}_{\varpi}(X,\Bbbk)}(\mathscr{F},\mathscr{F}^{\prime}[m])\to\operatorname{Hom}_{D^{\mathrm{b}}\mathsf{Sh}(\overline{P}^{X}_{n},\Bbbk)}(\mathscr{F}_{n},\mathscr{F}^{\prime}_{n}[m])$

is an isomorphism, see Remark 3.1. By construction, this amounts to proving
that for $n\gg 0$ the pullback functor induces an isomorphism

$\operatorname{Hom}_{D^{\mathrm{b}}\mathsf{Sh}_{\varpi}(X,\Bbbk)}(\mathscr{F},\mathscr{F}^{\prime}[m])\xrightarrow{\sim}\operatorname{Hom}_{D^{\mathrm{b}}\mathsf{Sh}_{\varpi}(P_{n}^{X},\Bbbk)}((p_{n}^{X})^{*}\mathscr{F},(p_{n}^{X})^{*}\mathscr{F}^{\prime}[m]).$

However by adjunction we have

$\operatorname{Hom}_{D^{\mathrm{b}}\mathsf{Sh}_{\varpi}(P_{n}^{X},\Bbbk)}((p_{n}^{X})^{*}\mathscr{F},(p_{n}^{X})^{*}\mathscr{F}^{\prime}[m])\\\
\cong\operatorname{Hom}_{D^{\mathrm{b}}\mathsf{Sh}_{\varpi}(X,\Bbbk)}(\mathscr{F},R(p_{n}^{X})_{*}(p_{n}^{X})^{*}\mathscr{F}^{\prime}[m]).$

Now the right-hand side can be replaced by

$\operatorname{Hom}_{D^{\mathrm{b}}\mathsf{Sh}_{\varpi}(X,\Bbbk)}(\mathscr{F},\tau_{\leq
0}(R(p_{n}^{X})_{*}(p_{n}^{X})^{*}\mathscr{F}^{\prime}[m]))\\\
\cong\operatorname{Hom}_{D^{\mathrm{b}}\mathsf{Sh}_{\varpi}(X,\Bbbk)}(\mathscr{F},\tau_{\leq
m}(R(p_{n}^{X})_{*}(p_{n}^{X})^{*}\mathscr{F}^{\prime})[m]).$

If $2n-2\geq m$ the morphism

$\mathscr{F}^{\prime}\to\tau_{\leq
m}(R(p_{n}^{X})_{*}(p_{n}^{X})^{*}\mathscr{F}^{\prime})$

induced by adjunction is an isomorphism, which concludes the proof of fully
faithfulness.

Once fully faithfulness is established, to conclude the proof it suffices to
prove that images of $\varpi$-equivariant sheaves on $X$ generate
$D^{\mathrm{b}}_{\varpi}(X,\Bbbk)$ as a triangulated category. This is however
clear from the construction of the “standard” t-structure in [BL,
§§2.5.1–2.5.2]. ∎

In particular, in the special case where the $\varpi$-action on $X$ is
trivial, using Proposition 3.3 combined with (2.3), we obtain a canonical
equivalence of triangulated categories

(3.3) $D^{\mathrm{b}}_{\varpi}(X,\Bbbk)\cong
D^{\mathrm{b}}\mathsf{Sh}(X,\Bbbk[\varpi]).$

We now consider two $\mathbb{F}$-schemes $X$ and $Y$ of finite type, with
admissible actions of $\varpi$, and a quasi-separated $\varpi$-equivariant
morphism $f:X\to Y$. We have considered functors

$\textstyle{D^{\mathrm{b}}\mathsf{Sh}_{\varpi}(X,\Bbbk)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Rf_{*}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
D^{\mathrm{b}}\mathsf{Sh}_{\varpi}(Y,\Bbbk)}$$\scriptstyle{f^{*}}$

in §2.2, and functors

$\textstyle{D^{\mathrm{b}}_{\varpi}(X,\Bbbk)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Rf_{*}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
D^{\mathrm{b}}_{\varpi}(Y,\Bbbk)}$$\scriptstyle{f^{*}}$

in §3.1. These functors are related in the natural way, as explained in the
following lemma.

######  Lemma 3.4.

The diagrams

$\textstyle{D^{\mathrm{b}}\mathsf{Sh}_{\varpi}(Y,\Bbbk)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\scriptstyle{\wr}$$\textstyle{D^{\mathrm{b}}\mathsf{Sh}_{\varpi}(X,\Bbbk)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\textstyle{D^{\mathrm{b}}_{\varpi}(Y,\Bbbk)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\textstyle{D^{\mathrm{b}}_{\varpi}(X,\Bbbk)}$
and
$\textstyle{D^{\mathrm{b}}\mathsf{Sh}_{\varpi}(X,\Bbbk)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Rf_{*}}$$\scriptstyle{\wr}$$\textstyle{D^{\mathrm{b}}\mathsf{Sh}_{\varpi}(Y,\Bbbk)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\textstyle{D^{\mathrm{b}}_{\varpi}(X,\Bbbk)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Rf_{*}}$$\textstyle{D^{\mathrm{b}}_{\varpi}(Y,\Bbbk)}$

are commutative, where the vertical arrows are the equivalences of Proposition
3.3.

###### Proof.

The commutativity of the left diagram can be seen on the definitions; the
commutativity of the right diagram follows by adjunction. ∎

###  3.3. The crucial lemma

We can now prove the lemma that will allow us to develop the “Smith theory for
sheaves” from [Tr] in our setting of étale sheaves.

We consider again an $\mathbb{F}$-scheme $X$ of finite type, with an
admissible action of $\varpi$. As in §2.4 we consider the fixed points
subscheme $X^{\varpi}$ and the closed, resp. open, embedding

$i:X^{\varpi}\hookrightarrow X,\quad\text{resp.}\quad j:X\smallsetminus
X^{\varpi}\hookrightarrow X.$

(Here $i$ and $j$ are automatically separated, see [SP, Tag 01L7].) For any
$\mathscr{F}$ in $D^{\mathrm{b}}_{\varpi}(X,\Bbbk)$ we have a canonical
morphism

(3.4) $i^{!}\mathscr{F}\to i^{*}\mathscr{F}$

in the category $D^{\mathrm{b}}_{\varpi}(X^{\varpi},\Bbbk)$, which can be
obtained by applying the functor $i^{*}$ to the adjunction morphism
$i_{!}i^{!}\mathscr{F}\to\mathscr{F}$.

From now on we will not consider the entire $\varpi$-equivariant derived
category $D^{\mathrm{b}}_{\varpi}(X,\Bbbk)$, but only the full triangulated
subcategory $D^{\mathrm{b}}_{\varpi,c}(X,\Bbbk)$ whose objects are those
$\mathscr{F}\in D^{\mathrm{b}}_{\varpi}(X,\Bbbk)$ such that the complex
$\mathsf{For}_{\varpi}(\mathscr{F})$ has constructible cohomology objects,
where $\mathsf{For}_{\varpi}$ is as in (3.1). For any $\mathbb{F}$-scheme $Y$
of finite type with trivial action of $\varpi$, we will say that an object
$\mathscr{F}$ in $D^{\mathrm{b}}_{\varpi,c}(Y,\Bbbk)$ has perfect geometric
stalks if, denoting by $\mathscr{F}^{\prime}$ the image of $\mathscr{F}$ under
the equivalence $D^{\mathrm{b}}_{\varpi}(Y,\Bbbk)\cong
D^{\mathrm{b}}\mathsf{Sh}(Y,\Bbbk[\varpi])$ from (3.3), for any geometric
point $\overline{y}$ of $Y$ the complex $\mathscr{F}^{\prime}_{\overline{y}}$
is a perfect complex of $\Bbbk[\varpi]$-modules.

######  Lemma 3.5.

For any $\mathscr{F}$ in $D^{\mathrm{b}}_{\varpi,c}(X,\Bbbk)$, the cone of
(3.4) has perfect geometric stalks.

###### Proof.

From the standard distinguished triangle in the “recollement” formalism we see
that the cone of (3.4) is isomorphic to $i^{*}Rj_{*}j^{*}(\mathscr{F})$. The
complex we want to consider is therefore

$\bigl{(}Rj_{*}j^{*}(\mathscr{F})\bigr{)}_{\overline{x}},$

where $\overline{x}$ is a geometric point of $X^{\varpi}$. In these terms, the
desired claim follows from Proposition 2.6 and Lemma 3.4. ∎

Later we will also need the following lemma, whose proof is close to that of
Lemma 3.5. Here we consider two $\mathbb{F}$-schemes of finite type $Z$ and
$Y$ with trivial actions of $\varpi$, and a quasi-separated morphism of
$\mathbb{F}$-schemes $f:Z\to Y$. Then we have a derived functor
$Rf_{*}:D^{\mathrm{b}}_{\varpi}(Z,\Bbbk)\to D^{\mathrm{b}}_{\varpi}(Y,\Bbbk)$,
see §3.1, which sends the subcategory $D^{\mathrm{b}}_{\varpi,c}(Z,\Bbbk)$
into $D^{\mathrm{b}}_{\varpi,c}(Y,\Bbbk)$ by [SGA4$\frac{1}{2}$, Th. finitude,
Théorème 1.1].

######  Lemma 3.6.

The functor

$Rf_{*}:D^{\mathrm{b}}_{\varpi,c}(Z,\Bbbk)\to
D^{\mathrm{b}}_{\varpi,c}(Y,\Bbbk)$

transforms objects with perfect geometric stalks into objects with perfect
geometric stalks.

###### Proof.

As in the proof of Proposition 2.6 it suffices to check that $Rf_{*}$
transforms objects of finite tor dimension into objects of finite tor
dimension, which follows from [SGA4, Exposé XVII, Théorème 5.2.11] and [SP,
Tag 0F10]. ∎

###  3.4. $\mathbb{G}_{\mathrm{m}}$-equivariant derived categories

Later we will also need to consider equivariant derived categories for actions
of the multiplicative group $\mathbb{G}_{\mathrm{m}}$ over $\mathbb{F}$. The
construction of this category is similar to, and in fact simpler than, the
construction in §3.1. Namely, for any $n\geq 1$ the $\varpi$-action on $V_{n}$
is obtained by restriction from a natural $\mathbb{G}_{\mathrm{m}}$-action,
and moreover we have a canonical map $V_{n}\to\mathbb{P}_{\mathbb{F}}^{n-1}$
which is a Zariski locally trivial principal $\mathbb{G}_{\mathrm{m}}$-bundle.
Therefore, given any $\mathbb{F}$-scheme $X$ endowed with an action of
$\mathbb{G}_{\mathrm{m}}$, we consider the diagonal action on $V_{n}\times X$,
and we have a Zariski locally trivial principal
$\mathbb{G}_{\mathrm{m}}$-bundle

$V_{n}\times X\to V_{n}\times^{\mathbb{G}_{\mathrm{m}}}X$

for some scheme $V_{n}\times^{\mathbb{G}_{\mathrm{m}}}X$ which can be
constructed by (Zariski) gluing over the natural cover which trivializes the
map $V_{n}\to\mathbb{P}_{\mathbb{F}}^{n-1}$. If we assume $X$ to be of finite
type, then as in §3.1 the map $p_{X,n}:V_{n}\times X\to X$ is
$(2n-2)$-acyclic, which allows to define a category
$D^{\mathrm{b}}(X,\mathbb{G}_{\mathrm{m}},n,\Bbbk)$ in terms similar to those
for $D^{\mathrm{b}}(-,\varpi,n,\Bbbk)$, and check that the subcategory
$D^{I}(X,\mathbb{G}_{\mathrm{m}},n,\Bbbk)$ does not depend on the choice of
$n$ as long as $2n-2\geq|I|$. We can finally define the equivariant derived
category $D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}}}(X,\Bbbk)$ as the direct
limit of the categories $D^{I}(X,\mathbb{G}_{\mathrm{m}},n,\Bbbk)$ (with $n\gg
0$) over the finite intervals $I\subset\mathbb{Z}$. These categories have the
same functoriality properties as the categories
$D^{\mathrm{b}}_{\varpi}(X,\Bbbk)$; in particular we have a natural
(triangulated) forgetful functor

$\mathsf{For}_{\mathbb{G}_{\mathrm{m}}}:D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}}}(X,\Bbbk)\to
D^{\mathrm{b}}\mathsf{Sh}(X,\Bbbk).$

As in the $\varpi$-equivariant setting, we will denote by
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(X,\Bbbk)$ the full subcategory of
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}}}(X,\Bbbk)$ whose objects are those
$\mathscr{F}$ such that $\mathsf{For}_{\mathbb{G}_{\mathrm{m}}}(\mathscr{F})$
has constructible cohomology sheaves.

The category $D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}}}(X,\Bbbk)$ has a
canonical object whose image under $\mathsf{For}_{\mathbb{G}_{\mathrm{m}}}$ is
the constant sheaf $\underline{\Bbbk}_{X}$; it will also be denoted
$\underline{\Bbbk}_{X}$. In these terms, the
$\mathbb{G}_{\mathrm{m}}$-equivariant cohomology of a complex $\mathscr{F}$ in
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}}}(X,\Bbbk)$ is defined as

$\mathsf{H}^{\bullet}_{\mathbb{G}_{\mathrm{m}}}(X,\mathscr{F})=\bigoplus_{n\in\mathbb{Z}}\operatorname{Hom}_{D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}}}(X,\Bbbk)}(\underline{\Bbbk}_{X},\mathscr{F}[n]).$

In the case $X=\operatorname{Spec}(\mathbb{F})=:\mathrm{pt}$, it is well known
(and easy to see) that we have a graded algebra isomorphism666To be more
precise, to get the isomorphism (3.5) one needs to fix a trivialization of the
Tate sheaf on $\mathrm{pt}$, see e.g. the proof of Lemma 3.9 below. This is
possible—though not canonical—since $\mathbb{F}$ is algebraically closed; we
fix such a trivialization once and for all.

(3.5)
$\mathsf{H}^{\bullet}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\underline{\Bbbk}_{\mathrm{pt}})=\Bbbk[x],$

where $x$ has degree $2$. If $n$ is even, we will denote by

$\mathrm{can}_{\mathrm{pt}}^{n}:\underline{\Bbbk}_{\mathrm{pt}}\to\underline{\Bbbk}_{\mathrm{pt}}[n]$

the morphism obtained as the inverse image of $x^{n/2}$.

By definition we have an embedding $\varpi\subset\mathbb{G}_{\mathrm{m}}$,
which provides a $\varpi$-action on $X$ by restriction, and we have a
canonical “restriction” triangulated functor

$\operatorname{Res}^{\mathbb{G}_{\mathrm{m}}}_{\varpi}:D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}}}(X,\Bbbk)\to
D^{\mathrm{b}}_{\varpi}(X,\Bbbk).$

This functor is compatible (in the obvious sense) with the pushforward and
pullback functors when they are defined, and moreover satisfies

$\mathsf{For}_{\varpi}\circ\operatorname{Res}^{\mathbb{G}_{\mathrm{m}}}_{\varpi}\cong\mathsf{For}_{\mathbb{G}_{\mathrm{m}}}.$

As a consequence, it must send
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(X,\Bbbk)$ into
$D^{\mathrm{b}}_{\varpi,c}(X,\Bbbk)$.

###  3.5. The “Smith category” of a point

In this subsection we consider the special case of the constructions of §3.4
where $X=\mathrm{pt}$. In this case, in view of (3.3) we have an equivalence
of triangulated categories

(3.6) $D^{\mathrm{b}}_{\varpi}(\mathrm{pt},\Bbbk)\cong
D^{\mathrm{b}}(\Bbbk[\varpi]\mathsf{-Mod}).$

Under this equivalence, the full subcategory
$D^{\mathrm{b}}_{\varpi,c}(\mathrm{pt},\Bbbk)$ corresponds to the full
subcategory of $D^{\mathrm{b}}(\Bbbk[\varpi]\mathsf{-Mod})$ whose objects are
the complexes whose cohomology is finite dimensional (or equivalently finitely
generated over $\Bbbk[\varpi])$, which itself is canonically equivalent to the
category $D^{\mathrm{b}}(\Bbbk[\varpi]\mathsf{-Mof})$, where
$\Bbbk[\varpi]\mathsf{-Mof}$ is the category of finite-dimensional
$\Bbbk[\varpi]$-modules.

We will denote by

$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)_{\varpi\mathsf{-perf}}\subset
D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)$

the full triangulated subcategory whose objects are the complexes
$\mathscr{F}$ such that
$\operatorname{Res}^{\mathbb{G}_{\mathrm{m}}}_{\varpi}(\mathscr{F})$,
considered as a complex of $\Bbbk[\varpi]$-modules through (3.6), is perfect
(i.e. isomorphic in $D^{\mathrm{b}}(\Bbbk[\varpi]\mathsf{-Mod})$ to a bounded
complex of finitely generated projective modules).

We then set

$\mathsf{Sm}(\mathrm{pt},\Bbbk):=D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)/D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)_{\varpi\mathsf{-perf}},$

where we consider the Verdier quotient category.

The following lemma will be crucial for us below, in that it will allow to use
some parity vanishing arguments in various variants of the category
$\mathsf{Sm}(\mathrm{pt},\Bbbk)$.

######  Lemma 3.7.

For any $n\in\mathbb{Z}$ we have

$\operatorname{Hom}_{\mathsf{Sm}(\mathrm{pt},\Bbbk)}(\underline{\Bbbk}_{\mathrm{pt}},\underline{\Bbbk}_{\mathrm{pt}}[n])=\begin{cases}\Bbbk&\text{if
$n$ is even;}\\\ 0&\text{otherwise,}\end{cases}$

where we omit the quotient functor
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)\to\mathsf{Sm}(\mathrm{pt},\Bbbk)$.

The proof of Lemma 3.7 will require some preparation. We start with the
following claim.

######  Lemma 3.8.

For any $\mathscr{F}$ in
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)$, there exists
a canonical isomorphism of graded $\Bbbk$-vector spaces

$\bigoplus_{m\in\mathbb{Z}}\operatorname{Hom}_{D^{\mathrm{b}}_{\varpi}(\mathrm{pt},\Bbbk)}\bigl{(}\underline{\Bbbk}_{\mathrm{pt}},\operatorname{Res}^{\mathbb{G}_{\mathrm{m}}}_{\varpi}(\mathscr{F})[m]\bigr{)}\cong\mathsf{H}^{\bullet}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\mathscr{F})\oplus\mathsf{H}^{\bullet}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\mathscr{F})[1].$

###### Proof.

We fix $\mathscr{F}$ in
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)$ and
$m\in\mathbb{Z}$. Then for $n\gg 0$ the object $\mathscr{F}$ is represented by
a triple $(\mathscr{F}_{n},\mathscr{F}_{X},\beta)$ in
$D^{\mathrm{b}}(\mathrm{pt},\mathbb{G}_{\mathrm{m}},n,\Bbbk)$, and by an
analogue of Remark 3.1 we have

$\mathsf{H}^{m}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\mathscr{F})=\operatorname{Hom}_{D^{\mathrm{b}}(\mathbb{P}_{\mathbb{F}}^{n-1},\Bbbk)}(\underline{\Bbbk}_{\mathbb{P}_{\mathbb{F}}^{n-1}},\mathscr{F}_{n}[m]),$

and similarly for
$\mathsf{H}^{m+1}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\mathscr{F})$. If we
denote by

$\pi_{n}:V_{n}/\varpi\to\mathbb{P}_{\mathbb{F}}^{n-1}$

the natural map, then
$\operatorname{Res}^{\mathbb{G}_{\mathrm{m}}}_{\varpi}(\mathscr{F})$ is
represented by the object
$((\pi_{n})^{*}\mathscr{F}_{n},\mathscr{F}_{X},\beta)$ in
$D^{\mathrm{b}}(\mathrm{pt},\varpi,n,\Bbbk)$, so that we have

$\operatorname{Hom}_{D^{\mathrm{b}}_{\varpi}(\mathrm{pt},\Bbbk)}\bigl{(}\underline{\Bbbk}_{\mathrm{pt}},\operatorname{Res}^{\mathbb{G}_{\mathrm{m}}}_{\varpi}(\mathscr{F})[m]\bigr{)}=\operatorname{Hom}_{D^{\mathrm{b}}(V_{n}/\varpi,\Bbbk)}(\underline{\Bbbk}_{V_{n}/\varpi},(\pi_{n})^{*}\mathscr{F}_{n}[m]).$

To prove the lemma, it therefore suffices to prove that for any $\mathscr{G}$
in $D^{\mathrm{b}}(\mathbb{P}_{\mathbb{F}}^{n-1},\Bbbk)$ we have a canonical
isomorphism

(3.7)
$\operatorname{Hom}_{D^{\mathrm{b}}(V_{n}/\varpi,\Bbbk)}(\underline{\Bbbk}_{V_{n}/\varpi},(\pi_{n})^{*}\mathscr{G})\cong\\\
\operatorname{Hom}_{D^{\mathrm{b}}(\mathbb{P}_{\mathbb{F}}^{n-1},\Bbbk)}(\underline{\Bbbk}_{\mathbb{P}_{\mathbb{F}}^{n-1}},\mathscr{G})\oplus\operatorname{Hom}_{D^{\mathrm{b}}(\mathbb{P}_{\mathbb{F}}^{n-1},\Bbbk)}(\underline{\Bbbk}_{\mathbb{P}_{\mathbb{F}}^{n-1}},\mathscr{G}[1]).$

We start by proving that

(3.8)
$(\pi_{n})_{*}\underline{\Bbbk}_{V_{n}/\varpi}=\underline{\Bbbk}_{\mathbb{P}_{\mathbb{F}}^{n-1}}\oplus\underline{\Bbbk}_{\mathbb{P}_{\mathbb{F}}^{n-1}}[-1].$

In fact, $V_{n}/\varpi$ is the complement of the zero section in the line
bundle $\tilde{\pi}_{n}:\mathscr{O}(\ell)\to\mathbb{P}_{\mathbb{F}}^{n-1}$. If
$i_{n}:\mathbb{P}_{\mathbb{F}}^{n-1}\hookrightarrow\mathscr{O}(\ell)$ is the
embedding of the zero section, and
$j_{n}:V_{n}/\varpi\hookrightarrow\mathscr{O}(\ell)$ is the complementary open
embedding, then we have a distinguished triangle

$(i_{n})_{*}\underline{\Bbbk}_{\mathbb{P}_{\mathbb{F}}^{n-1}}[-2]\to\underline{\Bbbk}_{\mathscr{O}(\ell)}\to
j_{*}\underline{\Bbbk}_{V_{n}/\varpi}\xrightarrow{[1]}.$

Applying the functor $(\tilde{\pi}_{n})_{*}$ we deduce a distinguished
triangle

$\underline{\Bbbk}_{\mathbb{P}_{\mathbb{F}}^{n-1}}[-2]\to\underline{\Bbbk}_{\mathbb{P}_{\mathbb{F}}^{n-1}}\to(\pi_{n})_{*}\underline{\Bbbk}_{V_{n}/\varpi}\xrightarrow{[1]},$

in which is the first map is (by definition) the (shift by $-2$ of the) Euler
class of $\mathscr{O}(\ell)$. Since $\Bbbk$ has characteristic $\ell$ this
Euler class vanishes, and we deduce the desired isomorphism (3.8).

Next, we claim that for any $\mathscr{G}$ in
$D^{\mathrm{b}}\mathsf{Sh}(\mathbb{P}_{\mathbb{F}}^{n-1},\Bbbk)$ we have a
canonical isomorphism

(3.9)
$\mathscr{G}\otimes_{\Bbbk}(\pi_{n})_{*}\underline{\Bbbk}_{V_{n}/\varpi}\xrightarrow{\sim}(\pi_{n})_{*}(\pi_{n})^{*}\mathscr{G}.$

In fact adjunction provides a canonical morphism from the left-hand side to
the right-hand side. To prove that this morphism is invertible it suffices to
check this property after pullback under the surjective morphism $\pi_{n}$.
However $\pi_{n}$ is a principal
$\mathbb{G}_{\mathrm{m}}/\varpi=\mathbb{G}_{\mathrm{m}}$-bundle, so that we
have a Cartesian diagram

$\textstyle{V_{n}/\varpi\times\mathbb{G}_{\mathrm{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{n}/\varpi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{n}}$$\textstyle{V_{n}/\varpi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{n}}$$\textstyle{\mathbb{P}_{\mathbb{F}}^{n-1},}$

hence the claim follows from the smooth base change theorem [SP, Tag 0EYU].

Combining (3.8) and (3.9) we obtain, for any $\mathscr{G}$ in
$D^{\mathrm{b}}\mathsf{Sh}(\mathbb{P}_{\mathbb{F}}^{n-1},\Bbbk)$, an
isomorphism

$(\pi_{n})_{*}(\pi_{n})^{*}\mathscr{G}\cong\mathscr{G}\oplus\mathscr{G}[-1].$

In view of the isomorphism

$\operatorname{Hom}_{D^{\mathrm{b}}(V_{n}/\varpi,\Bbbk)}(\underline{\Bbbk}_{V_{n}/\varpi},(\pi_{n})^{*}\mathscr{G})=\operatorname{Hom}_{D^{\mathrm{b}}(V_{n}/\varpi,\Bbbk)}((\pi_{n})^{*}\underline{\Bbbk}_{\mathbb{P}_{\mathbb{F}}^{n-1}},(\pi_{n})^{*}\mathscr{G})\\\
\cong\operatorname{Hom}_{D^{\mathrm{b}}(\mathbb{P}_{\mathbb{F}}^{n-1},\Bbbk)}(\underline{\Bbbk}_{\mathbb{P}_{\mathbb{F}}^{n-1}},(\pi_{n})_{*}(\pi_{n})^{*}\mathscr{G}),$

this implies (3.7), hence finishes the proof of the lemma. ∎

Using Lemma 3.8 we will be able to give a more explicit description of the
category
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)_{\varpi\mathsf{-perf}}$,
as follows.

######  Lemma 3.9.

The full subcategory
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)_{\varpi\mathsf{-perf}}\subset
D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)$ consists of the
complexes $\mathscr{F}$ such that

$\dim_{\Bbbk}\bigl{(}\mathsf{H}^{\bullet}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\mathscr{F})\bigr{)}<\infty.$

As a consequence, in $\mathsf{Sm}(\mathrm{pt},\Bbbk)$ we have a canonical
isomorphism of functors

$\mathrm{id}\cong[2].$

###### Proof.

Recall from (3.5) that there exists a canonical morphism

$\mathrm{can}_{\mathrm{pt}}^{2}:\underline{\Bbbk}_{\mathrm{pt}}\to\underline{\Bbbk}_{\mathrm{pt}}[2]$

in $D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\Bbbk)$. More
explicitly, this morphism can be constructed as follows: consider the natural
dilation action of $\mathbb{G}_{\mathrm{m}}$ on $\mathbb{A}_{\mathbb{F}}^{1}$.
If we denote by
$i:\mathrm{pt}=\\{0\\}\hookrightarrow\mathbb{A}_{\mathbb{F}}^{1}$ the
embedding, then we have
$i^{!}(\underline{\Bbbk}_{\mathbb{A}_{\mathbb{F}}^{1}})=\underline{\Bbbk}_{\mathrm{pt}}[-2]$.
Using adjunction, we deduce a canonical map

$i_{*}(\underline{\Bbbk}_{\mathrm{pt}}[-2])\to\underline{\Bbbk}_{\mathbb{A}_{\mathbb{F}}^{1}}.$

(Note that we have ignored a Tate twist here; see Footnote 6.) Applying
$i^{*}$ and shifting by $2$, we obtain the morphism
$\mathrm{can}_{\mathrm{pt}}^{2}$. Since
$(\mathbb{A}_{\mathbb{F}}^{1})^{\varpi}=\\{0\\}$, from this description and
Lemma 3.5 we obtain that the cone $C$ of $\mathrm{can}_{\mathrm{pt}}^{2}$
belongs to
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)_{\varpi\mathsf{-perf}}$.
In particular, since the tensor product of any bounded complex with a perfect
complex is perfect, this implies that for any $\mathscr{F}$ in
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\Bbbk)$ we have a
canonical isomorphism

$\mathscr{F}\xrightarrow{\sim}\mathscr{F}[2],$

providing the desired isomorphism of functors $\mathrm{id}\cong[2]$.

Now we claim that the triangulated subcategory $\langle C\rangle_{\Delta}$ of
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)$ generated by
$C$ is exactly the subcategory whose objects are the complexes $\mathscr{F}$
such that

$\dim_{\Bbbk}\bigl{(}\mathsf{H}^{\bullet}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\mathscr{F})\bigr{)}<\infty.$

Indeed we have
$\mathsf{H}^{\bullet}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},C)=\Bbbk[2]$, so
that $C$ belongs to this subcategory. To prove the opposite inclusion, we
prove by induction that for any $n\in\mathbb{Z}_{\geq 0}$, any complex
$\mathscr{F}$ such that
$\dim_{\Bbbk}\bigl{(}\mathsf{H}^{\bullet}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\mathscr{F})\bigr{)}=n$
belongs to $\langle C\rangle_{\Delta}$. In fact, if $n=0$ then using Lemma 3.8
we see that any object $\mathscr{F}$ such that
$\mathsf{H}^{\bullet}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\mathscr{F})=0$
satisfies
$\operatorname{Res}^{\mathbb{G}_{\mathrm{m}}}_{\varpi}(\mathscr{F})=0$, hence
$\mathsf{For}_{\mathbb{G}_{\mathrm{m}}}(\mathscr{F})=0$. From the definition,
we see that this implies that $\mathscr{F}=0$. Fix now $n\geq 0$, and assume
the result is known for $n$. If
$\dim_{\Bbbk}\bigl{(}\mathsf{H}^{\bullet}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\mathscr{F})\bigr{)}=n+1$,
and if $m$ is maximal such that

$\mathsf{H}^{m}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\mathscr{F})\neq 0,$

then any choice of a nonzero vector in this space provides a morphism

$\underline{\Bbbk}_{\mathrm{pt}}[-m]\to\mathscr{F}$

in $D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}}}(X,\Bbbk)$. By maximality the
composition of this map with $\mathrm{can}_{\mathrm{pt}}^{2}[-m-2]$ vanishes,
so that this map must factor through a morphism

$C[-m-2]\to\mathscr{F}.$

From the long exact sequence in equivariant cohomology we see that the cone
$\mathscr{G}$ of this map satisfies
$\dim_{\Bbbk}\bigl{(}\mathsf{H}^{\bullet}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\mathscr{G})\bigr{)}=n$,
which allows to conclude by induction.

The two claims we have proved so far show that the subcategory with objects
those complexes $\mathscr{F}$ such that
$\mathsf{H}^{\bullet}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\mathscr{F})$ is
finite dimensional is included in
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)_{\varpi\mathsf{-perf}}$.
On the other hand, if $\mathscr{F}$ is an object of
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)$ such that
$\operatorname{Res}^{\mathbb{G}_{\mathrm{m}}}_{\varpi}(\mathscr{F})$ is
perfect, then

$\dim_{\Bbbk}\left(\bigoplus_{n\in\mathbb{Z}}\operatorname{Hom}_{D^{\mathrm{b}}_{\varpi}(\mathrm{pt},\Bbbk)}(\underline{\Bbbk}_{\mathrm{pt}},\operatorname{Res}^{\mathbb{G}_{\mathrm{m}}}_{\varpi}(\mathscr{F})[n])\right)<\infty.$

From Lemma 3.8 we deduce that in this case
$\mathsf{H}^{\bullet}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\mathscr{F})$ is
finite dimensional, which concludes the proof. ∎

We can finally give the proof of Lemma 3.7.

###### Proof of Lemma 3.7.

Lemma 3.9 shows in particular that $\underline{\Bbbk}_{\mathrm{pt}}$ does not
belong to
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)_{\varpi\mathsf{-perf}}$,
hence has nonzero image in $\mathsf{Sm}(\mathrm{pt},\Bbbk)$. In view of the
isomorphism $\mathrm{id}\cong[2]$, this shows that
$\operatorname{Hom}_{\mathsf{Sm}(\mathrm{pt},\Bbbk)}(\underline{\Bbbk}_{\mathrm{pt}},\underline{\Bbbk}_{\mathrm{pt}}[n])\neq
0$ for any even $n$. Hence to conclude it only remains to prove that

$\dim\operatorname{Hom}_{\mathsf{Sm}(\mathrm{pt},\Bbbk)}(\underline{\Bbbk}_{\mathrm{pt}},\underline{\Bbbk}_{\mathrm{pt}}[n])\leq\begin{cases}1&\text{if
$n$ is even;}\\\ 0&\text{otherwise}\end{cases}$

A morphism $\varpi$ from $\underline{\Bbbk}_{\mathrm{pt}}$ to
$\underline{\Bbbk}_{\mathrm{pt}}$ in $\mathsf{Sm}(\mathrm{pt},\Bbbk)$ is
represented by a diagram

$\underline{\Bbbk}_{\mathrm{pt}}\xleftarrow{f}\mathscr{F}\xrightarrow{g}\underline{\Bbbk}_{\mathrm{pt}}[n]$

in which $\mathscr{F}$ belongs to
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)$, $f$ and $g$
are morphisms in
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)$, and the cone
of $f$ belongs to
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)_{\varpi\mathsf{-perf}}$,
i.e. has finite-dimensional cohomology (see Lemma 3.9). In particular, from
the long exact sequence in equivariant cohomology and (3.5) we obtain that
there exists $N\in 2\mathbb{Z}$ (which, for later use, we will assume to be at
least $-n$) such that for $m\geq N$ we have

$\mathsf{H}^{m}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\mathscr{F})=\begin{cases}\Bbbk&\text{if
$m$ is even;}\\\ 0&\text{otherwise.}\end{cases}$

If we choose a nonzero element in
$\mathsf{H}^{N}_{\mathbb{G}_{\mathrm{m}}}(\mathrm{pt},\mathscr{F})$,
considered as a morphism $h:\underline{\Bbbk}_{\mathrm{pt}}[-N]\to\mathscr{F}$
in $D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)$, then the
cone of $h$ has finite-dimensional equivariant cohomology, i.e. belongs to
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)_{\varpi\mathsf{-perf}}$
by Lemma 3.9. As a consequence, $\varpi$ can also be represented by the
diagram

$\underline{\Bbbk}_{\mathrm{pt}}\xleftarrow{f\circ
h}\underline{\Bbbk}_{\mathrm{pt}}[-N]\xrightarrow{g\circ
h}\underline{\Bbbk}_{\mathrm{pt}}[n].$

In case $n$ is odd we have

$\operatorname{Hom}_{D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)}(\underline{\Bbbk}_{\mathrm{pt}}[-N],\underline{\Bbbk}_{\mathrm{pt}}[n])=0,$

so that $g\circ h$ must be zero, which finishes the proof in this case.

On the other hand, if $n$ is even both spaces
$\operatorname{Hom}_{D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)}(\underline{\Bbbk}_{\mathrm{pt}}[-N],\underline{\Bbbk}_{\mathrm{pt}})$
and
$\operatorname{Hom}_{D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)}(\underline{\Bbbk}_{\mathrm{pt}}[-N],\underline{\Bbbk}_{\mathrm{pt}}[n])$
are $1$-dimensional, with a basis given by
$\mathrm{can}_{\mathrm{pt}}^{N}[-N]$ and
$\mathrm{can}_{\mathrm{pt}}^{n+N}[-N]$ respectively. Hence to conclude, it
only remains to prove that for $M,M^{\prime}\geq n$ even, the diagrams

$\underline{\Bbbk}_{\mathrm{pt}}\xleftarrow{\mathrm{can}_{\mathrm{pt}}^{M}[-M]}\underline{\Bbbk}_{\mathrm{pt}}[-M]\xrightarrow{\mathrm{can}_{\mathrm{pt}}^{n+M}[-M]}\underline{\Bbbk}_{\mathrm{pt}}[n]$

and

$\underline{\Bbbk}_{\mathrm{pt}}\xleftarrow{\mathrm{can}_{\mathrm{pt}}^{M^{\prime}}[-M^{\prime}]}\underline{\Bbbk}_{\mathrm{pt}}[-M^{\prime}]\xrightarrow{\mathrm{can}_{\mathrm{pt}}^{n+M^{\prime}}[-M^{\prime}]}\underline{\Bbbk}_{\mathrm{pt}}[n]$

represent the same morphism in $\mathsf{Sm}(\mathrm{pt},\Bbbk)$. However we
can assume that $M^{\prime}\geq M$; then the morphism
$\mathrm{can}^{M^{\prime}-M}_{\mathrm{pt}}[-M^{\prime}]:\underline{\Bbbk}_{\mathrm{pt}}[-M^{\prime}]\to\underline{\Bbbk}_{\mathrm{pt}}[-M]$
has a cone which belongs to
$D^{\mathrm{b}}_{\mathbb{G}_{\mathrm{m}},c}(\mathrm{pt},\Bbbk)_{\varpi\mathsf{-perf}}$,
and satisfies

$\displaystyle\mathrm{can}_{\mathrm{pt}}^{M^{\prime}}[-M^{\prime}]=(\mathrm{can}_{\mathrm{pt}}^{M}[-M])\circ(\mathrm{can}^{M^{\prime}-M}_{\mathrm{pt}}[-M^{\prime}]),$
$\displaystyle\mathrm{can}_{\mathrm{pt}}^{n+M^{\prime}}[-M^{\prime}]=(\mathrm{can}_{\mathrm{pt}}^{n+M}[-M])\circ(\mathrm{can}^{M^{\prime}-M}_{\mathrm{pt}}[-M^{\prime}]).$

The desired claim follows. ∎

##  4\. Fixed points of roots of unity on the affine Grassmannian

As in Section 3 we let $\mathbb{F}$ be an algebraically closed field of
characteristic $p>0$.

###  4.1. Affine Weyl group

Let $G$ be a connected reductive algebraic group over $\mathbb{F}$, and choose
a Borel subgroup $B\subset G$ and a maximal torus $T\subset B$. The Weyl group
of $(G,T)$ will be denoted $W_{\mathrm{f}}$. (Here, the subscript stands for
“finite,” and is here to avoid any confusion with the affine Weyl group
introduced below.) We will also denote by $U$ the unipotent radical of $B$, by
$B^{+}$ the Borel subgroup opposite to $B$ with respect to $T$, and by $U^{+}$
the unipotent radical of $B^{+}$.

We will denote by $\mathbf{X}:=X^{*}(T)$ the character lattice of $T$, and by
$\mathbf{X}^{\vee}:=X_{*}(T)$ its cocharacter lattice. Let
$\mathfrak{R}\subset\mathbf{X}$ be the root system of $(G,T)$, and let
$\mathfrak{R}^{+}\subset\mathfrak{R}$ be the system of positive roots
consisting of the $T$-weights in $\mathrm{Lie}(U^{+})$. Let also
$\mathfrak{R}^{\mathrm{s}}$ be the associated basis of $\mathfrak{R}$ (the
“simple roots”). These data define a set $S_{\mathrm{f}}$ of Coxeter
generators for $W_{\mathrm{f}}$, consisting of the reflections $s_{\alpha}$
with $\alpha\in\mathfrak{R}^{\mathrm{s}}$.

The _affine Weyl group_ is the semi-direct product

$W_{\mathrm{aff}}:=W_{\mathrm{f}}\ltimes\mathbb{Z}\mathfrak{R}^{\vee},$

where $\mathfrak{R}^{\vee}\subset\mathbf{X}^{\vee}$ is the coroot system, and
$\mathbb{Z}\mathfrak{R}^{\vee}$ is the coroot lattice. For
$\lambda\in\mathbb{Z}\mathfrak{R}^{\vee}$ we will denote by
$\mathsf{t}_{\lambda}$ the image of $\lambda$ in $W_{\mathrm{aff}}$. The group
$W_{\mathrm{aff}}$ admits a natural structure of Coxeter group extending that
of $W_{\mathrm{f}}$; the corresponding simple reflections
$S_{\mathrm{aff}}\subset W_{\mathrm{aff}}$ consist of $S_{\mathrm{f}}$
together with the elements $\mathsf{t}_{\beta^{\vee}}s_{\beta}$ with $\beta$ a
maximal root in $\mathfrak{R}$.

Given $n\in\mathbb{Z}$, we will consider two actions of $W_{\mathrm{aff}}$ on
$V:=\mathbf{X}^{\vee}\otimes_{\mathbb{Z}}\mathbb{R}$ defined, for $w\in
W_{\mathrm{f}}$ and $\lambda\in\mathbb{Z}\mathfrak{R}^{\vee}$, by

$(w\mathsf{t}_{\lambda})\cdot_{n}\mu=w(\mu-n\lambda),\qquad(w\mathsf{t}_{\lambda})\square_{n}\mu=w(\mu+n\lambda)$

for $\mu\in\mathbf{X}^{\vee}$, where in the right-hand side we consider the
natural action of $W_{\mathrm{f}}$ on $\mathbf{X}^{\vee}$. (Here the action
$\cdot_{n}$ appears due to the sign conventions in Bruhat–Tits theory; but the
action $\square_{n}$ is closer to the action which will be relevant when
considering Representation Theory.) Of course, these actions are related via

$w\square_{n}\mu=-(w\cdot_{n}(-\mu))$

for any $w\in W_{\mathrm{aff}}$ and $\mu\in V$.

We set

$\mathbf{a}_{n}:=\\{\lambda\in
V\mid\forall\alpha\in\mathfrak{R}^{+},\,-n<\langle\lambda,\alpha\rangle<0\\}.$

Then the closure $\overline{\mathbf{a}}_{n}$ of $\mathbf{a}_{n}$ is a
fundamental domain for the action of $W_{\mathrm{aff}}$ on $V$ via $\cdot_{n}$
and via $\square_{n}$. These actions stabilize $\mathbf{X}^{\vee}$, and a
fundamental domain for the action of $W_{\mathrm{aff}}$ on $\mathbf{X}^{\vee}$
(for each of these actions) is therefore
$\overline{\mathbf{a}}_{n}\cap\mathbf{X}^{\vee}$.

The _affine roots_ are the formal linear combinations $\alpha+m\hbar$ with
$\alpha\in\mathfrak{R}$ and $m\in\mathbb{Z}$. To such a combination we attach
an affine function $f^{n}_{\alpha+m\hbar}$ on $V$, determined by

$f^{n}_{\alpha+m\hbar}(v)=\langle\alpha,v\rangle+nm,$

and an element $s_{\alpha+m\hbar}\in W_{\mathrm{aff}}$ determined by

$s_{\alpha+m\hbar}=\mathsf{t}_{m\alpha^{\vee}}s_{\alpha}.$

We then have

$s_{\alpha+m\hbar}\cdot_{n}v=v-f^{n}_{\alpha+m\hbar}(v)\alpha^{\vee}$

for any $v\in V$.

######  Remark 4.1.

In practice, when considering these constructions in later sections, the
integer $n$ will be either $1$ or a prime number different from $p$. As this
assumption does not simplify the discussion in any way, we will not impose any
restriction on $n$ in this section.

###  4.2. Some Bruhat–Tits theory

For any positive integer $n$, we set $\mathscr{K}_{n}:=\mathbb{F}((z^{n}))$.
We will consider $\mathscr{K}:=\mathscr{K}_{1}$ as a valued field with its
natural valuation (such that $z$ has valuation $1$), and endow each
$\mathscr{K}_{n}$ with the valuation obtained by restriction. (In this way,
all the fields $\mathscr{K}_{n}$ are canonically isomorphic, but their
valuations differ.) We will denote by $\mathscr{O}_{n}$ the valuation ring of
$\mathscr{K}_{n}$, so that $\mathscr{O}_{n}:=\mathbb{F}[[z^{n}]]$. For any
$\lambda\in\mathbf{X}^{\vee}$ we have a point $z^{\lambda}\in G(\mathscr{K})$,
defined as the image of $z$ under the map $(\mathscr{K})^{\times}\to
G(\mathscr{K})$ induced by $\lambda$. If $\lambda\in n\mathbf{X}^{\vee}$, then
$z^{\lambda}$ belongs to $G(\mathscr{K}_{n})$.

The group scheme
$G\times_{\operatorname{Spec}(\mathbb{F})}\operatorname{Spec}(\mathscr{K}_{n})$
is a (split) connected reductive group scheme over $\mathscr{K}_{n}$, so that
one can consider the associated (enlarged) Bruhat–Tits building
$\mathfrak{B}_{n}$. Our choice of maximal torus in $G$ provides a split
maximal torus
$T\times_{\operatorname{Spec}(\mathbb{F})}\operatorname{Spec}(\mathscr{K}_{n})\subset
G\times_{\operatorname{Spec}(\mathbb{F})}\operatorname{Spec}(\mathscr{K}_{n})$,
which itself defines an apartment $A_{n}$ in $\mathfrak{B}_{n}$. This
apartment is an affine space with underlying vector space $V$, and it is
endowed with a canonical action of $N_{G}(T)(\mathscr{K}_{n})$ whose vectorial
part factors through the natural action of
$N_{G}(T)(\mathscr{K}_{n})/T(\mathscr{K}_{n})=W_{\mathrm{f}}$ on $V$, and such
that for $\lambda\in\mathbf{X}^{\vee}$ the element $z^{n\lambda}\in
T(\mathscr{K}_{n})$ acts by translation by $-n\lambda$. Let us choose, for any
$w\in W_{\mathrm{f}}$, a lift $\dot{w}$ of $w$ in $N_{G}(T)$. Then we will
consider the map

$\iota_{n}:W_{\mathrm{aff}}\to N_{G}(T)(\mathscr{K}_{n})$

defined by $\iota_{n}(\mathsf{t}_{\lambda}w)=z^{n\lambda}\dot{w}$ for $w\in
W_{\mathrm{f}}$ and $\lambda\in\mathbb{Z}\mathfrak{R}^{\vee}$.

If we choose a $W_{\mathrm{f}}$-fixed point in $A_{n}$, then the action of $V$
on this point defines an identification

(4.1) $V\xrightarrow{\sim}A_{n},$

under which the action of $N_{G}(T)(\mathscr{K}_{n})$ on $A_{n}$ identifies
with the action of $W_{\mathrm{aff}}$ on $V$ provided by $\cdot_{n}$. We will
fix such an identification once and for all, and use it to identify all the
data considered above related to $V$ as data related to $A_{n}$. (None of our
considerations below will depend on the choice of identification (4.1).)

The collection of fixed points of the reflections $s_{\alpha+m\hbar}$ (or in
other words of kernels of the functions $f^{n}_{\alpha+m\hbar}$) defines a
hyperplane arrangement in $A_{n}$, hence a collection of _facets_ (whose
closures are the nonempty intersections of such hyperplanes). In particular,
$\mathbf{a}_{n}$ is a facet of maximal dimension, i.e. an alcove. Another
example of facet is the intersection $\mathbf{o}_{n}$ of the reflection
hyperplanes associated with all the reflections $s_{\beta}$ with
$\beta\in\mathfrak{R}$, i.e. the set of $W_{\mathrm{f}}$-fixed points. The
facets we will mainly be interested in are those contained in the closure
$\overline{\mathbf{a}}_{n}$ of $\mathbf{a}_{n}$.

To any facet $\mathbf{f}$ in $A_{n}$, Bruhat–Tits theory associates a
“parahoric group scheme” $P_{\mathbf{f}}$ over
$\operatorname{Spec}(\mathscr{O}_{n})$, characterized as the unique (up to
isomorphism) smooth affine group scheme over
$\operatorname{Spec}(\mathscr{O}_{n})$ with connected geometric fibers such
that

(4.2)
$P_{\mathbf{f}}\times_{\operatorname{Spec}(\mathscr{O}_{n})}\operatorname{Spec}(\mathscr{K}_{n})=G\times_{\operatorname{Spec}(\mathbb{F})}\operatorname{Spec}(\mathscr{K}_{n}),$

and whose $\mathscr{O}_{n}$-points identify (via this isomorphism) with the
pointwise stabilizer of $\mathbf{f}$ in $G(\mathscr{K}_{n})$. In particular we
have

$P_{\mathbf{o}_{n}}=G\times_{\operatorname{Spec}(\mathbb{F})}\operatorname{Spec}(\mathscr{O}_{n}),$

and $P_{\mathbf{a}_{n}}$ is an Iwahori group scheme, whose group of
$\mathscr{O}_{n}$-points is the inverse image $\mathrm{Iw}_{n}$ of $B$ under
the map $G(\mathscr{O}_{n})\to G$ of evaluation at $z^{n}=0$. This
construction is compatible with inclusions of closures of facets in a natural
way; in particular for any facet $\mathbf{f}$ contained in
$\overline{\mathbf{a}}_{n}$ we have an inclusion

(4.3) $P_{\mathbf{a}_{n}}\subset P_{\mathbf{f}}.$

###  4.3. Loop groups and partial affine flag varieties

As above we fix a positive integer $n$. The $n$-th _loop group_ associated
with $G$ is the ind-affine group ind-scheme $L_{n}G$ over $\mathbb{F}$ which
represents the functor sending an $\mathbb{F}$-algebra $R$ to $G(R((z^{n})))$.
The associated _arc group_ (or positive loop group) is the affine group scheme
$L^{+}_{n}G$ over $\mathbb{F}$ which represents the functor sending $R$ to
$G(R[[z^{n}]])$.

The case we are mostly interested in is when $n=1$. In this case (here and in
later related notation), we will usually omit the subscript from the notation.
The case of a more general $n$ however naturally appears when considering the
action of $n$-th roots of unity by loop rotation. Namely, we have a natural
action of the multiplicative group $\mathbb{G}_{\mathrm{m}}$ over $\mathbb{F}$
on $LG$ by loop rotation. This action stabilizes the subgroup $L^{+}G$. Denote
now by ${\boldsymbol{\mu}}_{n}\subset\mathbb{G}_{\mathrm{m}}$ the subgroup
scheme of $n$-th roots of unity; we can then consider the fixed-points ind-
scheme $(LG)^{{\boldsymbol{\mu}}_{n}}$ and the fixed-points scheme
$(L^{+}G)^{{\boldsymbol{\mu}}_{n}}$ in the sense of §2.4.

######  Lemma 4.2.

We have identifications

$(LG)^{{\boldsymbol{\mu}}_{n}}=L_{n}G,\quad(L^{+}G)^{{\boldsymbol{\mu}}_{n}}=L_{n}^{+}G.$

###### Proof.

For any $\mathbb{F}$-algebra $R$, the $R$-points $(LG)(R)$ consist of the
$\mathbb{F}$-schemes morphisms $\operatorname{Spec}(R((z)))\to G$. Therefore,
the $R$-points of $(LG)^{{\boldsymbol{\mu}}_{n}}$ consist of the
${\boldsymbol{\mu}}_{n}$-invariant morphisms $\operatorname{Spec}(R((z)))\to
G$, i.e. the morphisms which factor through the quotient
$\operatorname{Spec}(R((z)))\to\operatorname{Spec}(R((z)))/{\boldsymbol{\mu}}_{n}=\operatorname{Spec}(R((z^{n})))$.
This proves the first identification. The proof of the second one is similar.
∎

It is well known that the fppf sheafification of the functor

$R\mapsto(LG)(R)/(L^{+}G)(R)$

is represented by an ind-projective ind-scheme of ind-finite type over
$\mathbb{F}$, which is called the _affine Grassmannian_ of $G$, and will be
denoted ${\mathcal{G}r}_{G}$. The main goal of this section is to describe the
ind-scheme $({\mathcal{G}r}_{G})^{{\boldsymbol{\mu}}_{n}}$, see Proposition
4.6 below. This will require discussing more general “partial affine flag
varieties” attached to $L_{n}G$, as follows.

If $\mathbf{f}\subset\overline{\mathbf{a}}_{n}$ is a facet, we can consider
the affine group scheme $L^{+}_{n}P_{\mathbf{f}}$ over $\mathbb{F}$ which
represents the functor sending $R$ to $P_{\mathbf{f}}(R[[z^{n}]])$. In view of
(4.2), $L^{+}_{n}P_{\mathbf{f}}$ is a subgroup of $L_{n}G$. The partial affine
flag variety

${\mathcal{F}l}^{n}_{\mathbf{f}}$

associated with $\mathbf{f}$ is the ind-projective ind-scheme of ind-finite
type over $\mathbb{F}$ which represents the fppf sheafification of the
presheaf

$R\mapsto L_{n}G(R)/L_{n}^{+}P_{\mathbf{f}}(R).$

These ind-schemes are the main object of study of [PR]. In particular, the
connected components of ${\mathcal{F}l}^{n}_{\mathbf{f}}$ are in a natural
bijection with the algebraic fundamental group of $G$ (see [PR, Theorem 0.1]);
the component corresponding to the neutral element will be denoted
${\mathcal{F}l}^{n,\circ}_{\mathbf{f}}$.

If $\alpha\in\mathfrak{R}$, we will denote by $U_{\alpha}$ the root subgroup
of $G$ attached to $\alpha$. Then, for an affine root $\alpha+m\hbar$, we will
denote by $U_{\alpha+m\hbar}$ the subgroup of $LG$ which, for any isomorphism
$u_{\alpha}:\mathbb{G}_{\mathrm{a}}\xrightarrow{\sim}U_{\alpha}$, identifies
with the image of the morphism $x\mapsto u_{\alpha}(xz^{m})$.

The following statements are easily checked.

######  Lemma 4.3.

Let $\alpha\in\mathfrak{R}$ and $m\in\mathbb{Z}$.

  1. (1)

The subgroup $U_{\alpha+m\hbar}$ is stable under the action of
${\boldsymbol{\mu}}_{n}$, and we have

$(U_{\alpha+m\hbar})^{{\boldsymbol{\mu}}_{n}}=\begin{cases}U_{\alpha+m\hbar}&\text{if
$m\in n\mathbb{Z}$;}\\\ \\{1\\}&\text{otherwise.}\end{cases}$

  2. (2)

If $\lambda\in\mathbf{X}^{\vee}$, we have

$z^{\lambda}\cdot U_{\alpha+m\hbar}\cdot
z^{-\lambda}=U_{\alpha+(m+\langle\lambda,\alpha\rangle)\hbar}.$

  3. (3)

If $\mathbf{f}\subset\overline{\mathbf{a}}_{n}$ is a facet, we have
$U_{\alpha+mn\hbar}\subset L_{n}^{+}P_{\mathbf{f}}$ iff
$f^{n}_{\alpha+m\hbar}$ takes nonnegative values on $\mathbf{f}$.

###  4.4. Big cells in partial affine flag varieties

Our arguments below will make use of the “big cell” in
${\mathcal{F}l}^{n}_{\mathbf{f}}$, whose construction we now recall following
de Cataldo–Haines–Li [dCHL]. We first consider the affine group ind-scheme
$L^{(-1)}_{n}G$ which represents the functor sending $R$ to the kernel of the
morphism

$G(R[z^{-n}])\to G(R)$

of evaluation at $z^{-n}=0$. Then $L^{(-1)}_{n}G$ is a subgroup ind-scheme of
$L_{n}G$, and we set

$L^{--}_{n}P_{\mathbf{a}_{n}}=L^{(-1)}_{n}G\cdot U^{+}.$

With this definition, it is well known (see e.g. [Fa, §2]) that the action on
the base point induces an open embedding

$L^{--}_{n}P_{\mathbf{a}_{n}}\to{\mathcal{F}l}^{n,\circ}_{\mathbf{a}_{n}},$

and that from this one can obtain an open cover of
${\mathcal{F}l}^{n,\circ}_{\mathbf{a}_{n}}$ parametrized by
$W_{\mathrm{aff}}$, where the open subset corresponding to $w$ is the image of

$\iota_{n}(w)\cdot L^{--}_{n}P_{\mathbf{a}_{n}}\cdot\iota_{n}(w)^{-1}$

under the map $g\mapsto g\cdot[\iota_{n}(w)]$. (Here, $[\iota_{n}(w)]$ is the
image of $\iota_{n}(w)$ in ${\mathcal{F}l}^{n,\circ}_{\mathbf{a}_{n}}$.)

For a general facet $\mathbf{f}\subset\overline{\mathbf{a}}_{n}$, we denote by
$W_{\mathrm{aff}}^{\mathbf{f}}$ the pointwise stabilizer of $\mathbf{f}$ in
$W_{\mathrm{aff}}$ (a finite parabolic subgroup) and set

$L^{--}_{n}P_{\mathbf{f}}=\bigcap_{w\in
W_{\mathrm{aff}}^{\mathbf{f}}}\iota_{n}(w)\cdot
L_{n}^{--}P_{\mathbf{a}_{n}}\cdot\iota_{n}(w)^{-1}.$

The following claim is easily checked.

######  Lemma 4.4.

For $\alpha\in\mathfrak{R}$ and $m\in\mathbb{Z}$, we have
$U_{\alpha+nm\hbar}\subset L^{--}_{n}P_{\mathbf{f}}$ iff
$f^{n}_{\alpha+m\hbar}$ takes negative values on $\mathbf{f}$.

With this definition, as explained in [dCHL, §3.8.1], the action on the base
point defines an open embedding

$L^{--}_{n}P_{\mathbf{f}}\to{\mathcal{F}l}^{n,\circ}_{\mathbf{f}}.$

One can obtain from this an open cover of
${\mathcal{F}l}^{n,\circ}_{\mathbf{f}}$ parametrized by the quotient
$W_{\mathrm{aff}}/W_{\mathrm{aff}}^{\mathbf{f}}$, where the open subset
attached to a coset $wW_{\mathrm{aff}}^{\mathbf{f}}$ is the image of the
subgroup

$\iota_{n}(w)\cdot L^{--}_{n}P_{\mathbf{f}}\cdot\iota_{n}(w)^{-1}$

under the morphism of action on the image of $\iota_{n}(w)$. (These data do
not depend on the choice of $w$ in its coset, and this claim can be deduced
from the corresponding fact for $\mathbf{a}_{n}$ by using the morphism
${\mathcal{F}l}^{n,\circ}_{\mathbf{a}_{n}}\to{\mathcal{F}l}^{n,\circ}_{\mathbf{f}}$
induced by (4.3).)

Note that in the special case $n=1$ and $\mathbf{f}=\mathbf{o}_{1}$, we have

$L^{--}_{1}P_{\mathbf{o}_{1}}=L^{(-1)}G.$

For $m\in\mathbb{Z}_{\geq 1}$, we will also denote by $L^{(-1)}G(m)$, resp.
$L_{n}^{(-1)}G(m)$, the subgroup of $L^{(-1)}G$, resp. $L_{n}^{(-1)}G$ which
represents the functor sending $R$ to the preimage of $T(R[t^{-1}]/t^{-m})$
under the composition

$L^{(-1)}G(R)\hookrightarrow G(R[t^{-1}])\to G(R[t^{-1}]/t^{-m}),$

resp. the preimage of $T(R[t^{-n}]/t^{-mn})$ under the composition

$L_{n}^{(-1)}G(R)\hookrightarrow G(R[t^{-n}])\to G(R[t^{-n}]/t^{-nm}).$

Below we will require the following properties of these subgroups:

  1. (1)

for fixed $\lambda\in\mathbf{X}^{\vee}$ and $m\in\mathbb{Z}_{>0}$, for
$m^{\prime}\gg 0$ we have $z^{\lambda}L^{(-1)}G(m^{\prime})z^{-\lambda}\subset
L^{(-1)}G(m)$;

  2. (2)

for any facet $\mathbf{f}\subset\overline{\mathbf{a}}_{n}$, for $m\gg 0$ we
have $L_{n}^{(-1)}G(m)\subset L_{n}^{--}P_{\mathbf{f}}$.

(Here, (2) follows from the fact that for any given $w\in W_{\mathrm{aff}}$,
for $m\gg 0$ we have $L_{n}^{(-1)}G(m)\subset\iota_{n}(w)\cdot
L_{n}^{--}P_{\mathbf{a}_{n}}\cdot\iota_{n}(w)^{-1}$, see [dCHL]. For (1), we
can use [dCHL, Remark 3.1.1] to reduce the claim to the case
$G=\mathrm{GL}_{n}(\mathbb{F})$, which is clear from a matrix calculation.)

######  Lemma 4.5.

For any $\lambda\in(-\overline{\mathbf{a}}_{n})\cap\mathbf{X}^{\vee}$, we have

$(z^{\lambda}\cdot L^{--}_{1}P_{\mathbf{o}_{1}}\cdot
z^{-\lambda})^{{\boldsymbol{\mu}}_{n}}=L^{--}_{n}P_{\mathbf{f}_{\lambda}},$

where $\mathbf{f}_{\lambda}\subset\overline{\mathbf{a}}_{n}$ is the facet
containing $-\lambda$.

###### Proof.

For any $\alpha\in\mathfrak{R}$, let us denote by $i_{\alpha}$ the largest
integer such that the function $f^{n}_{\alpha+i_{\alpha}\hbar}$ takes negative
values on $\mathbf{f}_{\lambda}$. In fact, since
$\mathbf{f}_{\lambda}\subset\overline{\mathbf{a}}_{n}$ we can describe this
integer very explicitly; namely:

  * •

if $\alpha\in\mathfrak{R}^{+}$ then
$\langle\lambda,\alpha\rangle\in\\{0,\cdots,n\\}$, and

(4.4) $i_{\alpha}=\begin{cases}0&\text{if
$\langle\lambda,\alpha\rangle>0$;}\\\ -1&\text{if
$\langle\lambda,\alpha\rangle=0$;}\end{cases}$

  * •

if $\alpha\in-\mathfrak{R}^{+}$ then
$\langle\lambda,\alpha\rangle\in\\{-n,\cdots,0\\}$, and

(4.5) $i_{\alpha}=\begin{cases}-1&\text{if
$\langle\lambda,\alpha\rangle>-n$;}\\\ -2&\text{if
$\langle\lambda,\alpha\rangle=-n$.}\end{cases}$

Recall from (1)–(2) above that we can choose $m$ large enough such that

$L_{n}^{(-1)}G(m)\subset
L^{--}_{n}P_{\mathbf{f}_{\lambda}}\quad\text{and}\quad z^{-\lambda}\cdot
L_{n}^{(-1)}G(m)\cdot z^{\lambda}\subset L^{(-1)}G.$

Then the arguments in [dCHL, Proofs of Lemma 3.6.3 and Proposition 3.6.4] show
that we have a direct product decomposition

$L^{--}_{n}P_{\mathbf{f}_{\lambda}}=L_{n}^{(-1)}G(m)\cdot\prod_{\alpha\in\mathfrak{R}^{+}}\prod_{j=-m}^{i_{\alpha}}U_{\alpha+jn\hbar}\cdot\prod_{\alpha\in-\mathfrak{R}^{+}}\prod_{j=-m}^{i_{\alpha}}U_{\alpha+jn\hbar}.$

(Here we use an arbitrary order on $\mathfrak{R}^{+}$ and on
$-\mathfrak{R}^{+}$.) Our choice of $m$ guarantees that $z^{-\lambda}\cdot
L_{n}^{(-1)}G(m)\cdot z^{\lambda}\subset L^{(-1)}G$, and in view of Lemma
4.3(2), for any affine root $\alpha+jn\hbar$ appearing in the decomposition
above, the fact that $f^{n}_{\alpha+j\hbar}(-\lambda)<0$ implies that

$z^{-\lambda}\cdot U_{\alpha+jn\hbar}\cdot z^{\lambda}\subset L^{(-1)}G.$

These considerations show that

$L^{--}_{n}P_{\mathbf{f}_{\lambda}}\subset z^{\lambda}\cdot
L^{--}_{1}P_{\mathbf{o}_{1}}\cdot z^{-\lambda},$

so that

$L^{--}_{n}P_{\mathbf{f}_{\lambda}}\subset(z^{\lambda}\cdot
L^{--}_{1}P_{\mathbf{o}_{1}}\cdot z^{-\lambda})^{{\boldsymbol{\mu}}_{n}}.$

To prove the reverse inclusion, we continue with some $m$ as above, and choose
$m^{\prime}\gg 0$ such that

$z^{\lambda}L^{(-1)}G(m^{\prime})z^{-\lambda}\subset L^{(-1)}G(nm)$

(see (1) above). We then have

$(z^{\lambda}L^{(-1)}G(m^{\prime})z^{-\lambda})^{{\boldsymbol{\mu}}_{n}}\subset(L^{(-1)}G(nm))^{{\boldsymbol{\mu}}_{n}}=L_{n}^{(-1)}G(m).$

As above we have a direct product decomposition

$L^{(-1)}G=L^{(-1)}G(m^{\prime})\cdot\prod_{\alpha\in\mathfrak{R}^{+}}\prod_{j=-m^{\prime}}^{-1}U_{\alpha+j\hbar}\cdot\prod_{\alpha\in-\mathfrak{R}^{+}}\prod_{j=-m^{\prime}}^{-1}U_{\alpha+j\hbar},$

where now (for notational convenience) we choose the order on
$\mathfrak{R}^{+}$ such that all the roots such that
$\langle\lambda,\alpha\rangle=0$ are bigger than the other ones, and the order
on $-\mathfrak{R}^{+}$ such that all the roots such that
$\langle\lambda,\alpha\rangle=-n$ are bigger that the other ones. From this
decomposition we see that $(z^{\lambda}\cdot L^{(-1)}G\cdot
z^{-\lambda})^{{\boldsymbol{\mu}}_{n}}$ is the product of
$(z^{\lambda}L^{(-1)}G(m^{\prime})z^{-\lambda})^{{\boldsymbol{\mu}}_{n}}$,
which is included in $L^{--}_{n}P_{\mathbf{f}_{\lambda}}$ by the choices of
$m$ and $m^{\prime}$, and of

$\left(\prod_{\alpha\in\mathfrak{R}^{+}}\prod_{j=-m^{\prime}+\langle\lambda,\alpha\rangle}^{-1+\langle\lambda,\alpha\rangle}U_{\alpha+j\hbar}\cdot\prod_{\alpha\in-\mathfrak{R}^{+}}\prod_{j=-m^{\prime}+\langle\lambda,\alpha\rangle}^{-1+\langle\lambda,\alpha\rangle}U_{\alpha+j\hbar}\right)^{{\boldsymbol{\mu}}_{n}},$

which by Lemma 4.3 is included in

$\prod_{\begin{subarray}{c}\alpha\in\mathfrak{R}^{+}\\\
\langle\lambda,\alpha\rangle>0\end{subarray}}\prod_{j=-N}^{0}U_{\alpha+jn\hbar}\cdot\prod_{\begin{subarray}{c}\alpha\in\mathfrak{R}^{+}\\\
\langle\lambda,\alpha\rangle=0\end{subarray}}\prod_{j=-N}^{-1}U_{\alpha+jn\hbar}\cdot\\\
\prod_{\begin{subarray}{c}\alpha\in-\mathfrak{R}^{+}\\\
\langle\lambda,\alpha\rangle>-n\end{subarray}}\prod_{j=-N}^{-1}U_{\alpha+jn\hbar}\cdot\prod_{\begin{subarray}{c}\alpha\in-\mathfrak{R}^{+}\\\
\langle\lambda,\alpha\rangle=-n\end{subarray}}\prod_{j=-N}^{-2}U_{\alpha+jn\hbar}$

for $N\gg 0$. Here all the affine root subgroups are included in
$L^{--}_{n}P_{\mathbf{f}_{\lambda}}$ by (4.4)–(4.5) and Lemma 4.4, which
finally proves that

$(z^{\lambda}\cdot L^{(-1)}G\cdot
z^{-\lambda})^{{\boldsymbol{\mu}}_{n}}\subset
L^{--}_{n}P_{\mathbf{f}_{\lambda}}$

and concludes the proof. ∎

###  4.5. Fixed points on the affine Grassmannian

The main result of the present section is the following claim. (Here, for
$\lambda\in\mathbf{X}^{\vee}$ we denote by $L_{\lambda}$ the coset of
$z^{\lambda}$ in ${\mathcal{G}r}_{G}$.)

######  Proposition 4.6.

For any $\lambda\in(-\overline{\mathbf{a}}_{n})\cap\mathbf{X}^{\vee}$, the map
$g\mapsto g\cdot L_{\lambda}$ factors through an open and closed embedding

${\mathcal{F}l}^{n,\circ}_{\mathbf{f}_{\lambda}}\to({\mathcal{G}r}_{G})^{{\boldsymbol{\mu}}_{n}},$

where $\mathbf{f}_{\lambda}$ is as in Lemma 4.5. Moreover, the induced map

$\bigsqcup_{\lambda\in(-\overline{\mathbf{a}}_{n})\cap\mathbf{X}^{\vee}}{\mathcal{F}l}^{n,\circ}_{\mathbf{f}_{\lambda}}\to({\mathcal{G}r}_{G})^{{\boldsymbol{\mu}}_{n}}$

is an isomorphism of ind-schemes.

###### Proof.

Arguments similar to those for Lemma 4.5 show that for
$\lambda\in(-\overline{\mathbf{a}}_{n})\cap\mathbf{X}^{\vee}$, the point
$L_{\lambda}$ is fixed under the action of $L_{n}^{+}P_{\mathbf{a}_{n}}$.
Since this point is also stable under the action of lifts of elements in
$W_{\mathrm{aff}}^{\mathbf{f}_{\lambda}}$, it is stabilized by
$L_{n}^{+}P_{\mathbf{f}_{\lambda}}$. Since ${\boldsymbol{\mu}}_{n}$ acts
trivially on $L_{n}G$, our morphism therefore indeed factors through a
morphism

${\mathcal{F}l}^{n}_{\mathbf{f}_{\lambda}}\to({\mathcal{G}r}_{G})^{{\boldsymbol{\mu}}_{n}},$

which we can then restrict to the wished-for morphism
${\mathcal{F}l}^{n,\circ}_{\mathbf{f}_{\lambda}}\to({\mathcal{G}r}_{G})^{{\boldsymbol{\mu}}_{n}}$.

We will now prove that the induced morphism

(4.6)
$\bigsqcup_{\lambda\in(-\overline{\mathbf{a}}_{n})\cap\mathbf{X}^{\vee}}{\mathcal{F}l}^{n,\circ}_{\mathbf{f}_{\lambda}}\to({\mathcal{G}r}_{G})^{{\boldsymbol{\mu}}_{n}}$

is an isomorphism, which will conclude the proof. For this we use the “big
cell” theory recalled in §4.4. Namely, recall that for
$\nu\in\mathbf{X}^{\vee}$ the morphism $g\mapsto g\cdot L_{\nu}$ defines an
open embedding

$z^{\nu}\cdot L^{--}_{1}P_{\mathbf{o}_{1}}\cdot
z^{-\nu}\to{\mathcal{G}r}_{G},$

and that the images of these maps constitute an open cover of
${\mathcal{G}r}_{G}$. These open subsets are stable under the
$\mathbb{G}_{\mathrm{m}}$-action by loop rotation, hence also under the
${\boldsymbol{\mu}}_{n}$-action we consider here; it follows that
$({\mathcal{G}r}_{G})^{{\boldsymbol{\mu}}_{n}}$ has an open cover parametrized
by $\mathbf{X}^{\vee}$, with the subset corresponding to $\nu$ naturally
isomorphic to
$(z^{\nu}L^{--}_{1}P_{\mathbf{o}_{1}}z^{-\nu})^{{\boldsymbol{\mu}}_{n}}$.

Similarly, for any
$\lambda\in(-\overline{\mathbf{a}}_{n})\cap\mathbf{X}^{\vee}$ and any coset
$wW_{\mathrm{aff}}^{\mathbf{f}_{\lambda}}$ in
$W_{\mathrm{aff}}/W_{\mathrm{aff}}^{\mathbf{f}_{\lambda}}$, we have considered
in §4.4 an open subset of ${\mathcal{F}l}^{n,\circ}_{\mathbf{f}_{\lambda}}$
naturally isomorphic to $\iota_{n}(w)\cdot
L^{--}_{n}P_{\mathbf{f}_{\lambda}}\cdot\iota_{n}(w)^{-1}$. Now since
$(-\overline{\mathbf{a}}_{n})\cap\mathbf{X}^{\vee}$ is a fundamental domain
for the action of $W_{\mathrm{aff}}$ on $\mathbf{X}^{\vee}$ (via
$\square_{n}$), we have a bijection

$\bigsqcup_{\lambda\in(-\overline{\mathbf{a}}_{n})\cap\mathbf{X}^{\vee}}W_{\mathrm{aff}}/W_{\mathrm{aff}}^{\mathbf{f}_{\lambda}}\xrightarrow{\sim}\mathbf{X}^{\vee}$

sending $wW_{\mathrm{aff}}^{\mathbf{f}_{\lambda}}\in
W_{\mathrm{aff}}/W_{\mathrm{aff}}^{\mathbf{f}_{\lambda}}$ to
$-(w\cdot_{n}(-\lambda))=w\square_{n}\lambda$.

To conclude the proof, we will show that the map (4.6) identifies the open
subset of ${\mathcal{F}l}^{n,\circ}_{\mathbf{f}_{\lambda}}$ associated with
the coset $wW_{\mathrm{aff}}^{\mathbf{f}_{\lambda}}$ with the open subset of
$({\mathcal{G}r}_{G})^{{\boldsymbol{\mu}}_{n}}$ corresponding to
$w\square_{n}\lambda$. For this it suffices to prove the equality

(4.7) $(z^{w\square_{n}\lambda}\cdot L^{--}_{1}P_{\mathbf{o}_{1}}\cdot
z^{-w\square_{n}\lambda})^{{\boldsymbol{\mu}}_{n}}=\iota_{n}(w)\cdot
L^{--}_{n}P_{\mathbf{f}_{\lambda}}\cdot\iota_{n}(w)^{-1}.$

In case $w=1$, the equality (4.7) was checked in Lemma 4.5. The deduce the
general case, write $w=\mathsf{t}_{\mu}v$ with $\mu\in\mathbf{X}^{\vee}$ and
$v\in W_{\mathrm{f}}$. Then we have

$(z^{w\square_{n}\lambda}\cdot L^{--}_{1}P_{\mathbf{o}_{1}}\cdot
z^{-w\square_{n}\lambda})^{{\boldsymbol{\mu}}_{n}}=(z^{v(\lambda)+n\mu}\cdot
L^{--}_{1}P_{\mathbf{o}_{1}}\cdot
z^{-v(\lambda)-n\mu})^{{\boldsymbol{\mu}}_{n}}\\\
=z^{n\mu}\dot{v}\cdot(z^{\lambda}\cdot L^{--}_{1}P_{\mathbf{o}_{1}}\cdot
z^{-\lambda})^{{\boldsymbol{\mu}}_{n}}\cdot\dot{v}^{-1}z^{-n\mu}=z^{n\mu}\dot{v}\cdot
L^{--}_{n}P_{\mathbf{f}_{\lambda}}\cdot\dot{v}^{-1}z^{-n\mu},$

which concludes the proof. ∎

For $\lambda\in(-\overline{\mathbf{a}}_{n})\cap\mathbf{X}^{\vee}$, we will
denote by ${\mathcal{G}r}_{G,(\lambda)}$ the image of
${\mathcal{F}l}^{n,\circ}_{\mathbf{f}_{\lambda}}$ in
$({\mathcal{G}r}_{G})^{{\boldsymbol{\mu}}_{n}}$ under the map of Proposition
4.6. We then have

$({\mathcal{G}r}_{G})^{{\boldsymbol{\mu}}_{n}}=\bigsqcup_{\lambda\in(-\overline{\mathbf{a}}_{n})\cap\mathbf{X}^{\vee}}{\mathcal{G}r}_{G,(\lambda)},$

which describes $({\mathcal{G}r}_{G})^{{\boldsymbol{\mu}}_{n}}$ as the union
of its connected components.

######  Remark 4.7.

The action of $L_{n}G$ on $L_{0}$ induces an embedding

$L_{n}G/L_{n}^{+}G\hookrightarrow({\mathcal{G}r}_{G})^{\varpi}.$

Here $L_{n}G/L_{n}^{+}G$ is of course isomorphic to ${\mathcal{G}r}_{G}$. In
terms of the decomposition in Proposition 4.6, this embedding identifies
$L_{n}G/L_{n}^{+}G$ with the union of the components
${\mathcal{G}r}_{G,(\lambda)}$ where $\lambda$ runs over
$(-\overline{\mathbf{a}}_{n})\cap n\mathbf{X}^{\vee}$.

###  4.6. Orbits on the affine Grassmannian

A crucial role in our discussion will be played by the following Iwahori
subgroups of $L^{+}G$, for which we introduce special notation:

$\mathrm{Iw}:=L_{1}^{+}P_{\mathbf{a}_{1}},\quad\mathrm{Iw}^{+}:=\dot{w}_{0}\cdot\mathrm{Iw}\cdot(\dot{w}_{0})^{-1}.$

(Here, $w_{0}$ is the longest element in $W_{\mathrm{f}}$.) More concretely,
$\mathrm{Iw}$, resp. $\mathrm{Iw}^{+}$, is the inverse image of $B$, resp.
$B^{+}$, under the map $\mathrm{ev}_{0}:L^{+}G\to G$ sending $z$ to $0$. We
also denote by $\mathrm{Iw}_{\mathrm{u}}$ and $\mathrm{Iw}_{\mathrm{u}}^{+}$
the pro-unipotent radicals of $\mathrm{Iw}$ and $\mathrm{Iw}^{+}$, i.e. the
inverse images of $U$ and $U^{+}$ under $\mathrm{ev}_{0}$.

The group scheme $L^{+}G$ acts on the affine Grassmannian ${\mathcal{G}r}_{G}$
(see §4.3), and the orbits of this action are parametrized by the subsemigroup
$\mathbf{X}^{\vee}_{+}\subset\mathbf{X}^{\vee}$ of dominant cocharacters. More
precisely, we have

(4.8)
$({\mathcal{G}r}_{G})_{\mathrm{red}}=\bigsqcup_{\lambda\in\mathbf{X}^{\vee}_{+}}{\mathcal{G}r}_{G}^{\lambda}\quad\text{with
${\mathcal{G}r}_{G}^{\lambda}:=L^{+}G\cdot L_{\lambda}$,}$

where the left-hand side denotes the reduced ind-scheme associated with
${\mathcal{G}r}_{G}$. Moreover, for any $\lambda\in\mathbf{X}^{\vee}_{+}$ the
closure $\overline{{\mathcal{G}r}_{G}^{\lambda}}$ is a projective
$\mathbb{F}$-scheme of finite type, on which the action of $L^{+}G$ factors
through an action of a quotient group scheme of finite type.

The orbits of $\mathrm{Iw}$ and $\mathrm{Iw}^{+}$ can be described similarly:
we have

$({\mathcal{G}r}_{G})_{\mathrm{red}}=\bigsqcup_{\lambda\in\mathbf{X}^{\vee}}{\mathcal{G}r}_{G,\lambda}\quad\text{with
${\mathcal{G}r}_{G,\lambda}:=\mathrm{Iw}\cdot L_{\lambda}$}$

and

$({\mathcal{G}r}_{G})_{\mathrm{red}}=\bigsqcup_{\lambda\in\mathbf{X}^{\vee}}{\mathcal{G}r}^{+}_{G,\lambda}\quad\text{with
${\mathcal{G}r}^{+}_{G,\lambda}:=\mathrm{Iw}^{+}\cdot L_{\lambda}$.}$

Moreover, each $\mathrm{Iw}$-orbit (resp. $\mathrm{Iw}^{+}$-orbit) is also an
$\mathrm{Iw}_{\mathrm{u}}$-orbit (resp. $\mathrm{Iw}_{\mathrm{u}}^{+}$-orbit),
and for any $\mu\in\mathbf{X}^{\vee}_{+}$ we have

${\mathcal{G}r}_{G}^{\mu}=\bigsqcup_{\lambda\in
W_{\mathrm{f}}\mu}{\mathcal{G}r}_{G,\lambda}=\bigsqcup_{\lambda\in
W_{\mathrm{f}}\mu}{\mathcal{G}r}^{+}_{G,\lambda}.$

For $\lambda\in\mathbf{X}^{\vee}$, the embedding of
${\mathcal{G}r}_{G,\lambda}^{+}$ in ${\mathcal{G}r}_{G}$ will be denoted
$j_{\lambda}^{+}$.

If $n\in\mathbb{Z}_{>0}$, we can also consider the Iwahori subgroups
$\mathrm{Iw}_{n},\mathrm{Iw}_{n}^{+}\subset L_{n}G$ defined as above, and
their pro-unipotent radicals
$\mathrm{Iw}_{\mathrm{u},n},\mathrm{Iw}_{\mathrm{u},n}^{+}$.

######  Lemma 4.8.

We have

$\mathrm{Iw}^{{\boldsymbol{\mu}}_{n}}=\mathrm{Iw}_{n},\quad(\mathrm{Iw}^{+})^{{\boldsymbol{\mu}}_{n}}=\mathrm{Iw}_{n}^{+},\quad(\mathrm{Iw}_{\mathrm{u}})^{{\boldsymbol{\mu}}_{n}}=\mathrm{Iw}_{\mathrm{u},n},\quad(\mathrm{Iw}_{\mathrm{u}}^{+})^{{\boldsymbol{\mu}}_{n}}=\mathrm{Iw}_{\mathrm{u},n}^{+}.$

For any $\lambda\in\mathbf{X}^{\vee}$ we have

$({\mathcal{G}r}_{G,\lambda})^{{\boldsymbol{\mu}}_{n}}=\mathrm{Iw}_{n}\cdot
L_{\lambda},\quad({\mathcal{G}r}^{+}_{G,\lambda})^{{\boldsymbol{\mu}}_{n}}=\mathrm{Iw}^{+}_{n}\cdot
L_{\lambda}.$

###### Proof.

The identifications in the first sentence are immediate consequences of Lemma
4.2.

For the description of
$({\mathcal{G}r}_{G,\lambda})^{{\boldsymbol{\mu}}_{n}}$, for any
$\alpha\in\mathfrak{R}$ we set $\delta_{\alpha}=1$ if
$\alpha\in\mathfrak{R}^{+}$, and $\delta_{\alpha}=0$ otherwise. Using the
notation introduced in §4.3, we set

$\mathrm{Iw}_{\mathrm{u}}^{\lambda}:=\prod_{\alpha\in\mathfrak{R}}\left(\prod_{\delta_{\alpha}\leq
m<\langle\lambda,\alpha\rangle}U_{\alpha+m\hbar}\right),$

where the products are taken in any chosen order. Then it is well known that
the map $u\mapsto u\cdot L_{\lambda}$ induces an isomorphism
$\mathrm{Iw}_{\mathrm{u}}^{\lambda}\xrightarrow{\sim}{\mathcal{G}r}_{G,\lambda}$.
We deduce an isomorphism
$(\mathrm{Iw}_{\mathrm{u}}^{\lambda})^{{\boldsymbol{\mu}}_{n}}\xrightarrow{\sim}({\mathcal{G}r}_{G,\lambda})^{{\boldsymbol{\mu}}_{n}}$,
and here by Lemma 4.3 we have

$(\mathrm{Iw}_{\mathrm{u}}^{\lambda})^{{\boldsymbol{\mu}}_{n}}=\prod_{\alpha\in\mathfrak{R}}\left(\prod_{\begin{subarray}{c}\delta_{\alpha}\leq
m<\langle\lambda,\alpha\rangle\\\ n\mid
m\end{subarray}}U_{\alpha+m\hbar}\right).$

It follows that
$({\mathcal{G}r}_{G,\lambda})^{{\boldsymbol{\mu}}_{n}}=\mathrm{Iw}_{n}\cdot
L_{\lambda}$, as desired. The proof that
$({\mathcal{G}r}_{G,\lambda}^{+})^{{\boldsymbol{\mu}}_{n}}=\mathrm{Iw}_{n}^{+}\cdot
L_{\lambda}$ is similar. ∎

######  Remark 4.9.

Standard considerations show that for any facet
$\mathbf{f}\subset\overline{\mathbf{a}}_{n}$, the $\mathrm{Iw}_{n}^{+}$-orbits
on ${\mathcal{F}l}^{n}_{\mathbf{f}}$ are parametrized in a natural way by the
quotient $W_{\mathrm{aff}}/W_{\mathrm{aff}}^{\mathbf{f}}$. On the other hand,
the $\mathrm{Iw}^{+}$-orbits on ${\mathcal{G}r}_{G}$ are naturally
parametrized by $\mathbf{X}^{\vee}$, so that by Lemma 4.8 the
$\mathrm{Iw}_{n}^{+}$-orbits on $({\mathcal{G}r}_{G})^{\mu_{n}}$ are also
parametrized by $\mathbf{X}^{\vee}$. Under the identification of Proposition
4.6, for any $\lambda\in(-\overline{\mathbf{a}}_{n})\cap\mathbf{X}^{\vee}$ the
orbit in ${\mathcal{F}l}^{n}_{\mathbf{f}_{\lambda}}$ corresponding to the
coset $wW_{\mathrm{aff}}^{\mathbf{f}}$ is mapped to the orbit in
$({\mathcal{G}r}_{G})^{\mu_{n}}$ parametrized by $w\square_{n}\lambda$.

##  5\. Iwahori–Whittaker sheaves on the affine Grassmannian

We continue with the setting of Section 4.

###  5.1. Iwahori–Whittaker sheaves

The category of sheaves on ${\mathcal{G}r}_{G}$ we will study is the
_Iwahori–Whittaker_ derived category, whose definition we briefly recall. (For
more details, see e.g. [AR1, Appendix A].)

From now on we let $\Bbbk$ be a finite field of positive characteristic
$\ell\neq p$ containing a nontrivial $p$-th root of unity. After choosing such
a root of unity $\zeta$, we obtain an Artin–Schreier local system
$\mathscr{L}_{\mathrm{AS}}$ on $\mathbb{G}_{\mathrm{a}}$, defined as the
direct summand of the local system
$\mathrm{AS}_{*}\underline{\Bbbk}_{\mathbb{G}_{\mathrm{a}}}$ on which
$\mathbb{F}_{p}$ acts via $n\mapsto\zeta^{n}$. (Here,
$\mathrm{AS}:\mathbb{G}_{\mathrm{a}}\to\mathbb{G}_{\mathrm{a}}$ is the map
$x\mapsto x^{p}-x$, a Galois cover of group $\mathbb{F}_{p}$.) We choose once
and for all a morphism of $\mathbb{F}$-algebraic groups
$\chi_{0}:U^{+}\to\mathbb{G}_{\mathrm{a}}$ which is nontrivial on any root
subgroup of $U^{+}$ associated with a simple root, and denote by

$\chi:\mathrm{Iw}_{\mathrm{u}}^{+}\to\mathbb{G}_{\mathrm{a}}$

the composition of $\chi_{0}$ with the morphism
$\mathrm{Iw}_{\mathrm{u}}^{+}\to U^{+}$ induced by $\mathrm{ev}_{0}$.

For $X\subset{\mathcal{G}r}_{G}$ a locally closed finite union of
$\mathrm{Iw}^{+}$-orbits, we can choose a smooth quotient $J$ of
$\mathrm{Iw}_{\mathrm{u}}^{+}$ of finite type such that the
$\mathrm{Iw}_{\mathrm{u}}^{+}$-action on $X$ factors through an action of $J$,
and such that $\chi$ factors through a morphism
$\chi_{J}:J\to\mathbb{G}_{\mathrm{a}}$. Then the
$(J,\chi_{J}^{*}\mathscr{L}_{\mathrm{AS}})$-equivariant derived category of
$\Bbbk$-sheaves on $X$ is by definition the full subcategory of
$D^{\mathrm{b}}\mathsf{Sh}(X,\Bbbk)$ whose objects are the complexes
$\mathscr{F}$ whose pullback under the action map $J\times X\to X$ is
isomorphic to $\chi_{J}^{*}\mathscr{L}_{\mathrm{AS}}\boxtimes\mathscr{F}$. It
is well known that this subcategory is triangulated, and that it does not
depend on the choice of $J$; it will be denoted

$D^{\mathrm{b}}_{\mathcal{IW}}(X,\Bbbk).$

It is known also that the perverse t-structure on
$D^{\mathrm{b}}\mathsf{Sh}(X,\Bbbk)$ restricts to a t-structure on
$D^{\mathrm{b}}_{\mathcal{IW}}(X,\Bbbk)$, which will also be called the
perverse t-structure. (Here, “$\mathcal{IW}$” stands for “Iwahori–Whittaker.”
We will use this expression as a replacement for
“$(J,\chi_{J}^{*}\mathscr{L}_{\mathrm{AS}})$-equivariant” where $J$ is as
above, in all circumstances where this notion does not depend on the choice of
$J$.)

One can also define the category

$D^{\mathrm{b}}_{\mathcal{IW}}({\mathcal{G}r}_{G},\Bbbk)$

of Iwahori–Whittaker sheaves on ${\mathcal{G}r}_{G}$ as the direct limit of
the categories $D^{\mathrm{b}}_{\mathcal{IW}}(X,\Bbbk)$ where $X$ runs over
the closed finite unions of $\mathrm{Iw}^{+}$-orbits, ordered by inclusion.
(Here, the transition functors are the—fully faithful—pushforward functors.)
Since, for $X\subset Y$, the pushforward functor
$D^{\mathrm{b}}_{\mathcal{IW}}(X,\Bbbk)\to
D^{\mathrm{b}}_{\mathcal{IW}}(Y,\Bbbk)$ is t-exact, from the perverse
t-structures on the categories $D^{\mathrm{b}}_{\mathcal{IW}}(X,\Bbbk)$ we
obtain a perverse t-structure on
$D^{\mathrm{b}}_{\mathcal{IW}}({\mathcal{G}r}_{G},\Bbbk)$, whose heart will be
denoted $\mathsf{Perv}_{\mathcal{IW}}({\mathcal{G}r}_{G},\Bbbk)$.

The considerations on stabilizers from the proof of Lemma 4.8 can be used to
see that for $\lambda\in\mathbf{X}^{\vee}$, the orbit
${\mathcal{G}r}_{G,\lambda}^{+}$ supports a nonzero Iwahori–Whittaker local
system iff $\lambda$ belongs to the subset

$\mathbf{X}^{\vee}_{++}:=\\{\mu\in\mathbf{X}^{\vee}\mid\forall\alpha\in\mathfrak{R}^{+},\,\langle\mu,\alpha\rangle>0\\}.$

Moreover, in this case there exists (up to isomorphism) exactly one such local
system of rank $1$; it will be denoted $\mathscr{L}_{\mathrm{AS}}^{\lambda}$.
This remark implies that for any
$\mu\in\mathbf{X}^{\vee}\smallsetminus\mathbf{X}^{\vee}_{++}$ the category
$D^{\mathrm{b}}_{\mathcal{IW}}({\mathcal{G}r}_{G,\mu}^{+},\Bbbk)$ is $0$; in
particular, the restriction and co-restriction of any object in
$D^{\mathrm{b}}_{\mathcal{IW}}(X,\Bbbk)$ (where $X$ is any locally closed
finite union of $\mathrm{Iw}^{+}$-orbits containing
${\mathcal{G}r}_{G,\mu}^{+}$) vanishes.

######  Remark 5.1.

If we assume that there exists an element $\varsigma\in\mathbf{X}^{\vee}$ such
that $\langle\varsigma,\alpha\rangle=1$ for all
$\alpha\in\mathfrak{R}^{\mathrm{s}}$, then we have
$\mathbf{X}^{\vee}_{++}=\varsigma+\mathbf{X}^{\vee}_{+}$.

For $\lambda\in\mathbf{X}^{\vee}_{++}$, we set

$\Delta^{\mathcal{IW}}_{\lambda}:=(j_{\lambda}^{+})_{!}\mathscr{L}_{\mathrm{AS}}^{\lambda}[\dim({\mathcal{G}r}_{G,\lambda}^{+})],\quad\nabla^{\mathcal{IW}}_{\lambda}:=(j_{\lambda}^{+})_{*}\mathscr{L}_{\mathrm{AS}}^{\lambda}[\dim({\mathcal{G}r}_{G,\lambda}^{+})].$

Since $j_{\lambda}^{+}$ is an affine embedding, these objects are perverse
sheaves by [BBDG, Corollaire 4.1.3]. Standard arguments (going back to [BGS])
show that the category
$\mathsf{Perv}_{\mathcal{IW}}({\mathcal{G}r}_{G},\Bbbk)$ admits a structure of
highest weight category (in the sense of [Ri, §7]) with weight set
$\mathbf{X}^{\vee}_{++}$, standard objects the objects
$(\Delta^{\mathcal{IW}}_{\lambda}:\lambda\in\mathbf{X}^{\vee}_{++})$, and
costandard objects the objects
$(\nabla^{\mathcal{IW}}_{\lambda}:\lambda\in\mathbf{X}^{\vee}_{++})$. In
particular, one can consider the tilting objects in this category, i.e. the
objects which admit both a filtration with standard subquotients, and a
filtration with costandard subquotients. Recall that, as remarked in [BBM],
this notion can also be characterized topologically: a perverse sheaf
$\mathscr{F}$ is tilting iff the complexes $(j_{\lambda}^{+})^{*}\mathscr{F}$
and $(j_{\lambda}^{+})^{!}\mathscr{F}$ are perverse (i.e. are direct sums of
copies of
$\mathscr{L}_{\mathrm{AS}}^{\lambda}[\dim({\mathcal{G}r}_{G,\lambda}^{+})]$)
for any $\lambda\in\mathbf{X}^{\vee}_{++}$.

The full subcategory of
$\mathsf{Perv}_{\mathcal{IW}}({\mathcal{G}r}_{G},\Bbbk)$ whose objects are the
tilting objects will be denoted
$\mathsf{Tilt}_{\mathcal{IW}}({\mathcal{G}r}_{G},\Bbbk)$. The general theory
of highest weight categories (reviewed e.g. in [Ri]) guarantees that the
indecomposable objects in this category are parametrized in a natural way by
$\mathbf{X}^{\vee}_{++}$. More precisely, for any
$\lambda\in\mathbf{X}^{\vee}_{++}$ there exists a unique (up to isomorphism)
indecomposable object $\mathscr{T}^{\mathcal{IW}}_{\lambda}$ in
$\mathsf{Tilt}_{\mathcal{IW}}({\mathcal{G}r}_{G},\Bbbk)$ which is supported on
$\overline{{\mathcal{G}r}_{G,\lambda}^{+}}$, and whose restriction to
${\mathcal{G}r}_{G,\lambda}^{+}$ is
$\mathscr{L}_{\mathrm{AS}}^{\lambda}[\dim({\mathcal{G}r}_{G,\lambda}^{+})]$;
then the assignment $\lambda\mapsto\mathscr{T}^{\mathcal{IW}}_{\lambda}$
induces a bijection between $\mathbf{X}^{\vee}_{++}$ and the set of
isomorphism classes of indecomposable objects in
$\mathsf{Tilt}_{\mathcal{IW}}({\mathcal{G}r}_{G},\Bbbk)$.

###  5.2. Loop rotation equivariant Iwahori–Whittaker sheaves

We will need to “add” the (loop rotation)
$\mathbb{G}_{\mathrm{m}}$-equivariance in the construction of §5.1. We
therefore consider a locally closed finite union of $\mathrm{Iw}^{+}$-orbits
$X\subset{\mathcal{G}r}_{G}$ as above. The $\mathbb{G}_{\mathrm{m}}$-action by
loop rotation on ${\mathcal{G}r}_{G}$ stabilizes each $\mathrm{Iw}^{+}$-orbit,