# From the Hecke Category to the Unipotent Locus

Minh-Tâm Quang Trinh  Massachusetts Institute of Technology, 77 Massachusetts
Avenue, Cambridge, MA 02139<EMAIL_ADDRESS>
###### Tóm tắt nội dung.

Let $W$ be the Weyl group of a split semisimple group $G$. Its Hecke category
$\mathsf{H}_{W}$ can be built from pure perverse sheaves on the double flag
variety of $G$. By developing a formalism of generalized realization functors,
we construct a monoidal trace from $\mathsf{H}_{W}$ to a category of bigraded
modules over a certain graded ring: namely, the endomorphisms of the
equivariant Springer sheaf over the unipotent locus of $G$. We prove that: (1)
On objects attached to positive braids $\beta$, the output is the weight-
graded, equivariant Borel–Moore homology of a generalized Steinberg scheme
$\mathcal{Z}(\beta)$. (2) Our functor contains, as a summand, one used by
Webster–Williamson to construct the Khovanov–Rozansky link invariant. In
particular, the Khovanov–Rozansky homology of the link closure of $\beta$ is
fully encoded in the Springer theory of $\mathcal{Z}(\beta)$.

Decategorifying, we get a trace on the Iwahori–Hecke algebra, valued in graded
virtual characters of $W$. We give a formula for it in terms of a pairing on
characters of $W$ called Lusztig’s exotic Fourier transform, which generalizes
it to all finite Coxeter groups. Using this formula, we establish properties
of the trace like rationality, symmetry, and compatibility with parabolic
induction. We also show that on periodic braids, the trace produces the
characters of explicit virtual modules over Cherendik’s rational double affine
Hecke algebra. For $W=S_{n}$, we recover an identity of
Gorsky–Oblomkov–Rasmussen–Shende relating these modules to the HOMFLY
polynomials of torus knots.

###### Mục lục

  1. 1 Purity, Mixedness, and Realization
  2. 2 Mixed Categories in Representation Theory
  3. 3 The $\mathbf{A}_{W}$ Trace
  4. 4 Steinberg Schemes
  5. 5 Steinberg Schemes of Braids
  6. 6 Khovanov–Rozansky Homology
  7. 7 The Decategorified Trace
  8. 8 Deligne–Lusztig Theory
  9. 9 Periodic Braids
  10. 10 The Rational DAHA
  11. A Coxeter Groups and Their Characters
  12. B Braid Varieties in the Literature
  13. C Examples of the Decategorified Trace

## Introduction

###  0.1. Summary

Let $\mathbf{V}$ be a finite-dimensional vector space over a subfield of
$\mathbf{R}$, and let $W$ be a finite group of automorphisms of $\mathbf{V}$
generated by reflections. Coxeter showed that $W$ admits a remarkable
presentation in terms of generators and relations [Co]. Via this presentation,
the group ring of $W$ can be deformed to a
$\mathbf{Z}[\bm{q}^{\pm\frac{1}{2}}]$-algebra known as the Iwahori–Hecke
algebra $H_{W}$. One can further define an infinite group $\mathit{Br}_{W}$,
generalizing the braid groups studied in geometric topology, such that $W$ and
$H_{W}$ are respective quotients of $\mathit{Br}_{W}$ and
$\mathbf{Z}[\bm{q}^{\pm\frac{1}{2}}][\mathit{Br}_{W}]$.

If $W=S_{n}$, the symmetric group on $n$ letters, then
$\mathit{Br}_{W}=\mathit{Br}_{n}$, the classical braid group on $n$ strands.
Any braid $\beta$ can be closed up, end-to-end, to yield a topological link
$\hat{\beta}$ in $3$-dimensional space. If
$\beta,\beta^{\prime}\in\mathit{Br}_{n}$ are conjugate, then
$\hat{\beta},\hat{\beta}^{\prime}$ are isotopic. Ocneanu, building on work of
Jones, used this observation to construct an isotopy invariant of links called
the HOMFLY series out of class functions on the groups $\mathit{Br}_{n}$.
These functions, in turn, factor through functions called Markov traces on the
algebras $H_{S_{n}}$ [HOMFLY, J]. Y. Gomi discovered that the latter admit a
uniform generalization to any finite Coxeter group $W$ [G].

It is now known that $H_{W}$ is the (triangulated) Grothendieck ring of a
monoidal triangulated category that we will call the Hecke category
$\mathsf{H}_{W}$. Gomi’s Markov trace is categorified by an additive functor

(1) $\displaystyle\mathsf{HHH}:\mathsf{H}_{W}\to\mathsf{Vect}_{3},$

where $\mathsf{Vect}_{3}$ denotes the category of triply-graded vector spaces.
As $W$ runs over the symmetric groups, these functors can be used to construct
a link invariant valued in $\mathsf{Vect}_{3}$, originally constructed by
Khovanov–Rozansky [KR, Kh]. More precisely, there is a map from braids
$\beta\in\mathit{Br}_{W}$ to objects $\mathcal{R}(\beta)\in\mathsf{H}_{W}$,
due to Rouquier [R06], and $\mathsf{HHH}(\mathcal{R}(\beta))$ can be used to
construct an isotopy invariant of $\hat{\beta}$ that decategorifies to its
HOMFLY series.

The most general definition of $\mathsf{H}_{W}$ is purely algebraic: the
homotopy category of Soergel bimodules over $\operatorname{{Sym}}(\mathbf{V})$
[EW14, EW16]. However, in representation theory, we are especially interested
in the case where $\mathbf{V}$ is the span of a root lattice and $W$ is the
Weyl group of the corresponding semisimple algebraic group $G$ over an
algebraically closed field, say, $\bar{\mathbf{F}}$ for a finite field
$\mathbf{F}$. Here, the Hecke category can be defined in terms of the geometry
of $G$, or rather, its split form $G_{0}$ over $\mathbf{F}$. We are led to
hope that $\mathsf{HHH}(\mathcal{R}(\beta))$ might be expressed in terms of
the (co)homology of an explicit variety over $\mathbf{F}$ attached to $\beta$.

In the series of papers [WW08, WW11, WW17], Webster–Williamson relate
$\mathsf{HHH}$ to the weight grading on the $G$-equivariant cohomology of
certain varieties with maps to $G$. The most inexplicit part of their work is
a procedure they call “chromatography,” which turns an object of the mixed
derived category
$\mathsf{D}_{G,m}^{b}(G_{0})=\mathsf{D}_{G,m}^{b}(G_{0},\bar{\mathbf{Q}}_{\ell})$
into a complex up to homotopy whose terms are pure complexes of sheaves. The
use of this procedure means that in general, their work only relates
$\mathsf{HHH}(\mathcal{R}(\beta))$ to a combination of several varieties,
rather than to a single one. These varieties encode the term-by-term structure
of $\mathcal{R}(\beta)$, when $\mathcal{R}(\beta)$ is viewed as a complex of
Soergel bimodules.

One of our goals is to simplify this situation. Instead of chromatography, we
will use a formalism of weight realization functors, inspired by works of
Bezrukavnikov–Yun [BY], Lusztig–Yun [LY20], and Rider [R]. For special braids
$\beta\in\mathit{Br}_{n}$ called Richardson braids, Galashin–Lam have recently
used Rider’s work to identify a certain summand of
$\mathsf{HHH}(\mathcal{R}(\beta))$, its so-called “lowest Hochschild degree,”
with the torus-equivariant compactly-supported cohomology of a positroid
variety [GL]. Their argument generalizes to any Weyl group $W$. We go further:
For any positive $\beta\in\mathit{Br}_{W}$, we realize the entirety of
$\mathsf{HHH}(\mathcal{R}(\beta))$ within the equivariant Borel–Moore homology
of an explicit $G$-variety over the unipotent locus of $G$.

Our viewpoint will lead us to introduce a refinement of $\mathsf{HHH}$, valued
in modules rather than vector spaces. Henceforth, we base-change $\mathbf{V}$
to a field of the form $\bar{\mathbf{Q}}_{\ell}$ for some prime $\ell$
invertible in $\mathbf{F}$. Form the semidirect-product ring

(2)
$\displaystyle\mathbf{A}_{W}=\bar{\mathbf{Q}}_{\ell}[W]\ltimes\operatorname{{Sym}}(\mathbf{V}),$

endowed with the grading where $W$ and $\mathbf{V}$ are respectively placed in
degrees $0$ and $2$. Let $\mathsf{Mod}_{2}^{b}(\mathbf{A}_{W})$ be the
category of $\mathbf{A}_{W}$-modules equipped with a bigrading where the
$\mathbf{A}_{W}$-action is diagonally graded, both gradings are bounded below,
and the difference between the gradings is uniformly bounded. We will
construct a contravariant(!) monoidal trace

(3)
$\displaystyle\mathsf{AH}:\mathsf{H}_{W}^{\mathrm{op}}\to\mathsf{Mod}_{2}^{b}(\mathbf{A}_{W}).$

We will show that $\mathsf{HHH}^{\vee}$ is the sum of the
$\operatorname{\text{$\Lambda$}}^{i}(\mathbf{V})$-isotypic components of
$\mathsf{AH}$, where $\operatorname{\text{$\Lambda$}}^{i}(\mathbf{V})$ denotes
the $i$th exterior power of $\mathbf{V}$, viewed as a representation of $W$.
The appearance of $\operatorname{\text{$\Lambda$}}^{i}(\mathbf{V})$ is
reminiscent of work of Bezrukavnikov–Tolmachov on a monodromic model of
$\mathsf{HHH}$ [BT]. For positive $\beta$, we will find that

(4)
$\displaystyle\mathsf{AH}(\mathcal{R}(\beta))\simeq\operatorname{{gr}}_{\ast}^{\bm{w}}\mathrm{H}_{-\ast}^{!,G}(\mathcal{Z}(\beta))$

for a certain $G$-scheme $\mathcal{Z}(\beta)$, where on the right-hand side,
$\mathrm{H}_{\ast}^{!,G}(-)=\mathrm{H}_{\ast}^{!,G}(-,\bar{\mathbf{Q}}_{\ell})$
denotes equivariant Borel–Moore homology and $\bm{w}$ denotes the weight
filtration coming from an $\mathbf{F}$-structure on $\mathcal{Z}(\beta)$. In
the case of the identity $\mathbf{1}\in\mathit{Br}_{W}$, the reduced scheme
$\mathcal{Z}(\mathbf{1})^{\mathrm{red}}$ is the variety Steinberg introduced
in [St76]. In particular,
$\mathsf{AH}(\mathcal{R}(\mathbf{1}))\simeq\mathbf{A}_{W}$ by a result of
Lusztig [L88].

(To explain the contravariance: The definition of $\mathsf{AH}$ will involve a
$\operatorname{{Hom}}$ functor. It turns out that we need a contravariant, not
covariant, $\operatorname{{Hom}}$ to get the right-hand side of (4) to match
$\mathsf{AH}(\mathcal{R}(\beta))$ rather than
$\mathsf{AH}(\mathcal{R}(\beta^{-1}))$.)

Let us sketch the geometry that underlies $\mathsf{H}_{W}$ and $\mathsf{AH}$.
Let $\mathcal{B}=\mathcal{B}_{0}\otimes_{\mathbf{F}}\bar{\mathbf{F}}$ and
$\mathcal{U}=\mathcal{U}_{0}\otimes_{\mathbf{F}}\bar{\mathbf{F}}$ denote the
flag variety and unipotent locus of
$G=G_{0}\otimes_{\mathbf{F}}\bar{\mathbf{F}}$. The Hecke category of $W$ can
be interpreted as

(5)
$\displaystyle\mathsf{H}_{W}=\mathsf{K}^{b}(\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})),$

where $\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})$ is a certain full,
additive subcategory of
$\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}\times\mathcal{B}_{0})$ generated by pure
shifted perverse sheaves. (More precisely, it is generated by the shift-twists
of the intersection complexes of the $G_{0}$-orbits of
$\mathcal{B}_{0}\times\mathcal{B}_{0}$.) The monoidal product on
$\mathsf{H}_{W}$ arises from a geometrically-defined convolution. The
operation of passing from a braid to its conjugacy class is _very roughly_
realized by pulling and pushing sheaves from
$\mathcal{B}_{0}\times\mathcal{B}_{0}$ to $G_{0}$ through the following
diagram, implicit in Lusztig’s work on character sheaves and now known as the
horocycle correspondence:

(6)
${\mathcal{B}_{0}\times\mathcal{B}_{0}}$${G_{0}\times\mathcal{B}_{0}}$${G_{0}}$$\mathit{act}$$\mathit{pr}$

Above, $\mathit{act}$ and $\mathit{pr}$ are the action and projection maps.

We introduce similar subcategories
$\mathsf{C}(G_{0})\subseteq\mathsf{D}_{G,m}^{b}(G_{0})$ and
$\mathsf{C}(\mathcal{U}_{0})\subseteq\mathsf{D}_{G,m}^{b}(\mathcal{U}_{0})$.
(The category $\mathsf{C}(G_{0})$ is generated by objects of the form
$E\otimes M$, where $E$ is a mixed version of a unipotent character sheaf and
$M$ is a pure weight-zero Frobenius module; the category
$\mathsf{C}(\mathcal{U}_{0})$ is generated by the pullbacks of these objects
along the inclusion $i:\mathcal{U}_{0}\to G_{0}$.) In particular,
$\mathsf{C}(G_{0})$ contains the Grothendieck–Springer sheaf $\mathcal{G}$,
while $\mathsf{C}(\mathcal{U}_{0})$ contains the Springer sheaf
$\mathcal{S}\simeq i^{\ast}\mathcal{G}\langle-{\dim\mathbf{V}}\rangle$. The
horocycle correspondence gives rise to a functor

(7)
$\displaystyle\mathsf{CH}:\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})\to\mathsf{C}(\mathcal{U}_{0}),$

while pullback gives rise to a functor
$i^{\ast}\langle-{\dim\mathbf{V}}\rangle:\mathsf{C}(G_{0})\to\mathsf{C}(\mathcal{U}_{0})$.

Crucially, $\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})$ and
$\mathsf{C}(\mathcal{U}_{0})$ satisfy a certain
$\operatorname{{Ext}}$-vanishing condition (see Section 1) that allows us to
define realization functors
$\rho:\mathsf{K}^{b}(\mathsf{C}(\mathcal{B}_{0}^{2}))\to\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}^{2})$
and
$\rho:\mathsf{K}^{b}(\mathsf{C}(\mathcal{U}_{0}))\to\mathsf{D}_{G,m}^{b}(\mathcal{U}_{0})$.
Even though we are unable to construct a similar functor for $G_{0}$, the
diagram

(8)
${\mathsf{K}^{b}(\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0}))}$${\mathsf{K}^{b}(\mathsf{C}(G_{0}))}$${\mathsf{K}^{b}(\mathsf{C}(\mathcal{U}_{0}))}$${\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}\times\mathcal{B}_{0})}$${\mathsf{D}_{G,m}^{b}(G_{0})}$${\mathsf{D}_{G,m}^{b}(\mathcal{U}_{0})}$$\mathsf{K}^{b}i^{\ast}\langle-r\rangle$$i^{\ast}\langle-r\rangle$$\mathsf{K}^{b}\mathsf{CH}$$\mathsf{CH}$$\rho$$\rho$

commutes. The monoidal trace $\mathsf{AH}$ is the composition

(9)
$\displaystyle\mathsf{H}_{W}=\mathsf{K}^{b}(\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0}))\to\mathsf{D}_{G,m}^{b}(\mathcal{U}_{0})\xrightarrow{\operatorname{{gr}}_{\ast}^{\bm{w}}\underline{\operatorname{{Hom}}}^{\ast}(-,\mathcal{S})}\mathsf{Mod}_{2}^{b}(\mathbf{A}_{W}),$

where the first arrow is the functor produced by the diagram, and
$\underline{\operatorname{{Hom}}}^{\ast}$ denotes an
$\operatorname{{Ext}}$-space endowed with a Frobenius action. We use:

  1. (1)

The bottom-left part of the diagram to match $\mathsf{AH}$ with the homology
of the generalized Steinberg scheme $\mathcal{Z}(\beta)$ (in Section 5).

  2. (2)

The upper-right part of the diagram to match
$\operatorname{{Hom}}_{W}(\operatorname{\text{$\Lambda$}}^{\ast}(\mathbf{V}),\mathsf{AH})$
with $\mathsf{HHH}$ (in Section 6).

The main obstacle with (2) is that Webster–Williamson interpret $\mathsf{HHH}$
using the hypercohomology of objects over $G$, whereas we define $\mathsf{AH}$
using objects over $\mathcal{U}$. Moreover, in their work, the Hochschild
degree of $\mathsf{HHH}$ arises from $\operatorname{{gr}}_{\ast}^{\bm{w}}$,
while in our work, it arises from
$\operatorname{\text{$\Lambda$}}^{\ast}(\mathbf{V})$. So the matching in (2)
is fairly indirect. Using orthogonality results of Lusztig [L85-II] and
Rider–Russell [RR], we reduce the problem to checking certain direct-summand
categories $\mathsf{C}_{\hat{W}}(G_{0})\subseteq\mathsf{C}(G_{0})$ and
$\mathsf{C}_{\hat{W}}(\mathcal{U}_{0})\subseteq\mathsf{C}(\mathcal{U}_{0})$,
which are roughly the “principal blocks” generated by $\mathcal{G}$ and
$\mathcal{S}$. On the level of these blocks, the key input is a geometrically-
defined isomorphism between
$\underline{\operatorname{{End}}}^{\ast}(\mathcal{S})$ and the pure part of
$\underline{\operatorname{{End}}}^{\ast}(\mathcal{G})$ (in Section 4).

As for (1), let us sketch the definition of $\mathcal{Z}(\beta)$. Let
$\mathit{Br}_{W}^{+}\subseteq\mathit{Br}_{W}$ be the monoid of positive
braids. The map $\mathit{Br}_{W}\to W$ admits a set-theoretic section
$W\to\mathit{Br}_{W}^{+}$ that we will denote $w\mapsto\sigma_{w}$. (In fact,
the images of the $\sigma_{w}$ in $H_{W}$ form a
$\mathbf{Z}[\bm{q}^{\pm\frac{1}{2}}]$-basis.) Deligne [D97] and Broué–Michel
[BM97] introduced a map

(10) $\displaystyle\beta\mapsto O(\beta)$

from elements of $\mathit{Br}_{W}^{+}$ to $G$-varieties over
$\mathcal{B}\times\mathcal{B}$, such that $O(\sigma_{w})$ is the $G$-orbit of
$\mathcal{B}\times\mathcal{B}$ indexed by $w$, and such that $O$ takes
compositions of braids to fiber products over $\mathcal{B}$, up to explicit
isomorphisms. We form a cartesian square:

(11)
${O(\beta)}$${\mathcal{U}(\beta)}$${\mathcal{B}\times\mathcal{B}}$${\mathcal{U}\times\mathcal{B}}$$\mathit{act}$

Then the map $\mathcal{U}(\mathbf{1})\to\mathcal{U}$ is the Springer
resolution. We define $\mathcal{Z}(\beta)$ be the fiber product
$\mathcal{U}(\mathbf{1})\times_{\mathcal{U}}\mathcal{U}(\beta)$.

In the rest of the paper, we study the decategorification of $\mathsf{AH}$.
Let $R(W)$ be the representation ring of $W$. Any
$M\in\mathsf{Mod}_{2}^{b}(\mathbf{A}_{W})$ decategorifies to an element

(12)
$\displaystyle[M]_{\bm{q}}=\sum_{i,j}{(-1)^{i-j}}\bm{q}^{\frac{j}{2}}M^{i,j}\in
R(W)(\\!(\bm{q}^{\frac{1}{2}})\\!)$

that we call its graded character. As a result, $\mathsf{AH}$ decategorifies
to a $\mathbf{Z}[\bm{q}^{\pm\frac{1}{2}}]$-linear function $H_{W}\to
R(W)(\\!(\bm{q}^{\frac{1}{2}})\\!)$. After multiplying by a normalization
factor that clears denominators, we get a function

(13) $\displaystyle\langle{-}\rangle_{\bm{q}}:H_{W}\to
R(W)[\bm{q}^{\pm\frac{1}{2}}]$

that admits an especially simple formula in terms of the character theory of
$H_{W}$. If we write $\hat{W}$ for the set of irreducible characters of $W$,
then it is

(14)
$\displaystyle\langle{\beta}\rangle_{\bm{q}}=\sum_{\phi,\psi\in\hat{W}}{\\{\phi,\psi\\}}\phi_{\bm{q}}(\beta)\psi,$

where $\phi_{\bm{q}}:H_{W}\to\mathbf{Z}[\bm{q}^{\pm\frac{1}{2}}]$ corresponds
to $\phi:\mathbf{Z}[W]\to\mathbf{Z}$ under Tits’s deformation theorem and
$\\{-,-\\}$ is a $\mathbf{Q}$-valued pairing on $\hat{W}$ derived from
Lusztig’s work on the character theory of $G(\mathbf{F})$ [L84-1]. The
function $\langle{-}\rangle_{\bm{q}}$ can be used to recover Gomi’s Markov
trace, and our formula is closely related to his formula. In particular, we
will show that various properties of the Markov trace extend to
$\langle{-}\rangle_{\bm{q}}$: its rationality, the palindromic symmetry of its
coefficients, and its compatibility with parabolic subgroups of $W$.

The function $w\mapsto\langle{\sigma_{w}}\rangle_{\bm{q}}$, evaluated on
elements $w\in W$ of minimal Bruhat length in their conjugacy classes,
recovers the function that Lusztig–Yun introduce in [LY19], up to a
normalization factor. Thus it also encodes the map from conjugacy classes in
$W$ to unipotent classes in $G$ that Lusztig constructs in [L11].

The definition of $\langle{-}\rangle_{\bm{q}}$ can be extended to any finite
Coxeter group $W$, as a consequence of [L93, L94, M]. In this generality, we
will relate special values of $\langle{-}\rangle_{\bm{q}}$ to the cyclotomic
block theory of $H_{W}$, _i.e._ , to the representation theory of the ring
obtained from $H_{W}$ by specializing $\bm{q}^{\frac{1}{2}}$ to a root of
unity. To this end, we need a central element $\pi\in\mathit{Br}_{W}^{+}$
known as the full twist. We will say that an element $\beta\in\mathit{Br}_{W}$
is periodic of slope $\nu=\frac{m}{n}\in\mathbf{Q}$ iff

(15) $\displaystyle\beta^{n}=\pi^{m}.$

For such $\beta$, we give an explicit formula for
$\langle{\beta}\rangle_{\bm{q}}$ in terms of constants of the form
$\mathrm{Deg}_{\phi}(e^{2\pi i\nu})$, where $\mathrm{Deg}_{\phi}(\bm{q})$ is
the so-called generic or unipotent degree of $\phi$ [GP, §8.1.8]. The
constants $\mathrm{Deg}_{\phi}(e^{2\pi i\nu})$ are integers that roughly
control the sizes of the blocks of $H_{W}(e^{\pi
i\nu})=H_{W}|_{\bm{q}^{1/2}=e^{\pi i\nu}}$.

As an application, we relate $[\mathsf{AH}(\mathcal{R}(\beta))]_{\bm{q}}$ to
the representations of a symplectic reflection algebra
$\mathbf{D}_{\nu}^{\mathit{rat}}\supseteq\mathbf{A}_{W}$ known as the rational
Cherednik algebra or rational double affine Hecke algebra (DAHA). We can view
it as a deformation of
$\mathbf{D}_{0}^{\mathit{rat}}=\bar{\mathbf{Q}}_{\ell}[W]\ltimes\mathcal{D}(\mathbf{V})$
that depends on $\nu$, where $\mathcal{D}(\mathbf{V})$ is the Weyl algebra of
differential operators on $\mathbf{V}$. Like a semisimple Lie algebra,
$\mathbf{D}_{\nu}^{\mathit{rat}}$ admits a “category $\mathsf{O}$” of well-
behaved modules, except that the simple objects are parametrized by $\hat{W}$.
We respectively write $L_{\nu}(\phi)$ and $\Delta_{\nu}(\phi)$ for the simple
and Verma objects indexed by $\phi\in\hat{W}$. There is a functor from
category $\mathsf{O}=\mathsf{O}_{\nu}$ to the category of $H_{W}(e^{\pi
i\nu})$-modules, known as the Knizhnik–Zamolodchikov functor, that sends
$\Delta_{\nu}(\phi)$ to the $H_{W}(e^{\pi i\nu})$-module associated with
$\phi_{\bm{q}}$.

Any object $M\in\mathsf{O}_{\nu}$ gives rise to a graded character
$[M]_{\bm{q}}\in R(W)(\\!(\bm{q}^{\frac{1}{2}})\\!)$. When $\beta$ is periodic
of slope $\nu$, we can express $[\mathsf{AH}(\mathcal{R}(\beta))]_{\bm{q}}$ as
an explicit virtual sum of the Verma characters
$[\Delta_{\nu}(\phi)]_{\bm{q}}$. At special values of $\nu$, we can say more.
For instance, at so-called cuspidal values, it simplifies to an explicit
_nonnegative_ sum of _simple_ characters $[L_{\nu}(\phi)]_{\bm{q}}$.

When $W$ is a Weyl group, we conjecture that this relation between periodic
braids and the rational DAHA is categorified by a
$\mathbf{D}_{\nu}^{\mathit{rat}}$-action on a modified version of
$\mathsf{AH}(\mathcal{R}(\beta))$. When $W=S_{n}$, our work recovers a theorem
of Gorsky–Oblomkov–Rasmussen–Shende relating the HOMFLY polynomials of torus
knots to the characters $[L_{\nu}(1)]_{\bm{q}}$, where $1\in\hat{W}$ is the
trivial character of $W$ [GORS].

###  0.2. Outline and Results

Throughout the paper, we use stacky conventions for (co)homological degree and
assume that $\mathbf{F}$ has large characteristic. In what follows, we also
assume that $\mathbf{V}^{W}=0$. We set $r=\dim\mathbf{V}$.

In Section 1, we review background on $G_{0}$-varieties and their derived
categories of $\ell$-adic complexes, with special attention to Deligne’s yoga
of weights. Then we introduce the formalism for realization functors that will
serve our purposes. In particular, we introduce a condition called
$\underline{\operatorname{{Hom}}}$-purity that slightly generalizes the
condition called Frobenius invariance in [R].

In Section 2, we give explicit definitions of the categories
$\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})$, $\mathsf{C}(G_{0})$,
$\mathsf{C}_{\hat{W}}(G_{0})$, $\mathsf{C}(\mathcal{U}_{0})$,
$\mathsf{C}_{\hat{W}}(\mathcal{U}_{0})$, as well as the functors
$\mathsf{CH}$, $i^{\ast}\langle-r\rangle$, $\mathsf{PR}$ that relate them. We
also introduce the objects $\mathcal{G}$ and $\mathcal{S}$, along with their
summands. We give a thorough exposition of the geometric interpretation of the
Hecke category that we were unable to find elsewhere. The main novelty in this
section is a proof that $\mathsf{C}(\mathcal{U}_{0})$ is
$\underline{\operatorname{{Hom}}}$-pure, using results from [L84-2, RR].

In Section 3, we introduce the ring $\mathbf{A}_{W}$ and the monoidal trace
$\mathsf{AH}$.

In Section 4, we first review equivariant Borel–Moore homology, then introduce
the Steinberg scheme $\mathcal{Z}\to\mathcal{U}$ and its multiplicative
analogue $\mathcal{Z}^{\prime}\to G$. Using an inductive argument inspired by
an unpublished note of Riche, we establish a comparison between the
Borel–Moore homologies of $\mathcal{Z}$ and $\mathcal{Z}^{\prime}$, which
leads to the following theorem:

######  Theorem 1.

The composition
$\bigoplus_{n}\operatorname{{gr}}_{n}^{\bm{w}}\underline{\operatorname{{End}}}^{n}(\mathcal{G})\to\underline{\operatorname{{End}}}^{\ast}(\mathcal{G})\xrightarrow{i^{\ast}}\underline{\operatorname{{End}}}^{\ast}(\mathcal{S})$
is an isomorphism of graded $\bar{\mathbf{Q}}_{\ell}$-algebras.

In Section 5, we first review the positive braid monoid
$\mathit{Br}_{W}^{+}\subseteq\mathit{Br}_{W}$. Then we review the map
$\beta\mapsto O(\beta)$ studied by Deligne and Broué–Michel and the map
$\beta\mapsto\mathcal{R}(\beta)$ introduced by Rouquier. Using our realization
formalism, we rederive the folklore comparison between the variety $O(\beta)$
and the complexes $\mathcal{R}(\beta)$, $\mathcal{R}(\beta^{-1})$ for positive
$\beta$. Using the work in Section 3, we prove:

######  Theorem 2.

We have

(16)
$\displaystyle\mathsf{AH}^{i,j}(\mathcal{R}(\beta))\simeq\operatorname{{gr}}_{j+2r}^{\bm{w}}\mathrm{H}_{-(i+2r)}^{!,G}(\mathcal{Z}(\beta))$

for all $\beta\in\mathit{Br}_{W}^{+}$ and $i,j\in\mathbf{Z}$.

In Section 6, we review Khovanov–Rozansky homology and its geometric
interpretation in terms of the horocycle correspondence due to
Webster–Williamson. We use the work of Sections 2, 3, and 4 to show:

######  Theorem 3.

With respect to the grading conventions on $\mathsf{AH}$ in Section 3 and on
$\mathsf{HHH}$ in Section 6, we have

(17)
$\displaystyle(\mathsf{HHH}^{i,i+j,k})^{\vee}\simeq\operatorname{{Hom}}_{W}(\operatorname{\text{$\Lambda$}}^{i}(\mathbf{V}),\mathsf{AH}^{j+k,j}(-))$

as functors $\mathsf{H}_{W}\to\mathsf{Vect}$ for all $i,j,k\in\mathbf{Z}$.

Let $\mathcal{X}(\beta)=\mathcal{X}(\beta)_{0}\otimes\bar{\mathbf{F}}$ be the
fiber of $\mathcal{U}(\beta)=\mathcal{U}(\beta)_{0}\otimes\bar{\mathbf{F}}$
above $1\in\mathcal{U}$. Using well-known properties of the $W$-action on the
Springer resolution, we deduce:

######  Corollary 4.

We have

(18) $\displaystyle\mathsf{HHH}^{0,j,k}(\mathcal{R}(\beta))^{\vee}$
$\displaystyle\simeq\operatorname{{gr}}_{j+2r}^{\bm{w}}\mathrm{H}_{-(j+k+2r)}^{!,G}(\mathcal{U}(\beta)),$
(19) $\displaystyle\mathsf{HHH}^{r,r+j,k}(\mathcal{R}(\beta))^{\vee}$
$\displaystyle\simeq\operatorname{{gr}}_{j+2(r-N)}^{\bm{w}}\mathrm{H}_{-(j+k+2(r-N))}^{!,G}(\mathcal{X}(\beta)).$

for all $\beta\in\mathit{Br}_{W}^{+}$, where $N=\dim\mathcal{B}$.

Combining the corollary with a “Serre duality” result for $\mathsf{HHH}$
[GHMN], we find that for $G=\operatorname{{SL}}_{n}$ and any
$\beta\in\mathit{Br}_{W}^{+}$, there is a weight-preserving isomorphism
between the equivariant Borel–Moore homologies of the varieties
$\mathcal{U}(\beta)$ and $\mathcal{X}(\beta\pi)$. We revisit this observation
in §0.3.2.

In Section 7, we introduce the function $\langle{-}\rangle_{\bm{q}}:H_{W}\to
R(W)[\bm{q}^{\pm\frac{1}{2}}]$ in terms of the background covered in Appendix
A. Its definition can be extended to any finite Coxeter group $W$. In the case
of Weyl groups, we show that $\langle{-}\rangle_{\bm{q}}$ is the
decategorification of $\mathsf{AH}$ up to a normalization factor. The key
ingredient is a multiplicity formula for character sheaves that involves
$\\{-,-\\}$, due to Lusztig [L85-III, Cor. 14.11] [L85-V, Thm. 23.1].

######  Theorem 5.

If $W$ is a Weyl group, and $\mathcal{K}\in\mathsf{H}_{W}$ decategorifies to
$\beta\in H_{W}$, then

(20)
$\displaystyle[\mathsf{AH}^{\vee}(\mathcal{K})]_{\bm{q}}=\langle{\beta}\rangle_{\bm{q}}\cdot[\operatorname{{Sym}}(\mathbf{V})]_{\bm{q}},$

where $(\mathsf{AH}^{\vee})^{i,j}=(\mathsf{AH}^{i,j})^{\vee}$ for all $i,j$
and
$[\operatorname{{Sym}}(\mathbf{V})]_{\bm{q}}=\sum_{i}{\bm{q}^{i}}\operatorname{{Sym}}^{i}(\mathbf{V})$.

Next, we explain the precise relationship between $\langle{-}\rangle_{\bm{q}}$
and the Markov trace studied by Ocneanu and Gomi: See Proposition 7.4.3. It is
most conveniently stated in terms of the class function
$\langle{-}\rangle_{\bm{q}}^{0}:\mathit{Br}_{W}\to
R(W)(\\!(\bm{q}^{\frac{1}{2}})\\!)$ defined by

(21)
$\displaystyle\langle{\beta}\rangle_{\bm{q}}^{0}=(-\bm{q}^{\frac{1}{2}})^{|\beta|}\langle{\beta}\rangle_{\bm{q}}\cdot\varepsilon[\operatorname{{Sym}}(\mathbf{V})]_{\bm{q}},$

where $|\beta|$ is the writhe of $\beta$ and $\varepsilon$ is the sign
character of $W$ (see §A.5).

Using properties of $\\{-,-\\}$, we show that certain rationality and symmetry
properties of the Markov trace generalize to $\langle{-}\rangle_{\bm{q}}$ and
$\langle{-}\rangle_{\bm{q}}^{0}$. For Weyl groups, the strongest form of the
rationality property will be Theorem 9, which we prove in Section 8. For
general Coxeter groups, the remaining properties are:

######  Theorem 6.

For all $\beta\in\mathit{Br}_{W}$, we have

(22) $\displaystyle\langle{\beta}\rangle_{\bm{q}}$
$\displaystyle\in\bm{q}^{-\frac{|\beta|}{2}}R(W)[\bm{q}]\cap
R(W)[\bm{q}^{\frac{1}{2}}-\varepsilon\bm{q}^{-\frac{1}{2}}],$ (23)
$\displaystyle\langle{\beta}\rangle_{\bm{q}}^{0}$ $\displaystyle\in
R(W)[\\![\bm{q}]\\!]\cap
R(W)(\bm{q})\cap(\bm{q}^{\frac{1}{2}})^{|\beta|-r}R(W)[(\bm{q}^{\frac{1}{2}}-\bm{q}^{-\frac{1}{2}})^{\pm
1}].$

In particular, $\langle{\beta}\rangle_{\bm{q}}$ and
$\langle{\beta}\rangle_{\bm{q}}^{0}$ are invariant under
$\bm{q}^{\frac{1}{2}}\mapsto-\varepsilon\bm{q}^{-\frac{1}{2}}$ and
$\bm{q}^{\frac{1}{2}}\mapsto-\bm{q}^{-\frac{1}{2}}$, respectively. Moreover,
$\langle{\beta}\rangle_{\bm{q}}^{0}$ has degree $|\beta|-r$ as a rational
function in $\bm{q}$.

In Section 8, we again assume that $W$ is the Weyl group of $G$. After
reviewing background on Deligne–Lusztig virtual representations, we use their
properties to express $\langle{-}\rangle_{\bm{q}}^{0}$ in terms of the
$\mathbf{F}$-point counts of the fibers of the map
$\mathcal{U}(\beta)\to\mathcal{U}$. Namely, we need results of
Deligne–Lusztig–Springer [DL, Thm. 6.9] [S76, Thm. 5.6], Kazhdan [Kaz],
Lusztig [L84-1, Cor. 4.24-4.25], and Geck–Lusztig [G92, Prop. 1.3]. To state
the formula, let $\mathcal{B}_{g}$ denote the Springer fiber above $g\in G$,
and let

(24) $\displaystyle
Q_{g}(w)=\sum_{n}{\operatorname{{tr}}(wF\mid\mathrm{H}^{2n}(\mathcal{B}_{g},\bar{\mathbf{Q}}_{\ell}))}$

for all $w\in W$. When $G$ is a _special_ algebraic group in the sense of
Serre [Gr], the formula below is a kind of virtual weight series for
$[\mathcal{Z}(\beta)/G]$, so it is a corollary of Theorem 5 via [HR, Appendix]
and [Jo, Thm. 4.10] (see Remark 8.5.2). However, not all algebraic groups are
special.

######  Theorem 7.

For all $\beta\in\mathit{Br}_{W}^{+}$, we have

(25) $\displaystyle\langle{\beta}\rangle_{\bm{q}}^{0}|_{\bm{q}=q}$
$\displaystyle=\frac{(-1)^{r-|\beta|}}{|G^{F}|}\sum_{u\in\mathcal{U}^{F}}|\mathcal{U}(\beta)_{u}^{F}|Q_{u}$

in $\mathbf{Q}\otimes R(W)$, where $q=|\mathbf{F}|$.

For $G=\operatorname{{PGL}}_{n}$, taking $\varepsilon$-isotypic components on
both sides and applying the Serre-duality result of [GHMN], or Kálmán’s
earlier result [Kál], recovers a theorem of Shende–Treumann–Zaslow, matching
the “lowest” $a$-degree of the HOMFLY series of the link closure of $\beta$
with the stacky point count of $[\mathcal{X}(\beta\pi)/G]$ [STZ, Thm. 6.34].
In fact, Kálmán’s result generalizes to a statement about Gomi’s Markov trace.
For general $G$, this generalization and Theorem 7 together imply [L21, Prop.
2.2], which Lusztig attributes to Kawanaka [Kaw]: See Corollary 8.6.3.

Recall that a subgroup $W^{\prime}\subseteq W$ is parabolic iff it is the
stabilizer of a vector in $\mathbf{V}$. In this case, there is an inclusion of
Iwahori–Hecke algebras $H_{W^{\prime}}\subseteq H_{W}$. Using Theorem 7 and
the compatibility of Deligne–Lusztig characters with parabolic induction, we
show that $\langle{-}\rangle_{\bm{q}}$ is similarly compatible with parabolic
induction.

######  Theorem 8.

If $W$ is a Weyl group of rank $r$ and $W^{\prime}\subseteq W$ is a parabolic
subgroup of rank $r^{\prime}$, then we have commutative diagrams:

(26)
${H_{W^{\prime}}}$${R(W^{\prime})[\bm{q}^{\pm\frac{1}{2}}]}$${H_{W}}$${R(W)[\bm{q}^{\pm\frac{1}{2}}]}$$\langle{-}\rangle_{\bm{q}}$$\operatorname{{Ind}}_{W^{\prime}}^{W}$$\langle{-}\rangle_{\bm{q}}$
${\mathit{Br}_{W^{\prime}}}$${R(W^{\prime})[\\![\bm{q}]\\!]}$${\mathit{Br}_{W}}$${R(W)[\\![\bm{q}]\\!]}$$\langle{-}\rangle_{\bm{q}}^{0}$$\frac{1}{(1-\bm{q})^{r-r^{\prime}}}\operatorname{{Ind}}_{W^{\prime}}^{W}$$\langle{-}\rangle_{\bm{q}}^{0}$

In particular, $\langle{\mathbf{1}}\rangle_{\bm{q}}$ is the character of the
regular representation of $W$, placed in $\bm{q}$-degree zero, and
$\langle{\mathbf{1}}\rangle_{\bm{q}}^{0}=(1-\bm{q})^{-r}\langle{\mathbf{1}}\rangle_{\bm{q}}$.

To conclude Section 8, we bound the pole of $\langle{-}\rangle_{\bm{q}}^{0}$
at $\bm{q}=1$ using Theorem 7 and an idea of Lusztig from [L11, §5]. For all
$w\in W$, let $r(w)$ be the dimension of the fixed subspace
$\mathbf{V}^{w}\subseteq\mathbf{V}$.

######  Theorem 9.

If $W$ is a Weyl group and $\beta\in\mathit{Br}_{W}^{+}$ maps to $w\in W$,
then

(27)
$\displaystyle\langle{\beta}\rangle_{\bm{q}}^{0}\in\frac{1}{(1-\bm{q})^{r(w)}}\,R(W)[\bm{q}].$

That is, the pole of $\langle{-}\rangle_{\bm{q}}^{0}$ at $\bm{q}=1$ is at most
$r(w)$.

In the case where $\beta=\sigma_{w}$ for some $w\in W$, this result follows
from work of Lusztig–Yun in [LY19]. Indeed, if $w\in W$ has minimal Bruhat
length in its conjugacy class, then Theorem 7 essentially implies

(28)
$\displaystyle\langle{\sigma_{w}}\rangle_{\bm{q}}^{0}=\frac{\Psi(w)}{(1-\bm{q})^{r(w)}},$

where $\Psi:W\to R(W)[\bm{q}]$ is the function introduced in _ibid._

In Sections 9-10, we allow $W$ to be any finite Coxeter group and $\mathbf{V}$
any realization of $W$ such that $\mathbf{V}^{W}=0$. In Section 9, we compute
$\langle{\beta}\rangle_{\bm{q}}$ for periodic $\beta$. The key input is that,
essentially by Schur’s lemma, the full twist $\pi$ acts on any simple
$H_{W}$-module by a monomial in $\bm{q}^{\frac{1}{2}}$. This, together with a
deformation argument of Jones [J, §9], allows us to determine the character
values $\phi_{\bm{q}}(\beta)$ whenever some nonzero power of $\beta$ equals a
power of the full twist.

######  Theorem 10.

If $\beta\in\mathit{Br}_{W}$ is periodic of slope $\nu$, then

(29)
$\displaystyle\langle{\beta}\rangle_{\bm{q}}=\sum_{\phi\in\hat{W}}{\bm{q}^{\nu\mathbf{c}(\phi)}}\mathrm{Deg}_{\phi}(e^{2\pi
i\nu})\phi,$

where $\mathrm{Deg}_{\phi}(\bm{q})\in\mathbf{Q}[\bm{q}]$ and
$\mathbf{c}(\phi)\in\mathbf{Z}$ are the unipotent degree and content of
$\phi$, respectively (see §A.8, §A.10).

In the setting of the above theorem, we will have $\mathrm{Deg}_{\phi}(e^{2\pi
i\nu})\in\mathbf{Z}$ for all $\phi$ and $\nu$. It will be convenient to set

(30)
$\displaystyle\Omega_{\nu}=\bigoplus_{\phi\in\hat{W}}\mathrm{Deg}_{\phi}(e^{2\pi
i\nu})\Delta_{\nu}(\phi),$

viewed as a virtual $\mathbf{D}_{\nu}^{\mathit{rat}}$-module. In Section 10,
we deduce:

######  Corollary 11.

If $\beta\in\mathit{Br}_{W}$ is periodic of slope $\nu$, then

(31)
$\displaystyle{[\Omega_{\nu}]_{\bm{q}}}=(\bm{q}^{\frac{1}{2}})^{r-|\beta|}\cdot\langle{\beta}\rangle_{\bm{q}}^{0}$

in $R(W)[(\bm{q}^{\frac{1}{2}}-\bm{q}^{-\frac{1}{2}})^{\pm 1}]$.

In what follows, let $d_{1},\ldots,d_{r}$ be the invariant degrees of the
$W$-action on $\mathbf{V}$ [GP, 149]. We say that $\nu\in\mathbf{Q}$ is:

  1. (1)

A singular slope iff $\nu\in\frac{1}{d_{i}}\mathbf{Z}$ for some $i$.

  2. (2)

A regular slope iff $W$ contains a $e^{2\pi i\nu}$-regular element $w$. This
means $w$ has an eigenvector in $\mathbf{V}$ with eigenvalue $e^{2\pi i\nu}$
and trivial $W$-stabilizer [S74].

  3. (3)

A regular elliptic slope iff, in (2), we can pick $w$ so that
$\mathbf{V}^{w}=0$.

  4. (4)

A cuspidal slope iff $\nu\in\frac{1}{d_{i}}\mathbf{Z}$ for a unique index $i$.

Conditions (1)-(4) are ordered by increasing strictness. By work of
Broué–Michel [BM97], condition (2) is equivalent to the existence of a
periodic braid of slope $\nu$.

In Section 10, we use tools from the representation theory of
$\mathbf{D}_{\nu}^{\mathit{rat}}$, especially the Knizhnik–Zamolodchikov
functor from category $\mathsf{O}_{\nu}$ to the category of $H_{W}(e^{\pi
i\nu})$-modules [GGOR], to prove:

######  Theorem 12.

Let $\nu\in\mathbf{Q}$.

  1. (1)

$L_{\nu}(1)$ occurs in $\Omega_{\nu}$ with multiplicity one.

  2. (2)

If $\nu$ is a singular slope, then $\Omega_{\nu}$ is a virtual linear
combination of torsion modules over the subring
$\operatorname{{Sym}}(\mathbf{V}^{\vee})\subseteq\mathbf{D}_{\nu}^{\mathit{rat}}$.

  3. (3)

If $\nu$ is a regular elliptic slope, $\nu>0$, and $W$ is a Weyl group, then

(32) $\displaystyle[\Omega_{\nu}]_{\bm{q}}\in R(W)[\bm{q}^{\pm\frac{1}{2}}].$

That is, $\Omega_{\nu}$ is a difference of finite-dimensional
$\mathbf{D}_{\nu}^{\mathit{rat}}$-modules.

  4. (4)

If $\nu$ is a cuspidal slope and $\nu>0$, then

(33)
$\displaystyle[\Omega_{\nu}]_{\bm{q}}=\sum_{\psi\in\hat{W}_{1}}{[L_{\nu}(\psi)]_{\bm{q}}},$

where $\hat{W}_{1}\subseteq\hat{W}$ is the set of elements of greatest
$\mathbf{a}$-value within their $e^{2\pi i\nu}$-blocks, among the $e^{2\pi
i\nu}$-blocks of $H_{W}$ of defect $1$ (see §A.11).

Parts (1) and (2) together imply a result of Dunkl–de Jeu–Opdam [DJO, Thm.
4.9(1)], while part (3) implies a result of Varagnolo–Vasserot [VV, Thm.
2.8.1(a)]. The key machinery in the proof of part (4) is work of Geck on the
block theory of $H_{W}$, which shows that when $\nu$ is cuspidal, the values
$\mathrm{Deg}_{\phi}(e^{2\pi i\nu})$ are computable and all belong to
$\\{-1,0,1\\}$ [G92]. Part (4) implies:

######  Corollary 13.

Keeping the hypotheses of Theorem 12, suppose that $W$ is irreducible (see
§A.1) and at least one of the following is true:

  * •

$W$ is not of type $E_{8}$ or $H_{4}$.

  * •

$\nu\notin\frac{1}{15}\mathbf{Z}$.

Then $[\Omega_{\nu}]_{\bm{q}}=[L_{\nu}(1)]_{\bm{q}}$.

For $W=S_{n}$, Theorem 3, Corollary 11, and Corollary 13 together recover
[GORS, Thm. 1.1], which was originally proved by explicit calculation in the
ring of symmetric functions.

In Appendix A, we review facts about the structure and representation theory
of $W$ and $H_{W}$ that we need elsewhere, including the definitions of the
pairing $\\{-,-\\}$ and the unipotent degrees $\mathrm{Deg}_{\phi}$.

In Appendix B, we state precise comparisons between the schemes
$\mathcal{U}(\beta)$, $\mathcal{Z}(\beta)$, $\mathcal{X}(\beta)$ and the
varieties in the works of Lusztig [L11], Shende–Treumann–Zaslow [STZ], Mellit
[Me], Casals–Gorsky–Gorsky–Simental [CGGS], and others.

In Appendix C, we list values for the characters
$\langle{\beta}\rangle_{\bm{q}}$ and $\langle{\beta}\rangle_{\bm{q}}^{0}$ in
low rank, as well as an example where $\beta$ is an iterated torus braid.

###  0.3. Conjectures and Questions

#### 0.3.1.

The appearance of the ring $\mathbf{A}_{W}$ suggests that $\mathsf{AH}$ is
closely related to a functor on $\mathsf{H}_{W}$ called the horizontal trace,
which [QRS, GW, GHW] studied in the case where $W=S_{n}$ via diagrammatic
rather than geometric methods. If $\star$ denotes the monoidal product on
$\mathsf{H}_{W}$, then the horizontal trace is roughly the universal additive
functor under which $\mathcal{K}\star\mathcal{L}$ and
$\mathcal{L}\star\mathcal{K}$ become isomorphic for all objects
$\mathcal{K},\mathcal{L}\in\mathsf{H}_{W}$.

In [QRS], Queffelec–Rose–Sartori show that the horizontal trace refines
$\mathsf{HHH}$, and in [GW], Gorsky–Wedrich roughly show that it takes values
in the homotopy category of $\mathbf{A}_{W}$-modules. (The difference is that
Gorsky–Wedrich take $\mathbf{V}$ to be the $n$-dimensional permutation
representation of $S_{n}$, rather than its $(n-1)$-dimensional irreducible
summand.)

######  Conjecture 14.

After we replace the semisimple group $G$ with the reductive group
$\mathrm{GL}_{n}$, our procedure to construct $\mathsf{AH}$ yields (the
objectwise dual of) (the homology of) the horizontal trace studied in [GW].

#### 0.3.2.

We expect the geometry of $\mathcal{U}(\beta)$ to manifest a “Serre duality”
phenomenon involving the full twist. Gorsky–Hogancamp–Mellit–Nakagane showed
that for $G=\operatorname{{SL}}_{n}$ and any $\beta\in\mathit{Br}_{W}$, the
lowest Hochschild degree of $\mathsf{HHH}(\mathcal{R}(\beta))$ and the highest
Hochschild degree of $\mathsf{HHH}(\mathcal{R}(\beta\pi))$ coincide [GHMN],
which categorifies a theorem of Kálmán about the HOMFLY series [Kál].
Corollary 4 leads us to expect a geometric explanation of this coincidence.

In earlier versions of this work, we hoped for the existence of an algebraic
homeomorphism from $[\mathcal{X}(\beta\pi)/G]$ onto $[\mathcal{U}(\beta)/G]$
that would explain the weight-preserving isomorphism between their Borel–Moore
homologies. Unfortunately, as we will explain in future work, this conjecture
already fails for $G=\operatorname{{SL}}_{3}$.

######  Question 15.

Is there a geometric explanation for the (shifted) isomorphism
$\operatorname{{gr}}_{\ast}^{\bm{w}}\mathrm{H}_{\ast}^{!,G}(\mathcal{X}(\beta\pi))\simeq\operatorname{{gr}}_{\ast}^{\bm{w}}\mathrm{H}_{\ast}^{!,G}(\mathcal{U}(\beta))$?

#### 0.3.3.

Oblomkov–Rozansky have recently proved that the Khovanov–Rozansky invariant of
a _knot_ enjoys a bigraded symmetry that categorifies the
$\bm{q}^{\frac{1}{2}}\mapsto-\bm{q}^{-\frac{1}{2}}$ symmetry of its HOMFLY
series [OR], resolving a conjecture from [DGR]. We expect this result to
extend to $\mathsf{AH}$. More precisely, we expect that for certain positive
braids $\beta$, the stack $[\mathcal{Z}(\beta)/G]$ enjoys a “curious
Lefschetz” property that implies a corresponding bigraded symmetry of the
module $\mathsf{AH}(\mathcal{R}(\beta))$, by way of Theorem 2.

Following Hausel–Rodriguez-Villegas [HR], if $X_{0}$ is (a stack quotient of)
a variety over $\mathbf{F}$ of dimension $d$ and
$X=X_{0}\otimes\bar{\mathbf{F}}$, then we say that the curious Lefschetz
property holds for $X$ iff there are isomorphisms

(34)
$\displaystyle\operatorname{{gr}}_{d-j}^{\bm{w}}\mathrm{H}_{!}^{i}(X,\bar{\mathbf{Q}}_{\ell})\xrightarrow{\sim}\operatorname{{gr}}_{d+j}^{\bm{w}}\mathrm{H}_{!}^{i+j}(X,\bar{\mathbf{Q}}_{\ell})$

of compactly-supported cohomology for all $i$ and $j$, induced by the cup
product with a fixed class in $\mathrm{H}_{!}^{2}$. This means the series

(35)
$\displaystyle(\bm{q}^{\frac{1}{2}})^{-d}\sum_{i,j}\bm{q}^{\frac{j}{2}}t^{i}\operatorname{{gr}}_{j}^{\bm{w}}\mathrm{H}_{!}^{i}(X,\bar{\mathbf{Q}}_{\ell})$

is invariant under $\bm{q}^{\frac{1}{2}}\mapsto\bm{q}^{-\frac{1}{2}}t^{-1}$.
Mellit has recently established this property for character varieties with
structure group $G=\operatorname{{GL}}_{n}$ [Me]. In the course of his proof,
he also establishes the property for certain varieties closely related to the
fibers of the maps $\mathcal{X}(\beta)\to\mathcal{B}$ (see Appendix B). His
results, as well as our Theorem 6, suggest evidence for:

######  Conjecture 16.

For all $\beta\in\mathit{Br}_{W}^{+}$, the curious Lefschetz property holds
for $\mathcal{U}(\beta)$ and $\mathcal{Z}(\beta)$. If, in addition, $\beta$
maps to an element $w\in W$ such that $\mathbf{V}^{w}=0$, then the property
also holds for their stack quotients by $G$.

Note that for $W=S_{n}$, the hypothesis that $w$ satsfies $\mathbf{V}^{w}=0$
is equivalent to the hypothesis that the link closure of $\beta$ is a knot.

In [GH], Gorsky–Hogancamp introduce a refinement of $\mathsf{HHH}$ called _$y$
-ified homology_ and denoted $\mathsf{HY}$. They conjecture that the bigraded
symmetry of the Khovanov–Rozansky invariant can be extended from knots to
arbitrary links by replacing $\mathsf{HHH}$ with $\mathsf{HY}$. In future
work, we will relate $\mathsf{HY}$ to a _monodromic_ refinement of
$\mathsf{AH}$.

#### 0.3.4.

Lastly, we propose a (partial) categorification of Theorem 10, which would
strengthen some of the conjectures from [GORS] relating knot invariants and
the rational DAHA.

Let $W=S_{n}$ and $\nu=\frac{m}{n}$, with $m,n>0$ coprime. Gorsky, Oblomkov,
Rasmussen, and Shende construct three filtrations on the simple
$\mathbf{D}_{\nu}^{\mathit{rat}}$-module $L_{\nu}(1)$ that are compatible with
the grading on its character. They conjecture that all three are the same
filtration $\bm{f}_{\leq\ast}$, and that their theorem can be promoted to an
isomorphism between the Khovanov–Rozansky homology of the $(m,n)$ torus knot
and the sum of the $\operatorname{\text{$\Lambda$}}^{i}(\mathbf{V})$-isotypics
of $\operatorname{{gr}}_{\ast}^{\bm{f}}L_{\nu}(1)$. One of the candidate
filtrations is constructed geometrically, as we now explain.

For any almost-simple, simply-connected group $G$ and regular elliptic slope
$\nu$, Oblomkov–Yun have related the DAHA module $L_{\nu}(1)$ to a complex
projective $\mathbf{G}_{m}$-variety that we will denote
$\mathcal{B}_{\nu}^{\mathit{aff}}$ [OY]. In technical terms,
$\mathcal{B}_{\nu}^{\mathit{aff}}$ is an affine Springer fiber of Iwahori type
with structure group $G$. By means of the topology of a generalized Hitchin
fibration, its equivariant cohomology is endowed with an increasing filtration
$\bm{p}_{\leq\ast}$ called the perverse filtration.

There is a flat deformation of $\mathbf{A}_{W}$ over
$\bar{\mathbf{Q}}_{\ell}[\epsilon]$ that we will denote
$\mathbf{A}_{W,\epsilon}$, known as Lusztig’s graded or degenerate affine
Hecke algebra and satisfying
$\mathbf{A}_{W,\epsilon}|_{\epsilon=0}=\mathbf{A}_{W}$. Oblomkov–Yun construct
an $\mathbf{A}_{W,\epsilon}$-action on
$\mathrm{H}_{\mathbf{G}_{m}}^{\ast}(\mathcal{B}_{\nu}^{\mathit{aff}},\bar{\mathbf{Q}}_{\ell})$
that sends $\epsilon$ to the equivariant parameter. Similarly, there is a flat
deformation of $\mathbf{D}_{\nu}^{\mathit{rat}}$ over
$\bar{\mathbf{Q}}_{\ell}[\epsilon]$ to a graded algebra that we will denote
$\mathbf{D}_{\nu,\epsilon}^{\mathit{rat}}$, satisfying
$\mathbf{D}_{\nu,\epsilon}^{\mathit{rat}}|_{\epsilon=1}=\mathbf{D}_{\nu}^{\mathit{rat}}$
[OY, §4.2.4]. The $\mathbf{A}_{W,\epsilon}$-action on
$\mathrm{H}_{\mathbf{G}_{m}}^{\ast}(\mathcal{B}_{\nu}^{\mathit{aff}},\bar{\mathbf{Q}}_{\ell})$
induces an $\mathbf{A}_{W}$-action that extends to a
$\mathbf{D}_{\nu,\epsilon}^{\mathit{rat}}$-action. For
$G=\operatorname{{SL}}_{n}$ and regular elliptic $\nu$, it turns out that

(36)
$\displaystyle\operatorname{{gr}}_{\ast}^{\bm{p}}\mathrm{H}_{\mathbf{G}_{m}}^{\ast}(\mathcal{B}_{\nu}^{\mathit{aff}},\bar{\mathbf{Q}}_{\ell})|_{\epsilon=1}\simeq
L_{\nu}(1)$

as $\mathbf{D}_{\nu}^{\mathit{rat}}$-modules, via an isomorphism that sends
the $\bm{p}$-grading to the grading on $[L_{\nu}(1)]_{\bm{q}}$. The authors of
[GORS] define the geometric candidate for $\bm{f}_{\leq\ast}$ to be the
difference of the perverse and cohomological filtrations on the left-hand side
of (36), up to a shift.

In forthcoming work, we will study a conjectural _transcendental_ retraction
from the stack $[\mathcal{Z}(\beta)/G]$ to the variety
$\mathcal{B}_{\nu}^{\mathit{aff}}$ for any periodic braid
$\beta\in\mathit{Br}_{W}^{+}$ of regular elliptic slope $\nu$. It is expected
to be a kind of nonabelian Hodge correspondence, where
$[\mathcal{Z}(\beta)/G]$ plays the role of the “Betti” side and
$\mathcal{B}_{\nu}^{\mathit{aff}}$ plays the role of the “Dolbeault” side. In
particular, the retraction should transport the doubled weight filtration
$\bm{w}_{\leq 2\ast}$ on the compactly-supported cohomology of
$[\mathcal{Z}(\beta)/G]$ to the perverse filtration $\bm{p}_{\leq\ast}$ on the
cohomology of $\mathcal{B}_{\nu}^{\mathit{aff}}$, up to a degree shift. We are
thus led to conjecture a Betti origin for the filtration $\bm{f}_{\leq\ast}$.

As evidence, note that the $G$-action on $\mathcal{Z}(\mathbf{1})$ extends to
a $(G\times\mathbf{G}_{m})$-action. The isomorphism
$\mathbf{A}_{W}\simeq\mathrm{H}_{-\ast}^{!,G}(\mathcal{Z}(\mathbf{1}))$ can be
lifted to an isomorphism

(37)
$\displaystyle\mathbf{A}_{W,\epsilon}\simeq\mathrm{H}_{-\ast}^{!,G\times\mathbf{G}_{m}}(\mathcal{Z}(\mathbf{1})),$

as shown in [L88]. Unfortunately, as Zhiwei Yun pointed out to me, there is no
obvious $(G\times\mathbf{G}_{m})$-action on $\mathcal{Z}(\beta)$ for other
$\beta$, even when $\beta$ is periodic. Even so, I hope for the following:

######  Conjecture 17.

There is a flat deformation of $\mathsf{AH}$ over
$\bar{\mathbf{Q}}_{\ell}[\epsilon]$ to a functor

(38)
$\displaystyle\mathsf{AH}_{\epsilon}:\mathsf{H}_{W}^{\mathrm{op}}\to\mathsf{Mod}_{2}^{b}(\mathbf{A}_{W,\epsilon})$

such that for all $\mathcal{K}\in\mathsf{H}_{W}$:

  1. (1)

The $\mathbf{A}_{W}$-action on $\mathsf{AH}(\mathcal{K})$ extends to a
$\bm{w}$-_filtered_ $\mathbf{A}_{W,\epsilon}$-action on
$\mathsf{AH}_{\epsilon}(\mathcal{K})$, where $\bm{w}$ denotes the second
grading. By [OY, Prop. 4.3.1], the latter action induces a $\bm{w}$-_graded_
$\mathbf{A}_{W}$-action on $\mathsf{AH}_{\epsilon}(\mathcal{K})$.

If $\beta\in\mathit{Br}_{W}^{+}$ is periodic of regular elliptic slope $\nu$,
then:

  1. (2)

The $\mathbf{A}_{W}$-action on $\mathsf{AH}_{\epsilon}(\mathcal{R}(\beta))$
extends to a $\mathbf{D}_{\nu,\epsilon}^{\mathit{rat}}$-action.

  2. (3)

The $\mathbf{D}_{\nu}^{\mathit{rat}}$-module formed by
$\mathsf{AH}_{1}(\mathcal{R}(\beta))$ contains $L_{\nu}(1)$ as a summand with
multiplicity one.

  3. (4)

The filtration $\bm{f}_{\leq\ast}$ on $L_{\nu}(1)$ defined in [GORS] arises
from the difference of the two gradings on
$\mathsf{AH}_{1}(\mathcal{R}(\beta))$.

Note that in part (1), the differences between the various specializations of
$\epsilon$ would still vanish at the level of graded characters: Thus,
$[\mathsf{AH}_{1}^{\vee}(\mathcal{R}(\beta))]_{\bm{q}}$ would remain the same
as $[\mathsf{AH}^{\vee}(\mathcal{R}(\beta))]_{\bm{q}}$.

######  Question 18.

Is there a $\mathbf{G}_{m}$-action on the stack $[\mathcal{Z}(\beta)/G]$ that
could give rise to the flat deformation of $\mathsf{AH}$ in Conjecture 17? If
not, does there still exist a non-algebraic $\mathbf{C}^{\times}$-action in
the complex setting?

###  0.4. Acknowledgments

I warmly thank Olivier Dudas, Ben Elias, Pavel Etingof, Pavel Galashin, Victor
Ginzburg, Eugene Gorsky, Quoc Hô, Matt Hogancamp, Oscar Kivinen, Thomas Lam,
George Lusztig, Andrei Neguţ, Châu Ngô, Emily Norton, Alexei Oblomkov, Jay
Taylor, David Treumann, and Zhiwei Yun for their interest and for many helpful
discussions.

Olivier Dudas, Emily Norton, and Jay Taylor provided references to the
literature that were helpful for Sections 7, 10, and 8, respectively.

Discussions with Quoc Hô and Zhiwei Yun were very helpful to me when I was
working out the formalism for Sections 1-2.

George Lusztig patiently helped me to correct many attributions throughout the
paper, especially in Section 8 and Appendix B, as well as other errata. The
proof of Theorem 8 is very close to an argument in [LY19].

This work expands and revises a portion of my PhD thesis at the University of
Chicago, advised by Châu Ngô and Victor Ginzburg. I am grateful to Laura Ball,
Oscar Kivinen, and Châu Ngô for their feedback on those chapters. I am also
grateful to Victor Ginzburg, Eugene Gorsky, and Andrei Neguţ for their
encouragement much earlier in the research process.

My work was supported by a Graduate Research Fellowship (grant #1746045) and a
Mathematical Sciences Postdoctoral Fellowship (grant #2002238) from the
National Science Foundation. In graduate school, I also benefited from
workshops held at the Mathematical Sciences Research Institute (MSRI) and the
American Institute for Mathematics (AIM).

##  1\. Purity, Mixedness, and Realization

### 1.1.

Fix a finite field $\mathbf{F}$ and an algebraic closure $\bar{\mathbf{F}}$.
Throughout the paper, _we assume that the characteristic of $\mathbf{F}$ is
large_. Recall that the Galois group of $\bar{\mathbf{F}}$ over $\mathbf{F}$
is pro-generated by the automorphism $a\mapsto a^{q}$. We write $F$ for the
inverse generator, also known as the geometric Frobenius element.

Fix a root datum $(\Phi,\mathbf{X},\Phi^{\vee},\mathbf{X}^{\vee})$. Let $G$ be
the reductive group over $\bar{\mathbf{F}}$ defined by the root datum, and let
$G_{0}$ be its split form over $\mathbf{F}$, so that
$G=G_{0}\otimes_{\mathbf{F}}\bar{\mathbf{F}}$. In some later sections, we will
require $G$ to be semisimple, not merely reductive.

### 1.2.

Let $X_{0}$ be a $G_{0}$-variety over $\mathbf{F}$, equipped with a fixed
$G_{0}$-stable stratification by locally closed subvarieties. Let
$X=X_{0}\otimes\bar{\mathbf{F}}$. In general, we use a subscript $0$ to denote
varieties over $\mathbf{F}$, and omit the subscript to denote their pullback
to $\bar{\mathbf{F}}$.

Fix a prime $\ell$ invertible in $\mathbf{F}$. We write

(39) $\displaystyle\mathsf{D}_{G}^{b}(X)$
$\displaystyle=\mathsf{D}_{G}^{b}(X,\bar{\mathbf{Q}}_{\ell}),$ (40)
$\displaystyle\emph{resp.}\qquad\mathsf{D}_{G,m}^{b}(X_{0})$
$\displaystyle=\mathsf{D}_{G,m}^{b}(X_{0},\bar{\mathbf{Q}}_{\ell}),$

for the bounded derived category of complexes, _resp._ mixed complexes, of
$\bar{\mathbf{Q}}_{\ell}$-sheaves on the lisse-étale site of the stack
$[X/G]$, _resp._ $[X_{0}/G_{0}]$, with constructible cohomology with respect
to the stratification. Mixed complexes were introduced by Deligne in the non-
stacky, étale setting [D80]; we refer to [BBD, Ch. 5] for an overview. We fix
a square root $q^{\frac{1}{2}}\in\bar{\mathbf{Q}}_{\ell}^{\times}$ to define a
half-Tate twist $(\frac{1}{2})$. The shift-twist $\langle
1\rangle=[1](\frac{1}{2})$ is a weight-preserving endofunctor of
$\mathsf{D}_{G,m}^{b}(X_{0})$. We write

(41) $\displaystyle\xi:\mathsf{D}_{G,m}^{b}(X_{0})\to\mathsf{D}_{G}^{b}(X)$

for the pullback functor.

In [LO08], Laszlo–Olsson extended the Grothendieck–Verdier yoga of the six
operations to lisse-étale sheaves on Artin stacks locally of finite type over
a finite-dimensional affine excellent base. In [LO09], they did the same for
the theory of $t$-structures and perverse sheaves developed in [BBD, Ch. 1-2].
In [Su], S. Sun extended the theory of mixed complexes to this setting, under
the hypothesis that _the stabilizers of the stack are affine algebraic
groups_. For all choices of $X_{0}$ that we will consider, this hypothesis
will hold for $[X_{0}/G_{0}]$ and $[X/G]$. Sun’s work covers all of the
results from [BBD, Ch. 5] that we need.

Throughout this paper, all maps between $G_{0}$-varieties or $G$-varieties
will be equivariant and compatible with the relevant stratifications. The six
operations on $\mathsf{D}_{G}^{b}$ and $\mathsf{D}_{G,m}^{b}$ are always
assumed to be derived: That is, we will drop $\mathrm{R}$ and $\mathrm{L}$
from our notation for derived functors.

######  Remark 1.2.1.

The categories $\mathsf{D}_{G,m}^{b}(X_{0})$ and $\mathsf{D}_{G}^{b}(X)$ need
not be closed under internal Hom. But for any
$K,L\in\mathsf{D}_{G,m}^{b}(X_{0})$, the object
$\underline{\operatorname{{Hom}}}(K,L)$ does exist in the _bounded-below_
derived category of mixed complexes over $[X_{0}/G_{0}]$, and similarly with
$\mathsf{D}_{G}^{b}(X)$ in place of $\mathsf{D}_{G,m}^{b}(X_{0})$ [LO08]. We
will not comment further on this issue since it will not raise problems,
except to note that the hypercohomology of
$\underline{\operatorname{{Hom}}}(K,L)$ will always be bounded below.

Let $\mathit{pt}_{0}=\operatorname{{Spec}}\mathbf{F}$, and let
$a:X_{0}\to\mathit{pt}_{0}$ be the structure map. For all objects
$K,L\in\mathsf{D}_{G,m}^{b}(X_{0})$, we write

(42) $\displaystyle\underline{\operatorname{{Hom}}}^{n}(K,L)$
$\displaystyle=\mathrm{H}^{n}(\xi
a_{\ast}\underline{\operatorname{{Hom}}}(K,L)).$

Similarly, we write
$\underline{\operatorname{{End}}}^{n}(K)=\underline{\operatorname{{Hom}}}^{n}(K,K)$.
We can view $a_{\ast}\underline{\operatorname{{Hom}}}(K,L)$ as a sheaf over
$\mathit{pt}_{0}$, so its pullback along $\xi$ can be viewed as a sheaf over
$\mathit{pt}$ endowed with an $F$-equivariant structure. In particular,
$\underline{\operatorname{{Hom}}}^{n}(K,L)$ is the vector space
$\operatorname{{Hom}}^{n}(\xi K,\xi L)$ endowed with an $F$-action. The
$\bar{\mathbf{Q}}_{\ell}[F]$-module formed by
$\underline{\operatorname{{Hom}}}^{\ast}(K,L)$ is related to the vector space
$\operatorname{{Hom}}^{\ast}(K,L)$ by short exact sequences

(43) $\displaystyle
0\to\underline{\operatorname{{Hom}}}^{n-1}(K,L)_{F}\to\operatorname{{Hom}}^{n}(K,L)\to\underline{\operatorname{{Hom}}}^{n}(K,L)^{F}\to
0$

for all $n$ [BBD, (5.1.2.5)]. The morphism
$\operatorname{{Hom}}^{n}(K,L)\to\operatorname{{Hom}}^{n}(\xi K,\xi L)$
factors through the morphism
$\operatorname{{Hom}}^{n}(K,L)\to\underline{\operatorname{{Hom}}}^{n}(K,L)^{F}$
above.

Throughout this paper, all $\bar{\mathbf{Q}}_{\ell}[F]$-modules will arise
from mixed complexes, and all of their eigenvalues under the action of $F$
will be pure of integer weight in the sense of [D80, §1.2.1]. We write
$\bm{w}_{\leq\ast}$ to denote the weight filtration on such a module $M$.
Explicitly, $\bm{w}_{\leq n}M\subseteq M$ is the sum of the generalized
eigenspaces of $F$ where the eigenvalues are pure of weight $\leq n$.

### 1.3.

We will introduce a mild generalization of the realization functors used by
Rider in [R]. Our formalism also resembles the ones used by Bezrukavnikov–Yun
in [BY, Appendix B] and Lusztig–Yun in [LY20, §9], though we do not need to
develop ours as extensively. All of these formalisms are ultimately based on
the work of Beilinson in [B].

Let $\mathsf{C}(X_{0})\subseteq\mathsf{D}_{G,m}^{b}(X_{0})$ be a full,
additive subcategory. We say that $\mathsf{C}(X_{0})$ is realizable iff, for
all $K,L\in\mathsf{C}(X_{0})$, we have

(44) $\displaystyle n\neq
0\implies\operatorname{{gr}}_{0}^{\bm{w}}\underline{\operatorname{{Hom}}}^{n}(K,L)=0,$

where $\operatorname{{gr}}_{0}^{\bm{w}}$ denotes the $0$th graded piece of the
weight filtration. (Compare to Assumption $\mathrm{C}_{1}$ in [BY, §B.1].) In
practice, the objects of $\mathsf{C}(X_{0})$ will be direct sums of pure
shifted perverse sheaves.

######  Lemma 1.3.1.

If $\mathsf{C}(X_{0})$ is realizable, then

(45) $\displaystyle n\neq 0,1\implies\operatorname{{Hom}}^{n}(K,L)=0$

for all $K,L\in\mathsf{C}(X_{0})$.

###### Chứng minh.

If $n\neq 0,1$, then
$\underline{\operatorname{{Hom}}}^{n-1}(K,L)_{F},\,\underline{\operatorname{{Hom}}}^{n}(K,L)^{F}=0$
by the realizability of $\mathsf{C}(X_{0})$, so
$\operatorname{{Hom}}^{n}(K,L)=0$ by the exactness of (43). ∎

We write $\underline{\mathsf{C}}(X_{0})$ for the category in which the objects
are the same as in $\mathsf{C}(X_{0})$, but the morphisms are given by

(46)
$\displaystyle\operatorname{{Hom}}_{\underline{\mathsf{C}}(X_{0})}(K,L)=\operatorname{{gr}}_{0}^{\bm{w}}\underline{\operatorname{{Hom}}}^{0}(K,L).$

By construction, it is a full, additive subcategory of
$\underline{\mathsf{D}}_{G,m}^{b}(X_{0})$. There is a functor

(47) $\displaystyle\kappa:\mathsf{C}(X_{0})\to\underline{\mathsf{C}}(X_{0})$

induced by the composition of morphisms

(48)
$\displaystyle\operatorname{{Hom}}(K,L)\to\underline{\operatorname{{Hom}}}(K,L)^{F}\to\bm{w}_{\leq
0}\,\underline{\operatorname{{Hom}}}(L,K)\to\operatorname{{Hom}}_{\underline{\mathsf{C}}(X_{0})}(K,L).$

If $\mathsf{C}(X_{0})$ is realizable, then (48) is injective by the exactness
of (43), so in this case, $\kappa$ is faithful.

### 1.4.

Given any additive category $\mathsf{A}$, we write
$\mathsf{C}^{b}(\mathsf{A})$ for the category of bounded complexes of objects
of $\mathsf{A}$ and $\mathsf{K}^{b}(\mathsf{A})$ for the homotopy category of
$\mathsf{C}^{b}(\mathsf{A})$. We write $[1]_{\triangle}$ for the endofunctors
of these categories that shift each complex down by one degree.

_Throughout the rest of this subsection, we assume that $\mathsf{C}(X_{0})$ is
realizable_. We will construct realization functors

(49) $\displaystyle\rho:\mathsf{K}^{b}(\mathsf{C}(X_{0}))$
$\displaystyle\to\mathsf{D}_{G,m}^{b}(X_{0}),$ (50)
$\displaystyle\emph{resp.}\qquad\rho:\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))$
$\displaystyle\to\underline{\mathsf{D}}_{G,m}^{b}(X_{0}),$

that are triangulated in the sense that $\rho([1]_{\triangle})=[1]$ and
restrict to the identity functor on $\mathsf{C}(X_{0})$, _resp._
$\underline{\mathsf{C}}(X_{0})$, viewed as the full subcategory of
$\mathsf{K}^{b}(\mathsf{C}(X_{0}))$, _resp._
$\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))$, of complexes concentrated in
degree zero. The explicit construction of these functors will show that the
diagram

(51)
${\mathsf{K}^{b}(\mathsf{C}(X_{0}))}$${\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))}$${\mathsf{D}_{G,m}^{b}(X_{0})}$${\underline{\mathsf{D}}_{G,m}^{b}(X_{0})}$$\mathsf{K}^{b}\kappa$$\mathsf{K}^{b}\kappa$$\rho$$\rho$

commutes.

Let $\mathsf{D}$ and $\mathsf{C}$ be defined by one of the following
situations:

  1. (1)

$\mathsf{D}=\mathsf{D}_{G,m}^{b}(X_{0})$ and $\mathsf{C}=\mathsf{C}(X_{0})$.

  2. (2)

$\mathsf{D}=\underline{\mathsf{D}}_{G,m}^{b}(X_{0})$ and
$\mathsf{C}=\underline{\mathsf{C}}(X_{0})$.

Let $\tilde{\mathsf{D}}$ be a filtered version of $\mathsf{D}$ in the sense of
Beilinson [B], and let $\omega:\tilde{\mathsf{D}}\to\mathsf{D}$ be the
forgetful functor. We will construct a full and faithful embedding
$\tilde{\rho}:\mathsf{C}^{b}(\mathsf{C})\to\tilde{\mathsf{D}}$, fitting into a
commutative diagram:

(52)
${\mathsf{C}^{b}(\mathsf{C})}$${\mathsf{K}^{b}(\mathsf{C})}$${\tilde{\mathsf{D}}}$${\mathsf{D}}$$\tilde{\rho}$$\omega$$\rho$

More precisely, we will show that $\omega\circ\tilde{\rho}$ factors through
the quotient functor
$\mathsf{C}^{b}(\mathsf{C})\to\mathsf{K}^{b}(\mathsf{C})$, allowing us to
define $\rho$.

The filtered structure of $\tilde{\mathsf{D}}$ is given by full triangulated
subcategories $\tilde{\mathsf{D}}^{\leq n},\tilde{\mathsf{D}}^{\geq n}$ for
each integer $n$, subject to certain axioms [R, Def. 3.1]. Let

(53) $\displaystyle\bm{w}_{\leq n}:\tilde{\mathsf{D}}$
$\displaystyle\to\tilde{\mathsf{D}}^{\leq n},$ (54)
$\displaystyle\emph{resp.}\qquad\bm{w}_{\geq n}:\tilde{\mathsf{D}}$
$\displaystyle\to\tilde{\mathsf{D}}^{\geq n},$

be right adjoint to the inclusion $\tilde{\mathsf{D}}^{\leq
n}\to\tilde{\mathsf{D}}$, _resp._ left adjoint to the inclusion
$\tilde{\mathsf{D}}^{\geq n}\to\tilde{\mathsf{D}}$. We set

(55) $\displaystyle\operatorname{{gr}}_{n}=\bm{w}_{\leq n}\,\bm{w}_{\geq
n}=\bm{w}_{\geq n}\,\bm{w}_{\leq
n}:\tilde{\mathsf{D}}\to\tilde{\mathsf{D}}^{\leq
n}\cap\tilde{\mathsf{D}}^{\geq n}\subseteq\tilde{\mathsf{D}}.$

Let $s:\tilde{\mathsf{D}}\to\tilde{\mathsf{D}}$ be the shift-of-filtration
functor, so that $s(\tilde{\mathsf{D}}^{\leq n})=\tilde{\mathsf{D}}^{\leq
n+1}$ and $s(\tilde{\mathsf{D}}^{\geq n})=\tilde{\mathsf{D}}^{\geq n+1}$.

Let $\tilde{\mathsf{A}}\subseteq\tilde{\mathsf{D}}$ be the full, additive
subcategory of objects $K$ such that $\operatorname{{gr}}_{n}(K)\in
s^{n}\mathsf{C}[n]$ for all $n$. We construct an additive functor
$\alpha:\tilde{\mathsf{A}}\to\mathsf{C}^{b}(\mathsf{C})$ as follows. Let

(56) $\displaystyle\alpha(K)^{n}=\omega\operatorname{{gr}}_{-n}(K)[n],$

and let $\delta^{n}:\alpha(K)^{n}\to\alpha(K)^{n+1}$ be the morphism we obtain
by applying $\omega(-)[n]$ to the boundary morphism in the exact triangle

(57)
$\displaystyle\begin{split}\operatorname{{gr}}_{-(n+1)}(K)\to\bm{w}_{\leq-n}\,\bm{w}_{\geq-(n+1)}(K)&\to\operatorname{{gr}}_{-n}(K)\xrightarrow{\delta^{n}}\operatorname{{gr}}_{-(n+1)}(K)[1].\end{split}$

The map $K\mapsto\alpha(K)$ is functorial and additive, so it remains to show
that:

######  Lemma 1.4.1.

For all $K\in\tilde{\mathsf{A}}$, the sequence $\alpha(K)$ is a well-defined
object of $\mathsf{C}^{b}(\mathsf{C})$, _i.e._ ,
$\delta^{n+1}\circ\delta^{n}=0$ for all $n$.

###### Chứng minh.

The proof is analogous to that of [R, Lem. 3.6], except that we use (44) or
(45) in place of [R, Lem. 2.4] to ensure that
$\operatorname{{Hom}}(\operatorname{{gr}}_{-n-2}(K),\operatorname{{gr}}_{-n}(K))$
vanishes for all $n$. ∎

We want to show that $\alpha$ is an equivalence. That $\alpha$ is faithful
amounts to the following statement:

######  Lemma 1.4.2.

If $\phi:K\to L$ is a morphism in $\tilde{\mathsf{A}}$, then $\phi=0$ if and
only if $\operatorname{{gr}}_{n}(\phi)=0$ for all $n$.

###### Chứng minh.

The proof is analogous to that of [R, Lem. 3.5], except that we use (44) or
(45) in place of [R, Lem. 2.4] to ensure that
$\operatorname{{Hom}}(\operatorname{{gr}}_{n}(K),\bm{w}_{\leq n-1}(L))$
vanishes for all $K,L\in\tilde{\mathsf{A}}$ and $n$. ∎

######  Lemma 1.4.3.

The functor $\alpha:\tilde{\mathsf{A}}\to\mathsf{C}^{b}(\mathsf{C})$ is an
equivalence. Moreover, $\tilde{\rho}\vcentcolon=\alpha^{-1}$ factors through
the quotient functor
$\mathsf{C}^{b}(\mathsf{C})\to\mathsf{K}^{b}(\mathsf{C})$.

###### Chứng minh.

The proof is analogous to that of [R, Prop. 3.7], with Lemma 1.4.2 in place of
[R, Lem. 3.5]. ∎

######  Remark 1.4.4.

Realization functors are compatible with other functors in the following
sense. We keep the notation above, but assume that we have two copies of each
category and functor, _e.g._ , $\mathsf{D}_{1},\mathsf{D}_{2}$ in place of
$\mathsf{D}$. Let $\Phi:\mathsf{D}_{1}\to\mathsf{D}_{2}$ be an exact functor
that admits a canonical lift
$\tilde{\Phi}:\tilde{\mathsf{D}}_{1}\to\tilde{\mathsf{D}}_{2}$ compatible with
the filtered structures on these categories in the sense of [B]. In
particular, this condition means $\tilde{\Phi}$ is compatible with the
functors $\operatorname{{gr}}_{n}$. If $\Phi$ restricts to a functor
$\mathsf{C}_{1}\to\mathsf{C}_{2}$, then there is a commutative diagram:

(58)
${\tilde{\mathsf{A}}_{1}}$${\tilde{\mathsf{A}}_{2}}$${\mathsf{C}^{b}(\mathsf{C}_{1})}$${\mathsf{C}^{b}(\mathsf{C}_{2})}$$\tilde{\Phi}$$\Phi$$\alpha_{1}$$\alpha_{2}$

It in turn induces a commutative diagram:

(59)
${\mathsf{K}^{b}(\mathsf{C}_{1})}$${\mathsf{K}^{b}(\mathsf{C}_{2})}$${\mathsf{D}_{1}}$${\mathsf{D}_{2}}$$\mathsf{K}^{b}\Phi$$\Phi$$\rho$$\rho$

So in this situation, $\Phi$ is compatible with realization.

In practice, it will be easier to check the conditions on $\Phi$ when the
categories $\mathsf{D}_{i}$ take the form $\mathsf{D}_{G,m}^{b}(X_{0})$ than
when they take the form $\underline{\mathsf{D}}_{G,m}^{b}(X_{0})$.

### 1.5.

We say that a full, additive subcategory
$\mathsf{C}(X_{0})\subseteq\mathsf{D}_{G,m}^{b}(X_{0})$ is
$\underline{\operatorname{{Hom}}}$-pure iff:

  1. (1)

$\mathsf{C}(X_{0})$ is stable under $\langle n\rangle$ for all
$n\in\mathbf{Z}$.

  2. (2)

$\underline{\operatorname{{Hom}}}^{0}(K,L)$ is concentrated in weight zero for
all $K,L\in\mathsf{C}(X_{0})$.

These hypotheses imply that

(60)
$\displaystyle\underline{\operatorname{{Hom}}}^{n}(K,L)\simeq\underline{\operatorname{{Hom}}}^{0}(K,L\langle
n\rangle)(-\tfrac{n}{2})$

is concentrated in weight $n$ for all $K,L\in\mathsf{C}(X_{0})$ and
$n\in\mathbf{Z}$, so $\underline{\operatorname{{Hom}}}$-purity implies
realizability. The following result is inspired by [R, Thm. 4.3]:

######  Proposition 1.5.1.

If $\mathsf{C}(X_{0})$ is $\underline{\operatorname{{Hom}}}$-pure, then
$\rho:\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))\to\underline{\mathsf{D}}_{G,m}^{b}(X_{0})$
induces an isomorphism of vector spaces

(61)
$\displaystyle\operatorname{{Hom}}_{\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))}(\mathcal{K},\mathcal{L}\langle
n\rangle[-n]_{\triangle})\xrightarrow{\sim}\operatorname{{gr}}_{n}^{\bm{w}}\underline{\operatorname{{Hom}}}^{0}(\rho(\mathcal{K}),\rho(\mathcal{L}))$

for all
$\mathcal{K},\mathcal{L}\in\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))$ and
$n\in\mathbf{Z}$.

###### Chứng minh.

First, suppose that $\mathcal{K}=K$ and $\mathcal{L}=L[i]_{\triangle}$, where
$K,L\in\underline{\mathsf{C}}(X_{0})$, viewed as objects of
$\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))$ represented by complexes
concentrated in degree zero. In this case, the right-hand side of (61) becomes
$\operatorname{{gr}}_{n}^{\bm{w}}\underline{\operatorname{{Hom}}}^{i}(K,L)$.
If $i\neq n$, then

(62)
$\displaystyle\operatorname{{Hom}}_{\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))}(\mathcal{K},\mathcal{L}\langle
n\rangle[-n]_{\triangle})\simeq
0\simeq\operatorname{{gr}}_{n}^{\bm{w}}\underline{\operatorname{{Hom}}}^{i}(K,L),$

where the second isomorphism uses the
$\underline{\operatorname{{Hom}}}$-purity of $\mathsf{C}(X_{0})$. If $i=n$,
then

(63)
$\displaystyle\begin{split}\operatorname{{Hom}}_{\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))}(\mathcal{K},\mathcal{L}\langle
n\rangle[-n]_{\triangle})&\simeq\operatorname{{Hom}}_{\underline{\mathsf{C}}(X_{0})}(K,L\langle
n\rangle)\\\
&\simeq\operatorname{{gr}}_{0}^{\bm{w}}\underline{\operatorname{{Hom}}}^{0}(K,L\langle
n\rangle)\\\
&\simeq\operatorname{{gr}}_{n}^{\bm{w}}\underline{\operatorname{{Hom}}}^{n}(K,L).\end{split}$

Now, we allow $\mathcal{K}$ and $\mathcal{L}$ to be arbitrary, and proceed by
double induction on their amplitudes. As in [R], we only illustrate the
induction on the amplitude of $\mathcal{K}$.

By taking (brutal) truncations of $\mathcal{K}$, we can build an exact
triangle
$\mathcal{K}^{\prime}\to\mathcal{K}\to\mathcal{K}^{\prime\prime}\to\mathcal{K}^{\prime}[1]$
in which
$\mathcal{K}^{\prime},\mathcal{K}^{\prime\prime}\in\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))$
are of strictly smaller amplitude than $\mathcal{K}$. We get the following
commutative diagram in which the rows are exact:

(64) ${\operatorname{{Hom}}(\mathcal{K}^{\prime},\mathcal{L}\langle
n\rangle[-n-1]_{\triangle})}$${\operatorname{{Hom}}(\mathcal{K}^{\prime\prime},\mathcal{L}\langle
n\rangle[-n]_{\triangle})}$${\operatorname{{Hom}}(\mathcal{K},\mathcal{L}\langle
n\rangle[-n]_{\triangle})}$${\operatorname{{gr}}_{n}^{\bm{w}}\underline{\operatorname{{Hom}}}^{-1}(\rho(\mathcal{K}^{\prime}),\rho(\mathcal{L}))}$${\operatorname{{gr}}_{n}^{\bm{w}}\underline{\operatorname{{Hom}}}^{0}(\rho(\mathcal{K}^{\prime\prime}),\rho(\mathcal{L}))}$${\operatorname{{gr}}_{n}^{\bm{w}}\underline{\operatorname{{Hom}}}^{0}(\rho(\mathcal{K}),\rho(\mathcal{L}))}$${{\cdots}}$${\operatorname{{Hom}}(\mathcal{K}^{\prime},\mathcal{L}\langle
n\rangle[-n]_{\triangle})}$${\operatorname{{Hom}}(\mathcal{K}^{\prime\prime},\mathcal{L}\langle
n\rangle[n+1]_{\triangle})}$${{\cdots}}$${\operatorname{{gr}}_{n}^{\bm{w}}\underline{\operatorname{{Hom}}}^{0}(\rho(\mathcal{K}^{\prime}),\rho(\mathcal{L}))}$${\operatorname{{gr}}_{n}^{\bm{w}}\underline{\operatorname{{Hom}}}^{1}(\rho(\mathcal{K}^{\prime\prime}),\rho(\mathcal{L}))}$

By the inductive hypothesis, the first, second, fourth, and fifth vertical
arrows are isomorphisms, so by the five lemma, the third vertical arrow is an
isomorphism as well, completing the induction. ∎

Generalizing [R, Def. 2.3], we say that $\mathsf{C}(X_{0})$ is Frobenius
invariant iff it is $\underline{\operatorname{{Hom}}}$-pure and, strengthening
hypothesis (2), $\underline{\operatorname{{Hom}}}^{0}(K,L)$ is $F$-invariant
for all $K,L$. In this case,
$\underline{\mathsf{C}}(X_{0})\simeq\mathsf{C}(X_{0})$. In particular, by the
commutativity of (51), Proposition 1.5.1 is actually a generalization of [R,
Thm. 4.3].

### 1.6.

It is not obvious when functors between categories of the form
$\mathsf{C}(X_{0})$ can be extended to functors between categories of the form
$\underline{\mathsf{C}}(X_{0})$, because
$\operatorname{{Hom}}_{\underline{\mathsf{C}}(X_{0})}$ is defined in terms of
the global sections of a sheaf. However, we will only need this property in
the case of pullback functors, where we can make use of the well-known
adjunctions. The arguments below are essentially the same as the arguments
showing that the morphisms on homology/cohomology induced by pullbacks and
pushforwards between $G_{0}$-varieties are weight-preserving, _cf._ Section 4.

Namely, if $f:Y_{0}\to X_{0}$ is an (equivariant, stratification-preserving)
map of $G_{0}$-varieties over $\mathbf{F}$, then we have a composition of
natural transformations of sheaf-valued bifunctors on
$\mathsf{D}_{G,m}^{b}(X_{0})$:

(65)
$\displaystyle\underline{\operatorname{{Hom}}}(-,-)\to\underline{\operatorname{{Hom}}}(-,f_{\ast}f^{\ast}(-))\xrightarrow{\sim}f_{\ast}\underline{\operatorname{{Hom}}}(f^{\ast}(-),f^{\ast}(-)).$

Let $a:X_{0}\to\mathit{pt}_{0}$ and $b:Y_{0}\to\mathit{pt}_{0}$ be the
structure maps, so that $b=a\circ f$. Then we have
$b_{\ast}=a_{\ast}f_{\ast}$, so applying $a_{\ast}$ gives:

(66) $\displaystyle a_{\ast}\underline{\operatorname{{Hom}}}(-,-)\to
b_{\ast}\underline{\operatorname{{Hom}}}(f^{\ast}(-),f^{\ast}(-)).$

Applying $\operatorname{{gr}}_{0}^{\bm{w}}\,\mathrm{H}^{0}\circ\xi$ gives:

(67) $\displaystyle
f^{\ast}:\operatorname{{Hom}}_{\underline{\mathsf{D}}_{G,m}^{b}(X_{0})}(-,-)\to\operatorname{{Hom}}_{\underline{\mathsf{D}}_{G,m}^{b}(Y_{0})}(f^{\ast}(-),f^{\ast}(-)).$

We can check that these natural transformations together define a functor

(68) $\displaystyle
f^{\ast}:\underline{\mathsf{D}}_{G,m}^{b}(X_{0})\to\underline{\mathsf{D}}_{G,m}^{b}(Y_{0})$

that sends $K\mapsto f^{\ast}K$ at the level of objects.

######  Lemma 1.6.1.

For $f:Y_{0}\to X_{0}$ as above, we have a commutative diagram:

(69)
${\mathsf{D}_{G,m}^{b}(X_{0})}$${\mathsf{D}_{G,m}^{b}(Y_{0})}$${\underline{\mathsf{D}}_{G,m}^{b}(X_{0})}$${\underline{\mathsf{D}}_{G,m}^{b}(Y_{0})}$$f^{\ast}$$\kappa$$\kappa$$f^{\ast}$

###### Chứng minh.

After we apply $\mathrm{H}^{0}$ to (66), the resulting natural transformation
is $F$-equivariant and commutes with shifts. So the result follows from the
exactness of (43). ∎

##  2\. Mixed Categories in Representation Theory

### 2.1.

In this section, we introduce the various choices of $X_{0}$ and
$\mathsf{C}(X_{0})$ that we need in the rest of the paper. They, and the
functors between them, will be summarized by the following commutative
diagram:

(70)
${\mathsf{K}^{b}(\mathsf{C}_{\hat{W}}(G_{0}))}$${\mathsf{K}^{b}(\mathsf{C}_{\hat{W}}(\mathcal{U}_{0}))}$${\mathsf{K}^{b}(\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0}))}$${\mathsf{K}^{b}(\mathsf{C}(G_{0}))}$${\mathsf{K}^{b}(\mathsf{C}(\mathcal{U}_{0}))}$${\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}\times\mathcal{B}_{0})}$${\mathsf{D}_{G,m}^{b}(G_{0})}$${\mathsf{D}_{G,m}^{b}(\mathcal{U}_{0})}$$\mathsf{K}^{b}i^{\ast}\langle-r\rangle$$\mathsf{K}^{b}i^{\ast}\langle-r\rangle$$i^{\ast}\langle-r\rangle$$\oplus$$\oplus$$\mathsf{K}^{b}\mathsf{CH}$$\mathsf{CH}$$\rho$$\rho$$\mathsf{K}^{b}\mathsf{PR}$$\mathsf{K}^{b}\mathsf{PR}$

Note that we will not discuss the “missing” functor
$\mathsf{K}^{b}(\mathsf{C}(G_{0}))\to\mathsf{D}_{G,m}^{b}(G_{0})$.

### 2.2.

Let $W$ be the Weyl group of $G$. Henceforth, we fix a Coxeter presentation of
$W$ (see §A.1), so that $W$ is endowed with a Bruhat order. As in §A.5, we
write $\hat{W}$ for the set of irreducible characters of $W$. In particular,
$1\in\hat{W}$ is the trivial character.

It will also be convenient to set

(71) $\displaystyle r$
$\displaystyle=\operatorname{{rk}}_{\mathbf{Z}}\mathbf{X},$ (72)
$\displaystyle N$ $\displaystyle=\tfrac{1}{2}|\Phi|.$

We have $\dim G=r+2N$.

### 2.3.

Let $\mathcal{B}_{0}$ be the flag variety of $G_{0}$, _i.e._ , the variety
that parametrizes its Borel subgroups. Let $G_{0}$ act on $\mathcal{B}_{0}$ by
right conjugation.

We stratify $\mathcal{B}_{0}\times\mathcal{B}_{0}$ by its diagonal
$G_{0}$-orbits. The poset formed by the orbits under the closure relation is
isomorphic to the poset formed by $W$ under the Bruhat order. For each $w\in
W$, let

(73) $\displaystyle j_{w}:O_{w,0}\to\mathcal{B}_{0}\times\mathcal{B}_{0}$

denote the inclusion of the corresponding orbit. We have $\dim O_{w}=|w|+N$,
where $|w|$ is the Bruhat length of $w$. Thus the (shift-twisted) intersection
complex

(74)
$\displaystyle\mathit{IC}_{w}=j_{w,!\ast}\bar{\mathbf{Q}}_{\ell}\langle|w|-r-N\rangle\in\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}\times\mathcal{B}_{0})$

is a simple perverse sheaf of weight zero on
$[(\mathcal{B}_{0}\times\mathcal{B}_{0})/G_{0}]$.

######  Definition 2.3.1.

Let
$\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})\subseteq\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}\times\mathcal{B}_{0})$
be the full, replete, additive subcategory generated by the objects
$\mathit{IC}_{w}\langle n\rangle$ with $w\in W$ and $n\in\mathbf{Z}$.

######  Lemma 2.3.2 (Achar–Riche).

$\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})$ is Frobenius-invariant,
hence $\underline{\operatorname{{Hom}}}$-pure.

###### Chứng minh.

[AR, Lem. 7.8] proves a Frobenius-invariance statement for any variety
equipped with an affine even stratification. Their argument generalizes to the
stacks in [Su] as long as the same condition holds on the stratification.

Fix a Borel $B_{0}\subseteq G_{0}$. The stratification of the coset space
$B_{0}\backslash G_{0}$ into right $B_{0}$-orbits, _i.e._ , into double
cosets, is affine even. We have an isomorphism

(75) $\displaystyle[B_{0}\backslash
G_{0}/B_{0}]\simeq[(\mathcal{B}_{0}\times\mathcal{B}_{0})/G_{0}],$

so the stacky version of [AR, Lem. 7.8] implies the desired result. ∎

### 2.4.

The convolution on
$\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}\times\mathcal{B}_{0})$ is the monoidal
product

(76)
$\displaystyle(-)\star(-):\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}\times\mathcal{B}_{0})\times\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}\times\mathcal{B}_{0})\to\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}\times\mathcal{B}_{0})$

defined by

(77) $\displaystyle K\star
L=\mathit{pr}_{1,3,\ast}(\mathit{pr}_{1,2}\times\mathit{pr}_{2,3})^{\ast}(K\boxtimes
L)\langle r+N\rangle,$

where
$\mathit{pr}_{1,2},\mathit{pr}_{1,3},\mathit{pr}_{2,3}:\mathcal{B}_{0}^{3}\to\mathcal{B}_{0}^{2}$
are the three projection maps.

We claim that $\star$ is weight-preserving by the results of [D80]. Indeed,
$\mathit{pr}_{1,3}$ is proper, and it can be checked that the map of stacks
$[\mathcal{B}_{0}^{3}/G_{0}]\to[\mathcal{B}_{0}^{2}/G_{0}]\times[\mathcal{B}_{0}^{2}/G_{0}]$
induced by $\mathit{pr}_{1,2}\times\mathit{pr}_{2,3}$ is smooth of relative
dimension $\dim G-\dim\mathcal{B}=r+N$. More strongly, we have [BY, Prop.
3.2.5]:

######  Lemma 2.4.1 (Bezrukavnikov–Yun).

The convolution $\star$ restricts to a monoidal product on
$\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})$.

######  Remark 2.4.2.

Let $\mathsf{C}(\mathcal{B}\times\mathcal{B})$ be the the essential image of
$\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})$ in
$\mathsf{D}_{G}^{b}(\mathcal{B}\times\mathcal{B})$. Prior to [BY], it was
known that the convolution on
$\mathsf{D}_{G}^{b}(\mathcal{B}\times\mathcal{B})$ preserves the class of
objects in $\mathsf{C}(\mathcal{B}\times\mathcal{B})$, _cf._ [S82, Prop.
2.13]. As explained in [BY, Rem. 3.2.6], the force of the lemma is to assert
that for all $K,L\in\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})$, the
object $K\star L$ is semisimple _before_ its pullback to
$\mathcal{B}\times\mathcal{B}$. The proof involves developing a Frobenius-
enriched version of some of the results of [S82].

The convolution on $\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})$ induces
one on $\mathsf{K}^{b}(\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0}))$. In
the case where _$G$ is semisimple_, we define the Hecke category of $W$ to be
the resulting monoidal triangulated category, and abbreviate

(78)
$\displaystyle\begin{split}\mathsf{H}_{W}&=\mathsf{K}^{b}(\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})).\end{split}$

By Remark 1.4.4, the convolution on $\mathsf{H}_{W}$ is compatible with
realization functors in the sense that the diagram

(79)
${\mathsf{H}_{W}\times\mathsf{H}_{W}}$${\mathsf{H}_{W}}$${\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}\times\mathcal{B}_{0})\times\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}\times\mathcal{B}_{0})}$${\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}\times\mathcal{B}_{0})}$$\star$$\star$$\rho\times\rho$$\rho$

commutes.

######  Remark 2.4.3.

This definition of the Hecke category originates in work of Beilinson,
Ginzburg, Soergel, and others on Koszul duality patterns in the geometry of
mixed complexes. See [BGS, §4] and the references therein.

### 2.5.

Let $H_{W}$ be the Iwahori–Hecke algebra of $W$ (see §A.3). Below, we review
how $\mathsf{H}_{W}$ decategorifies to $H_{W}$.

If $\mathsf{A}$ is an additive category, then we write $[\mathsf{A}]_{\oplus}$
for its _split_ Grothendieck group. The elements of $[\mathsf{A}]_{\oplus}$
are the isomorphism classes $[K]$ of the objects $K$, and the relations are
$[K\oplus L]=[K]+[L]$ for all objects $K,L$. If $\mathsf{A}$ is given a
monoidal product $\star$, then $[\mathsf{A}]_{\oplus}$ forms a ring under the
multiplication $[K]\cdot[L]=[K\star L]$.

We endow $[\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})]_{\oplus}$ with
the action of $\mathbf{Z}[\bm{q}^{\pm\frac{1}{2}}]$ in which
$\bm{q}^{\frac{1}{2}}$ acts by $\langle-1\rangle$. For each
$K\in\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})$, let

(80)
$\displaystyle\chi_{w}(K)=\sum_{i,j}{(-1)^{i+j}}\bm{q}^{\frac{j}{2}}\dim\operatorname{{gr}}_{j}^{\bm{w}}\mathrm{H}^{i}(j_{w}^{\ast}K),$

the virtual weight polynomial of the stalk of $K$ over $[O_{w,0}/G]$. Thm. 2.8
of [S82] implies if $G$ is _semisimple_ , then the map

(83)
$\displaystyle\begin{array}[]{rcl}{[\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})]_{\oplus}}&\to&H_{W}\\\
{[K]}&\mapsto&\displaystyle\sum_{w\in
W}\bm{q}^{\frac{|w|}{2}}\sigma_{w}\chi_{w}(K)\end{array}$

is a morphism of $\mathbf{Z}[\bm{q}^{\pm\frac{1}{2}}]$-algebras. In fact, it
is an isomorphism, and if $\gamma_{w}$ is the image of $[\mathit{IC}_{w}]$,
then the set $\\{\gamma_{w}\\}_{w\in W}$ forms a free
$\mathbf{Z}[\bm{q}^{\pm\frac{1}{2}}]$-linear basis of $H_{W}$, _cf._ [WW11,
407]. The element $\gamma_{w}$ is called the Kazhdan–Lusztig element attached
to $w$.

If $\mathsf{T}$ is a triangulated (additive) category, then we write
$[\mathsf{T}]_{\triangle}$ for its _triangulated_ Grothendieck group. The
elements of $[\mathsf{T}]_{\triangle}$ are the isomorphism classes of the
objects of $\mathsf{T}$, and the relations are
$[\mathcal{K}]=[\mathcal{K}^{\prime}]+[\mathcal{K}^{\prime\prime}]$ for each
exact triangle
$\mathcal{K}^{\prime}\to\mathcal{K}\to\mathcal{K}^{\prime\prime}\to\mathcal{K}^{\prime}[1]$
in $\mathsf{T}$.

If $\mathsf{A}$ is any additive category, then there is an isomorphism
$[\mathsf{K}^{b}(\mathsf{A})]_{\triangle}\xrightarrow{\sim}[\mathsf{A}]_{\oplus}$
that sends any complex to the alternating sum of its terms [Ro]. In
particular,

(84)
$\displaystyle\begin{split}[\mathsf{H}_{W}]_{\triangle}&\simeq[\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})]_{\oplus}.\end{split}$

Since the convolutions on these categories are compatible with each other,
this is an isomorphism of rings, not just groups. We endow
$[\mathsf{H}_{W}]_{\triangle}$ with the action of
$\mathbf{Z}[\bm{q}^{\pm\frac{1}{2}}]$ in which $\bm{q}^{\frac{1}{2}}$ acts by
$\langle-1\rangle$. Then we obtain a
$\mathbf{Z}[\bm{q}^{\pm\frac{1}{2}}]$-algebra isomorphism
$[\mathsf{H}_{W}]_{\triangle}\xrightarrow{\sim}H_{W}$ that again sends
$[\mathit{IC}_{w}]\mapsto\gamma_{w}$.

######  Remark 2.5.1.

Recall that $q^{\frac{1}{2}}\in\bar{\mathbf{Q}}_{\ell}^{\times}$ is a fixed
square root of $q=|\mathbf{F}|$. In [I], Iwahori gave an isomorphism of
$\bar{\mathbf{Q}}_{\ell}$-algebras:

(85) $\displaystyle\bar{\mathbf{Q}}_{\ell}\otimes
H_{W}|_{\bm{q}^{1/2}=q^{1/2}}\simeq\operatorname{{End}}_{G^{F}}(\bar{\mathbf{Q}}_{\ell}[\mathcal{B}^{F}]).$

The isomorphism
$H_{W}\simeq[\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})]_{\oplus}$ is an
integral version in the following sense. We have a commutative diagram

(86)
${{[\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0})]_{\oplus}}}$${H_{W}}$${\mathbf{Z}[q^{\pm\frac{1}{2}}][\mathcal{B}^{F}\times\mathcal{B}^{F}]^{G^{F}}}$${H_{W}|_{\bm{q}^{1/2}=q^{1/2}}}$$\sim$$\sim$

where the left-hand arrow is induced by the function-sheaf dictionary.
Iwahori’s isomorphism can be factored as

(87)
$\displaystyle\operatorname{{End}}_{G^{F}}(\bar{\mathbf{Q}}_{\ell}[\mathcal{B}^{F}])\xrightarrow{\sim}\bar{\mathbf{Q}}_{\ell}[\mathcal{B}^{F}\times\mathcal{B}^{F}]^{G^{F}}\xrightarrow{\sim}\bar{\mathbf{Q}}_{\ell}\otimes
H_{W}|_{\bm{q}^{1/2}=q^{1/2}},$

where the second arrow is induced by the bottom arrow of the diagram.

### 2.6.

Let $G_{0}$ act on itself by right conjugation. There is a $G_{0}$-stable
stratification of $G_{0}$ by locally-closed subvarieties [L84-2, §3.1] [L85-I,
§3.11], now known as Jordan classes. Let

(88) $\displaystyle\mathit{act}:G_{0}\times\mathcal{B}_{0}$
$\displaystyle\to\mathcal{B}_{0}\times\mathcal{B}_{0},$ (89)
$\displaystyle\mathit{pr}:G_{0}\times\mathcal{B}_{0}$ $\displaystyle\to G_{0}$

be defined by

(90) $\displaystyle\mathit{act}(g,B)$ $\displaystyle=(g^{-1}Bg,B),$ (91)
$\displaystyle\mathit{pr}(g,B)$ $\displaystyle=g.$

The maps above are $G_{0}$-equivariant with respect to the diagonal
$G_{0}$-actions on $G_{0}\times\mathcal{B}_{0}$ and
$\mathcal{B}_{0}\times\mathcal{B}_{0}$. Moreover, we can stratify
$G_{0}\times\mathcal{B}_{0}$ so that $\mathit{act}$, _resp._ $\mathit{pr}$, is
stratification-preserving with respect to the Bruhat stratification of
$\mathcal{B}_{0}\times\mathcal{B}_{0}$, _resp._ the Jordan stratification of
$G_{0}$.

Following [L85-I], we define the character functor

(92)
$\displaystyle\mathsf{CH}:\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}\times\mathcal{B}_{0})\to\mathsf{D}_{G,m}^{b}(G_{0})$

to be the composition

(93)
$\displaystyle\begin{split}\mathsf{CH}=\bigoplus_{i}{}^{p}\mathcal{H}^{i}[-i]\circ\mathit{pr}_{\ast}\,\mathit{act}^{!}\langle
r+N\rangle.\end{split}$

Since $\mathit{act}$ is smooth of relative dimension $r+N$ and $\mathit{pr}$
is proper, $\mathsf{CH}$ preserves the class of pure complexes of weight zero
[D80]. We define a mixed unipotent character sheaf to be a simple perverse
sheaf $E\in\mathsf{D}_{G,m}^{b}(G_{0})$ such that $E\langle-i\rangle$ is a
subquotient of $\mathsf{CH}(\mathit{IC}_{w})$, _resp._
$\xi\mathsf{CH}(\mathit{IC}_{w})$, for some $w\in W$ and $i\in\mathbf{Z}$.
This means $E$ is also pure of weight zero.

Recall that by [BBD, Prop. 5.3.9], every indecomposable pure perverse sheaf
takes the form $E\otimes M$, where $E$ is a simple perverse sheaf and $M$ is
the pullback of a sheaf on $\mathit{pt}_{0}$ that corresponds to a
$\bar{\mathbf{Q}}_{\ell}[F]$-module on which $F$ acts by a single unipotent
Jordan block. Thus, a simple perverse sheaf $E\in\mathsf{D}_{G,m}^{b}(G_{0})$
is a mixed unipotent character sheaf if and only if $E\langle-i\rangle\otimes
M$ occurs as a direct summand of $\mathsf{CH}(\mathit{IC}_{w})$ for some
$w,i,M$ as above.

######  Definition 2.6.1.

Let $\mathsf{C}(G_{0})\subseteq\mathsf{D}_{G,m}^{b}(G_{0})$ be the full,
replete, additive subcategory generated by the objects $E\otimes M$, where $E$
is a mixed unipotent character sheaf and $M$ is the pullback of a pure object
of $\mathsf{D}_{G,m}^{b}(\mathit{pt}_{0})$ of weight zero.

######  Remark 2.6.2.

The objects $\xi\mathsf{CH}(\mathit{IC}_{w})\in\mathsf{D}_{G}^{b}(G)$ coincide
with the objects $\bar{K}_{w}^{\mathcal{L}}$ studied by Lusztig in [L85-III]
when $\mathcal{L}$ is the trivial local system on a fixed maximal torus of
$G$. They also coincide with the objects $\mathbf{K}_{w}$ studied by
Webster–Williamson in [WW11] up to a shift-twist.

### 2.7.

The character functor is a monoidal trace on
$\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}\times\mathcal{B}_{0})$ in the following
sense. Abbreviate

(94)
$\displaystyle\mathsf{CH}(-\star-)\vcentcolon=\mathsf{CH}\circ((-)\star(-)):\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}\times\mathcal{B}_{0})^{2}\to\mathsf{D}_{G,m}^{b}(G_{0}).$

The result below can be deduced from a paper of Lusztig [L15], identifying the
monoidal center of $\mathsf{D}_{G}^{b}(\mathcal{B}\times\mathcal{B})$ with a
certain category of unmixed unipotent character sheaves, but we give a self-
contained proof.

######  Lemma 2.7.1.

We have
${\mathsf{CH}(-\star-)\circ(\mathit{pr}_{2}\times\mathit{pr}_{1})^{\ast}}\,\simeq\,{\mathsf{CH}(-\star-)}$,
where $\mathit{pr}_{2}\times\mathit{pr}_{1}$ is the involution of
$[(\mathcal{B}_{0}\times\mathcal{B}_{0})/G_{0}]^{2}$ that swaps the factors.

###### Chứng minh.

Observe that $\mathsf{CH}(-\star-)$ is induced by pullback and pushforward
along the solid arrows in the diagram below:

(95)
${{[(G_{0}\times\mathcal{B}_{0}\times\mathcal{B}_{0})/G_{0}]}}$${{[(G_{0}\times\mathcal{B}_{0})/G_{0}]}}$${{[G_{0}/G_{0}]}}$${{[(\mathcal{B}_{0}\times\mathcal{B}_{0}\times\mathcal{B}_{0})/G_{0}]}}$${{[(\mathcal{B}_{0}\times\mathcal{B}_{0})/G_{0}]}}$${{[(\mathcal{B}_{0}\times\mathcal{B}_{0})/G_{0}]^{2}}}$$\mathit{pr}$$\mathit{act}$$\mathit{pr}_{1,3}$$\mathit{pr}_{1,2}\times\mathit{pr}_{2,3}$

Let the square in the diagram be cartesian. By proper base change across that
square, we see that

(96)
$\displaystyle\mathsf{CH}(-\star-)=\bigoplus_{i}{}^{p}\mathcal{H}^{i}[-i]\circ\Psi_{\ast}\circ\Phi^{\ast}\langle
2(r+N)\rangle,$

where the functors

(97)
$\displaystyle\Phi:[(G_{0}\times\mathcal{B}_{0}\times\mathcal{B}_{0})/G_{0}]$
$\displaystyle\to[(\mathcal{B}_{0}\times\mathcal{B}_{0})/G_{0}]^{2},$ (98)
$\displaystyle\Psi:[(G_{0}\times\mathcal{B}_{0}\times\mathcal{B}_{0})/G_{0}]$
$\displaystyle\to[G_{0}/G_{0}]$

are defined by

(99) $\displaystyle\Phi([g,B,B^{\prime}])$
$\displaystyle=([g^{-1}B^{\prime}g,B],[B,B^{\prime}]),$ (100)
$\displaystyle\Psi([g,B,B^{\prime}])$ $\displaystyle=[g].$

Now consider the diagram

(101)
${{[(\mathcal{B}_{0}\times\mathcal{B}_{0})/G_{0}]^{2}}}$${{[(G_{0}\times\mathcal{B}_{0}\times\mathcal{B}_{0})/G_{0}]}}$${{[G_{0}/G_{0}]}}$${{[(\mathcal{B}_{0}\times\mathcal{B}_{0})/G_{0}]^{2}}}$${{[(G_{0}\times\mathcal{B}_{0}\times\mathcal{B}_{0})/G_{0}]}}$$\Phi$$\Psi$$\Phi$$\Psi$$\mathit{pr}_{2}\times\mathit{pr}_{1}$$\Xi$

in which $\Xi([g,B,B^{\prime}])=[g,g^{-1}B^{\prime}g,B]$. We can check that
the square on the left and the triangle on the right commute. (The former is
where it is essential to use orbits, not orbit representatives.) Thus the
whole diagram commutes, giving us the desired equivalence of functors. ∎

######  Remark 2.7.2.

The cocenter of an associative algebra $A$ is the largest commutative
quotient, _i.e._ , the algebra $A/[A,A]$. We expect that the functor
$\mathsf{CH}$ categorifies the map from the Iwahori–Hecke algebra to its
cocenter, by way of the function-sheaf dictionary (see Remark 2.5.1).

### 2.8.

Let $\tilde{G}_{0}$ be defined by the cartesian diagram:

(102)
${\tilde{G}_{0}}$${G_{0}\times\mathcal{B}_{0}}$${O_{1,0}=\mathcal{B}_{0}}$${\mathcal{B}_{0}\times\mathcal{B}_{0}}$$\mathit{act}$$j_{1}$

The composition $p^{\prime}:\tilde{G}_{0}\to G_{0}\times\mathcal{B}_{0}\to
G_{0}$ is a small, hence proper, map called the Grothendieck–Springer map. The
Grothendieck–Springer sheaf is

(103) $\displaystyle\mathcal{G}=p^{\prime}_{\ast}\bar{\mathbf{Q}}_{\ell}.$

By base change and smallness, $\mathcal{G}\simeq\mathsf{CH}(\mathit{IC}_{1})$.

The Grothendieck–Springer map restricts to a Galois $W$-cover over the regular
semisimple locus of $G$. Therefore, the decomposition of $\mathcal{G}$ into
indecomposable perverse sheaves takes the form

(104)
$\displaystyle\mathcal{G}\simeq\bigoplus_{\phi\in\hat{W}}\mathcal{G}_{\phi}\otimes
V_{\phi},$

where each $\mathcal{G}_{\phi}$ is a simple perverse sheaf and $V_{\phi}$ is
the irreducible representation of $W$ over $\bar{\mathbf{Q}}_{\ell}$ of
character $\phi$. We pin down $\mathcal{G}_{\phi}$ up to unique isomorphism by
declaring that Frobenius acts trivially on $V_{\phi}$.

######  Definition 2.8.1.

Let $\mathsf{C}_{\hat{W}}(G_{0})\subseteq\mathsf{C}(G_{0})$ be the full,
replete, additive subcategory generated by the objects
$\mathcal{G}_{\phi}\otimes M$, where $\phi\in\hat{W}$ and $M$ is the pullback
of a pure object of $\mathsf{D}_{G}^{b}(\mathit{pt}_{0})$ of weight zero.

######  Remark 2.8.2.

The work in Section 4 will show that $\mathsf{C}_{\hat{W}}(G_{0})$ and thus
$\mathsf{C}(G_{0})$ are not realizable, let alone
$\underline{\operatorname{{Hom}}}$-pure.

######  Remark 2.8.3.

If $G$ is of type $A$, meaning $W=S_{n}$ for some $n$, then every (unmixed)
unipotent character sheaf on $G$ is isomorphic to a summand of
$\xi\mathcal{G}$ (_e.g._ , by Lusztig’s classification of character sheaves).
So in this case, we have $\mathsf{C}_{\hat{W}}(G_{0})\simeq\mathsf{C}(G_{0})$.
In other types, a unipotent character sheaf that does not occur as a summand
of $\xi\mathcal{G}$ is said to be cuspidal.

### 2.9.

Let $i:\mathcal{U}_{0}\to G_{0}$ be the inclusion of the unipotent locus. The
action of $G_{0}$ on itself by conjugation restricts to an action on
$\mathcal{U}_{0}$. We stratify $\mathcal{U}_{0}$ by its $G_{0}$-orbits. Every
orbit is a Jordan class of $G_{0}$, so the inclusion $i:\mathcal{U}_{0}\to
G_{0}$ is stratification-preserving. We have $\dim\mathcal{U}=2N$, from which
$\dim\mathcal{U}-\dim G=-r$.

######  Definition 2.9.1.

Let
$\mathsf{C}(\mathcal{U}_{0})\subseteq\mathsf{D}_{G,m}^{b}(\mathcal{U}_{0})$ be
the full, replete, additive subcategory generated by the objects
$i^{\ast}E\otimes M$, where $E$ is a mixed unipotent character sheaf and $M$
is the pullback of a pure object of $\mathsf{D}_{G,m}^{b}(\mathit{pt}_{0})$ of
weight zero.

Since the map $i$ is merely a closed immersion, there is a priori no reason
that $i^{\ast}$ sends unipotent character sheaves to well-behaved objects.
Nevertheless:

######  Theorem 2.9.2 (Lusztig).

If $E$ is a mixed unipotent character sheaf, then there exist a $G_{0}$-orbit
$j_{C}:C_{0}\to\mathcal{U}_{0}$ and a pure $\ell$-adic local system
$\mathcal{L}\in\mathsf{D}_{G,m}^{b}(C_{0})$ of weight zero such that
$i^{\ast}E\simeq j_{C,!\ast}\mathcal{L}\langle n_{C}\rangle$ for some
$n_{C}\in\mathbf{Z}$ that only depends on $C$.

###### Chứng minh.

The unmixed version of this statement is [L84-2, Thm. 6.5(c)]. Below, we
explain how to deduce our mixed version. The notation $\mathit{IC}(-,-)$ will
mean the intersection complex of a local system (_not_ necessarily shifted to
be perverse).

First, $\xi E$ is an object of $\mathsf{D}_{G}^{b}(G)$ that, in Lusztig’s
notation, takes the form $\phi_{!}K[\dim\bar{Y}]$, where $\phi$ is a map
$X\to\bar{Y}\subseteq G$ and $K=\mathit{IC}(X,\bar{\mathcal{E}}_{1})$ for some
local system $\bar{\mathcal{E}}_{1}$ over $X$ [L84-2, §4]. As long as we start
with $\mathbf{F}$-structures on $X,Y,\phi,\bar{\mathcal{E}}_{1}$ such that
$\bar{\mathcal{E}}_{1}$ is pure of weight zero, the construction descends to
$G_{0}$ to yield $E=\phi_{!}K\langle\dim\bar{Y}\rangle$. In the notation of
[L84-2, §6.6], we have $i^{\ast}K\simeq K^{a}$, where
$K^{a}=\mathit{IC}(X_{0}^{a},\bar{\mathcal{E}}_{1}^{a})$ and
$\bar{\mathcal{E}}_{1}^{a}=i^{\ast}\bar{\mathcal{E}}_{1}$ (and the subscript
$0$ means the $\mathbf{F}$-structure on $X^{a}$ coming from the
$\mathbf{F}$-structure we chose on $X$). Therefore,
$i^{\ast}E\simeq\phi_{!}^{a}K^{a}\langle\dim\bar{Y}\rangle$. Since $\phi^{a}$
is proper, $\bar{\mathcal{E}}_{1}^{a}$ being pure implies $\phi_{!}^{a}K^{a}$
being pure of the same weight [D80], _cf._ (6.6.1) of [L84-2].

In our situation, $E$ is a unipotent character sheaf, which means that
$\bar{\mathcal{E}}_{1}$ is a constant sheaf [L85-I, §3.1, §4.4]. Therefore,
$\bar{\mathcal{E}}_{1}^{a}$ is the constant sheaf on $X_{0}^{a}$, hence pure
of weight zero. Therefore $i^{\ast}E$ is pure of weight zero. Now, the rest of
[L84-2, Thm. 6.5(c)] implies that at the unmixed level, $i^{\ast}E$ is a shift
of some intersection complex $j_{C,!\ast}\mathcal{E}$. Since $\mathcal{E}$ is
the shifted restriction of $i^{\ast}E$ to $C$, it must be pure. So up to
twisting $\mathcal{E}$, we can find a pure local system $\mathcal{L}$ of
weight zero such that $i^{\ast}E\simeq j_{C,!\ast}\mathcal{L}\langle n\rangle$
for some $n$. Finally, [L84-2, Thm. 6.5(c)] also implies that $n$ only depends
on $\dim\bar{Y}^{a}$ and $\dim C$, and the former is determined by a variety
$C_{1}$ that only depends on $C$. ∎

### 2.10.

Let $p:\tilde{\mathcal{U}}_{0}\to\mathcal{U}_{0}$ be defined by the cartesian
diagram:

(105)
${\tilde{\mathcal{U}}_{0}}$${\tilde{G}_{0}}$${\mathcal{U}_{0}}$${G_{0}}$$p$$p^{\prime}$$i$

Then $p$ (or rather,
$p^{\mathrm{red}}:\tilde{\mathcal{U}}_{0}^{\mathrm{red}}\to\mathcal{U}_{0}$)
is a semismall resolution of singularities called the Springer resolution. The
Springer sheaf is

(106)
$\displaystyle\mathcal{S}=p_{\ast}\bar{\mathbf{Q}}_{\ell}\langle-r\rangle.$

By proper base change, $\mathcal{S}=i^{\ast}\mathcal{G}\langle-r\rangle$.
Setting $\mathcal{S}_{\phi}=i^{\ast}\mathcal{G}_{\phi}\langle-r\rangle$, we
have that

(107)
$\displaystyle\mathcal{S}\simeq\bigoplus_{\phi\in\hat{W}}\mathcal{S}_{\phi}\otimes
V_{\phi}$

is the decomposition of $\mathcal{S}$ into indecomposable perverse sheaves.

######  Definition 2.10.1.

Let
$\mathsf{C}_{\hat{W}}(\mathcal{U}_{0})\subseteq\mathsf{C}(\mathcal{U}_{0})$ be
the full, replete, additive subcategory generated by the objects
$\mathcal{S}_{\phi}\otimes M$, where $\phi\in\hat{W}$ and $M$ is the pullback
of a pure object of $\mathsf{D}_{G,m}^{b}(\mathit{pt}_{0})$ of weight zero.

######  Lemma 2.10.2.

$\mathsf{C}(\mathcal{U}_{0})$ and thus $\mathsf{C}_{\hat{W}}(\mathcal{U}_{0})$
are $\underline{\operatorname{{Hom}}}$-pure.

###### Chứng minh.

$\mathsf{C}(\mathcal{U}_{0})$ is closed under $\langle n\rangle$ for all $n$
because the same is true of the class of pure objects of weight zero in
$\mathsf{D}_{G,m}^{b}(\mathit{pt}_{0})$. It remains to check that
$\underline{\operatorname{{Hom}}}^{0}(i^{\ast}E,i^{\ast}E^{\prime})$ is
concentrated in weight zero for all mixed unipotent character sheaves
$E,E^{\prime}$.

In the notation of Theorem 2.9.2, we can write $i^{\ast}E\simeq
j_{C,!\ast}\mathcal{L}\langle n_{C}\rangle$ and $i^{\ast}E^{\prime}\simeq
j_{C^{\prime},!\ast}\mathcal{L}^{\prime}\langle n_{C^{\prime}}\rangle$ for
some $C,C^{\prime},\mathcal{L},\mathcal{L}^{\prime},n_{C},n_{C^{\prime}}$. It
remains to show that

(108)
$\displaystyle\begin{split}&\underline{\operatorname{{Hom}}}^{0}(j_{C,!\ast}\mathcal{L}\langle
n_{C}\rangle,j_{C^{\prime},!\ast}\mathcal{L}^{\prime}\langle
n_{C^{\prime}}\rangle)\\\
&=\underline{\operatorname{{Hom}}}^{d}(j_{C,!\ast}\mathcal{L}\langle\dim
C\rangle,j_{C^{\prime},!\ast}\mathcal{L}^{\prime}\langle\dim
C^{\prime}\rangle)(\tfrac{d}{2})\end{split}$

is concentrated in weight zero, where $d=(n_{C^{\prime}}-\dim
C^{\prime})-(n_{C}-\dim C)$. Note that $j_{C,!\ast}\mathcal{L}\langle\dim
C\rangle$ and $j_{C^{\prime},!\ast}\mathcal{L}^{\prime}\langle\dim
C^{\prime}\rangle$ are both pure perverse of weight zero. So the desired claim
follows from [RR, Cor. 4.8]. ∎

### 2.11.

To conclude this section, we discuss the relationships between the categories
$\mathsf{C}(G_{0})$, $\mathsf{C}_{\hat{W}}(G_{0})$,
$\mathsf{C}(\mathcal{U}_{0})$, and $\mathsf{C}_{\hat{W}}(\mathcal{U}_{0})$.

######  Theorem 2.11.1 (Lusztig, Rider–Russell).

The horizontal arrows in the commutative diagrams below are inclusions of
direct summands:

(109)
${\mathsf{C}_{\hat{W}}(G_{0})}$${\mathsf{C}(G_{0})}$${\underline{\mathsf{C}}_{\hat{W}}(G_{0})}$${\underline{\mathsf{C}}(G_{0})}$$\kappa$$\kappa$
${\mathsf{C}_{\hat{W}}(\mathcal{U}_{0})}$${\mathsf{C}(\mathcal{U}_{0})}$${\underline{\mathsf{C}}_{\hat{W}}(\mathcal{U}_{0})}$${\underline{\mathsf{C}}(\mathcal{U}_{0})}$$\kappa$$\kappa$

###### Chứng minh.

In the diagram for $G_{0}$, the statement for the bottom arrow is essentially
a theorem of Lusztig [L85-II, Prop. 7.2]. The only modification needed is
$G$-equivariance, which is done in [WW08, Prop. 8]. In the diagram for
$\mathcal{U}_{0}$, the statement for the bottom arrow follows from [RR, Thm.
3.5]. Finally, the statements for the top arrows follow from the exactness of
(43). ∎

We define the functors

(110) $\displaystyle\mathsf{PR}:\mathsf{C}(G_{0})$
$\displaystyle\to\mathsf{C}_{\hat{W}}(G_{0}),$ (111)
$\displaystyle\mathsf{PR}:\mathsf{C}(\mathcal{U}_{0})$
$\displaystyle\to\mathsf{C}_{\hat{W}}(\mathcal{U}_{0}),$

as well as their analogues with $\underline{\mathsf{C}}$ in place of
$\mathsf{C}$, to be the projections that correspond to these summand
inclusions. By Lemma 1.6.1, we have:

######  Lemma 2.11.2.

We have a commutative diagram:

(112)
${\mathsf{C}(G_{0})}$${\mathsf{C}(\mathcal{U}_{0})}$${\underline{\mathsf{C}}(G_{0})}$${\underline{\mathsf{C}}(\mathcal{U}_{0})}$$i^{\ast}\langle-r\rangle$$\kappa$$\kappa$$i^{\ast}\langle-r\rangle$

Here, the bottom arrow is defined by (68).

##  3\. The $\mathbf{A}_{W}$ Trace

### 3.1.

If $A$ is a $\mathbf{Z}$-graded $\bar{\mathbf{Q}}_{\ell}$-algebra, then we
write:

  1. (1)

$\mathsf{Mod}_{1}(A)$ for the category of graded left $A$-modules in which the
grading is bounded below and each graded piece is finite-dimensional.

  2. (2)

$\mathsf{Mod}_{2}^{b}(A)$ for the category of left $A$-modules $M$ equipped
with a $\mathbf{Z}^{2}$-grading such that $M\in\mathsf{Mod}_{1}(A)$ with
respect to both gradings and the value of $i-j$ is uniformly bounded among the
nonzero components $M^{i,j}$.

If $R$ is an (ungraded) $\bar{\mathbf{Q}}_{\ell}$-algebra and $A$ forms an
$R$-module, then we write $R\ltimes A$ for the vector space $R\otimes A$
endowed with the ring structure where:

  * •

$R$ and $A$ form subrings.

  * •

The only relations between $R$ and $A$ are those of the form $ra=(r\cdot a)r$
for all $r\in R$ and $a\in A$.

If the $R$-action on $A$ preserves the grading, then we extend the grading on
$A$ to a grading on $R\ltimes A$ in which $R$ is placed in degree zero, so
that $R\ltimes A$ forms a graded $A$-algebra.

Recall that $\mathbf{X}$ is the character lattice in the root datum of $G$. We
set

(113) $\displaystyle\mathbf{V}=\mathbf{X}\otimes\bar{\mathbf{Q}}_{\ell}.$

Let the symmetric algebra $\operatorname{{Sym}}(\mathbf{V})$ be endowed with
the grading where $\operatorname{{Sym}}^{i}(\mathbf{V})$ is placed in degree
$2i$, and let

(114)
$\displaystyle\mathbf{A}_{W}=\bar{\mathbf{Q}}_{\ell}[W]\ltimes\operatorname{{Sym}}(\mathbf{V}).$

Below, we construct a contravariant, monoidal trace from the Hecke category to
$\mathsf{Mod}_{2}^{b}(\mathbf{A}_{W})$.

### 3.2.

Suppose that $X_{0}$ is a $G_{0}$-variety over $\mathbf{F}$ and that
$\mathsf{C}(X_{0})$ is a full, additive subcategory of
$\mathsf{D}_{G,m}^{b}(X_{0})$ stable under $\langle n\rangle$ for all
$n\in\mathbf{Z}$. Any object $K\in\underline{\mathsf{C}}(X_{0})$ gives rise to
a graded $\bar{\mathbf{Q}}_{\ell}$-algebra:

(115)
$\displaystyle\mathbf{A}_{K}=\bigoplus_{n}{\operatorname{{Hom}}_{\underline{\mathsf{C}}(X_{0})}(K,K\langle
n\rangle)}.$

We write the multiplication in $\mathbf{A}_{K}$ as right-to-left composition:
$g\cdot f=g\circ f$. Then we have additive functors

(116)
$\displaystyle\bigoplus_{n}{\operatorname{{Hom}}_{\underline{\mathsf{C}}(X_{0})}(-,K\langle
n\rangle)}:\underline{\mathsf{C}}(X_{0})^{\mathrm{op}}$
$\displaystyle\to\mathsf{Mod}_{1}(\mathbf{A}_{K}),$ (117)
$\displaystyle\bigoplus_{n}{\operatorname{{Hom}}_{\underline{\mathsf{C}}(X_{0})}(K,(-)\langle
n\rangle)}:\underline{\mathsf{C}}(X_{0})$
$\displaystyle\to\mathsf{Mod}_{1}(\mathbf{A}_{K}^{\mathrm{op}}).$

Above, the gradings are bounded below by Remark 1.2.1. These functors can be
respectively extended to

(118)
$\displaystyle\bigoplus_{i,j}{\operatorname{{Hom}}_{\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))}(-,K\langle
j\rangle[i-j]_{\triangle})}:\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))^{\mathrm{op}}$
$\displaystyle\to\mathsf{Mod}_{2}^{b}(\mathbf{A}_{K}),$ (119)
$\displaystyle\bigoplus_{i,j}{\operatorname{{Hom}}_{\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))}(K,(-)\langle
j\rangle[i-j]_{\triangle})}:\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))$
$\displaystyle\to\mathsf{Mod}_{2}^{b}(\mathbf{A}_{K}^{\mathrm{op}}),$

once we embed $\underline{\mathsf{C}}(X_{0})$ into
$\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))$ as the full subcategory of
complexes concentrated in degree zero. Above, in either direct sum, the value
of $i-j$ is uniformly bounded among the nonzero terms precisely because the
objects of $\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))$ can be represented
by bounded complexes.

In the rest of this section, we only use the contravariant functors; the
covariant ones will appear again in Section 6. The following lemma is a
consequence of Proposition 1.5.1.

######  Lemma 3.2.1.

If $\mathsf{C}(X_{0})$ is $\underline{\operatorname{{Hom}}}$-pure, then we
have a commutative diagram:

(120)
${\mathsf{K}^{b}(\mathsf{C}(X_{0}))^{\mathrm{op}}}$${\mathsf{K}^{b}(\underline{\mathsf{C}}(X_{0}))^{\mathrm{op}}}$${\mathsf{D}_{G,m}^{b}(X_{0})^{\mathrm{op}}}$${\mathsf{Mod}_{2}^{b}(\mathbf{A}_{K})}$$\mathsf{K}^{b}\kappa$$\rho$$\bigoplus_{i,j}\operatorname{{Hom}}(-,K\langle
j\rangle[i-j]_{\triangle})$$\bigoplus_{i,j}\operatorname{{gr}}_{j}^{\bm{w}}\underline{\operatorname{{Hom}}}^{i}(-,K)$

### 3.3.

We now set $X_{0}=\mathcal{U}_{0}$ and
$\mathsf{C}(X_{0})=\mathsf{C}(\mathcal{U}_{0})$ as in Definition 2.9.1. Recall
that $\mathcal{S}\in\mathsf{C}(\mathcal{U}_{0})$ denotes the (mixed) Springer
sheaf.

######  Theorem 3.3.1 (Lusztig).

We have $\mathbf{A}_{W}\simeq\mathbf{A}_{\mathcal{S}}$ as graded
$\bar{\mathbf{Q}}_{\ell}$-algebras.

###### Chứng minh.

By [L88, Cor. 6.4],
$\mathbf{A}_{W}\simeq\operatorname{{End}}^{\ast}(\xi\mathcal{S})$. By Lemma
2.10.2 (or [R, Lem. 5.4]), we have
$\operatorname{{End}}^{i}(\xi\mathcal{S})\simeq\underline{\operatorname{{Hom}}}^{0}(\mathcal{S},\mathcal{S}\langle
i\rangle)$ for all $i$ after we forget Frobenius. ∎

######  Remark 3.3.2.

Although we will not use it, the proof of [R, Prop. 6.5] implies that the
functor $\bigoplus_{i}\operatorname{{Hom}}(\mathcal{S},(-)\langle i\rangle)$
induces a triangulated equivalence

(121)
$\displaystyle\mathsf{K}^{b}(\mathsf{C}_{\hat{W}}(\mathcal{U}_{0}))\simeq\mathsf{D}^{b}(\mathsf{Mod}_{1}(\mathbf{A}_{W})),$

where the right-hand side is the bounded derived category of
$\mathsf{Mod}_{1}(\mathbf{A}_{W})$.

######  Definition 3.3.3.

The $\mathbf{A}_{W}$ trace on the Hecke category is the contravariant functor

(122)
$\displaystyle\mathsf{AH}:\mathsf{H}_{W}^{\mathrm{op}}\to\mathsf{Mod}_{2}^{b}(\mathbf{A}_{W})$

defined by the commutative diagram

(123)
${\mathsf{K}^{b}(\mathsf{C}(\mathcal{B}_{0}\times\mathcal{B}_{0}))^{\mathrm{op}}}$${\mathsf{K}^{b}(\mathsf{C}(\mathcal{U}_{0}))^{\mathrm{op}}}$${\mathsf{K}^{b}(\underline{\mathsf{C}}(\mathcal{U}_{0}))^{\mathrm{op}}}$${\mathsf{D}_{G,m}^{b}(\mathcal{B}_{0}\times\mathcal{B}_{0})^{\mathrm{op}}}$${\mathsf{D}_{G,m}^{b}(\mathcal{U}_{0})^{\mathrm{op}}}$${\mathsf{Mod}_{2}^{b}(\mathbf{A}_{W})}$$\mathsf{K}^{b}(i^{\ast}\langle-r\rangle\circ\mathsf{CH})$$i^{\ast}\langle-r\rangle\circ\mathsf{CH}$$\mathsf{K}^{b}\kappa$$\rho$$\rho$

that we obtain by combining Remark 1.4.4, Lemma 2.10.2, Lemma 3.2.1, and
Theorem 3.3.1. Explicitly,

(124)
$\displaystyle\begin{split}\mathsf{AH}^{i,j}&=\operatorname{{Hom}}_{\mathsf{K}^{b}(\underline{\mathsf{C}}(\mathcal{U}_{0}))}(\mathsf{K}^{b}(\kappa\circ
i^{\ast}\langle-r\rangle\circ\mathsf{CH})(-),\mathcal{S}\langle
j\rangle[i-j]_{\triangle})\\\
&\simeq\operatorname{{gr}}_{j}^{\bm{w}}\underline{\operatorname{{Hom}}}^{i}((\rho\circ\mathsf{K}^{b}(i^{\ast}\langle-r\rangle\circ\mathsf{CH}))(-),\mathcal{S})\\\
&\simeq\operatorname{{gr}}_{j}^{\bm{w}}\underline{\operatorname{{Hom}}}^{i}((i^{\ast}\langle-r\rangle\circ\mathsf{CH}\circ\rho)(-),\mathcal{S})\end{split}$

for all $i,j$. It is a monoidal trace in the sense of Lemma 2.7.1 because it
factors through $\mathsf{CH}$.

##  4\. Steinberg Schemes

### 4.1.

In Section 2, we reviewed the (mixed) Springer sheaf
$\mathcal{S}\in\mathsf{D}_{G,m}^{b}(\mathcal{U}_{0})$ and the (mixed)
Grothendieck–Springer sheaf $\mathcal{G}\in\mathsf{D}_{G,m}^{b}(G_{0})$. In
Section 3, we gave a construction of a graded algebra $\mathbf{A}_{K}$ from a
mixed complex $K$: explicitly,

(125)
$\displaystyle\begin{split}\mathbf{A}_{K}&=\bigoplus_{n}{\operatorname{{gr}}_{0}^{\bm{w}}}\underline{\operatorname{{Hom}}}^{0}(K,K\langle
n\rangle)\\\
&\simeq\bigoplus_{n}{\operatorname{{gr}}_{n}^{\bm{w}}}\underline{\operatorname{{End}}}^{n}(K).\end{split}$

In this section, we use topological methods to prove Theorem 1: The map

(126) $\displaystyle
i^{\ast}:\mathbf{A}_{\mathcal{G}}\to\mathbf{A}_{\mathcal{S}}$

induced by $i:\mathcal{U}_{0}\to\mathcal{G}_{0}$ via Lemma 1.6.1 is an
isomorphism of graded $\bar{\mathbf{Q}}_{\ell}$-algebras.

######  Remark 4.1.1.

By Lemma 2.10.2, $\underline{\operatorname{{End}}}^{n}(\mathcal{S})$ is
concentrated in weight $n$ (see also [R, Lem. 5.4].) However, if $n>0$, then
$\underline{\operatorname{{End}}}^{n}(\mathcal{G})$ is not concentrated in
weight $n$, essentially because the cohomology of the Grothendieck–Springer
resolution of $G$ contains the cohomology of a torus. See Example 6.2.2.

######  Remark 4.1.2.

Our proof of Theorem 1 is inspired by an unpublished note of Riche about the
analogous statement at the level of Lie algebras, which is well-known. Riche
states that his note was inspired by [CG, §3.4].

### 4.2.

If $X_{0}$ is a $G_{0}$-variety over $\mathbf{F}$, with structure map
$a:X_{0}\to\mathit{pt}_{0}$, then its mixed equivariant dualizing complex is

(127)
$\displaystyle\omega=\omega_{X_{0}/G_{0}}\vcentcolon=a^{!}\bar{\mathbf{Q}}_{\ell}\in\mathsf{D}_{G,m}^{b}(X_{0}).$

Its equivariant Borel–Moore homology (with coefficients in
$\bar{\mathbf{Q}}_{\ell}$) is

(128)
$\displaystyle\mathrm{H}_{-\ast}^{!,G}(X)\vcentcolon=\underline{\operatorname{{Hom}}}^{\ast}(\bar{\mathbf{Q}}_{\ell},\omega),$

viewed as a graded $\bar{\mathbf{Q}}_{\ell}[F]$-module. Borel–Moore homology
is functorial in the following senses:

  1. (1)

If $f:Y\to X$ is proper, then it induces a pushforward

(129) $\displaystyle
f_{\ast}:\mathrm{H}_{-\ast}^{!,G}(Y)\to\mathrm{H}_{-\ast}^{!,G}(X)$

via the composition
$f_{\ast}\omega_{Y/G}\xrightarrow{\sim}f_{!}f^{!}\omega_{X/G}\to\omega_{X/G}$.

  2. (2)

If $f:Y\to X$ is smooth of relative dimension $d$, then it induces a pullback

(130) $\displaystyle
f^{\ast}:\mathrm{H}_{-\ast}^{!,G}(X)\to\mathrm{H}_{-\ast}^{!,G}(Y)\langle-2d\rangle$

via the composition $\omega_{X/G}\to
f_{\ast}f^{\ast}\omega_{X/G}\xrightarrow{\sim}f_{\ast}\omega_{Y/G}\langle-2d\rangle$.
If $f$ is an affine-space bundle, then $f^{\ast}$ is a shifted isomorphism.

  3. (3)

If $i:Z\to X$ is a closed immersion with complement $j:U\to X$, then it
induces a long exact sequence

(131)
$\displaystyle\cdots\to\mathrm{H}_{i}^{!,G}(Z)\to\mathrm{H}_{i}^{!,G}(X)\to\mathrm{H}_{i}^{!,G}(U)\to\mathrm{H}_{i-1}^{!,G}(Z)\to\cdots$

via the exact triangle $i_{!}i^{!}\to\operatorname{{id}}\to
j_{\ast}j^{\ast}\to i_{!}i^{!}[1]$.

  4. (4)

If we have a cartesian square

(132) ${Y}$${Y^{\prime}}$${X}$${X^{\prime}}$$f_{Y}$$g$$g^{\prime}$$f$

in which $[X/G],[X^{\prime}/G]$ are smooth of respective dimensions
$d,d^{\prime}$, then $f$ induces a pullback with restricted support

(133) $\displaystyle
f_{Y}^{\ast}:\mathrm{H}_{-\ast}^{!,G}(Y^{\prime})\to\mathrm{H}_{-\ast}^{!,G}(Y)\langle
2(d^{\prime}-d)\rangle$

via the composition

(134)
$\displaystyle\omega_{Y^{\prime}/G}\langle-2d^{\prime}\rangle\xrightarrow{\sim}{g^{\prime}}^{!}\bar{\mathbf{Q}}_{\ell}\to
f_{Y,\ast}g^{!}f^{\ast}\bar{\mathbf{Q}}_{\ell}\xrightarrow{\sim}f_{Y,\ast}\omega_{Y/G}\langle-2d\rangle.$

Here, we use proper base change [LO08, §5] to identify
${g^{\prime}}^{!}f_{\ast}\simeq f_{Y,\ast}g^{!}$.

### 4.3.

We can express $\underline{\operatorname{{Hom}}}$-spaces between pushforward
sheaves, like $\mathbf{A}_{\mathcal{S}}$ and $\mathbf{A}_{\mathcal{G}}$, in
terms of the Borel–Moore homology of associated schemes.

######  Lemma 4.3.1.

If we have a cartesian square of $G$-varieties

(135) ${\tilde{Y}}$${\tilde{X}}$${Y}$${X}$$\tilde{f}$$p_{Y}$$p$$f$

in which $[\tilde{X}/G]$ is smooth of dimension $\tilde{d}$, then

(136)
$\displaystyle\underline{\operatorname{{Hom}}}^{\ast}(f_{!}\bar{\mathbf{Q}}_{\ell},p_{\ast}\bar{\mathbf{Q}}_{\ell})\simeq\mathrm{H}_{-\ast}^{!,G}(\tilde{Y})\langle-2\tilde{d}\rangle.$

In particular, if $p$ is proper and $K=p_{\ast}\bar{\mathbf{Q}}_{\ell}$, then

(137)
$\displaystyle\mathbf{A}_{K}^{n}\simeq\operatorname{{gr}}_{n-2\tilde{d}}^{\bm{w}}\mathrm{H}_{-(n-2\tilde{d})}^{!,G}(\tilde{X}\times_{X}\tilde{X})$

for all $n$, and $\mathrm{H}_{-\ast}^{!,G}(\tilde{Y})$ forms a left graded
$\mathbf{A}_{K}$-module.

###### Chứng minh.

By proper base change,

(138) $\displaystyle\begin{split}f^{!}p_{\ast}\bar{\mathbf{Q}}_{\ell}&\simeq
p_{Y,\ast}\tilde{f}^{!}\bar{\mathbf{Q}}_{\ell}\simeq
p_{Y,\ast}\tilde{f}^{!}\omega_{\tilde{X}/G}\langle-2\tilde{d}\rangle\simeq
p_{Y,\ast}\omega_{\tilde{Y}/G}\langle-2\tilde{d}\rangle,\end{split}$

giving
$\underline{\operatorname{{Hom}}}(f_{!}\bar{\mathbf{Q}}_{\ell},p_{\ast}\bar{\mathbf{Q}}_{\ell})\simeq
f_{\ast}\underline{\operatorname{{Hom}}}(\bar{\mathbf{Q}}_{\ell},f^{!}p_{\ast}\bar{\mathbf{Q}}_{\ell})\simeq
f_{\ast}p_{Y,\ast}\underline{\operatorname{{Hom}}}(\bar{\mathbf{Q}}_{\ell},\omega_{\tilde{Y}/G})\langle-2\tilde{d}\rangle$.
∎

### 4.4.

In the notation of Section 2, let

(139) $\displaystyle\mathcal{Z}_{0}$
$\displaystyle=\tilde{\mathcal{U}}_{0}\times_{\mathcal{U}_{0}}\tilde{\mathcal{U}}_{0},$
(140) $\displaystyle\mathcal{Z}^{\prime}_{0}$
$\displaystyle=\tilde{G}_{0}\times_{G_{0}}\tilde{G}_{0}.$

In the literature, the underlying reduced scheme of $\mathcal{Z}$ is called
the Steinberg variety, since it was introduced in [St76]. Lemma 4.3.1 implies
that

(141) $\displaystyle\mathbf{A}_{\mathcal{S}}^{n}$
$\displaystyle\simeq\operatorname{{gr}}_{n+2r}^{\bm{w}}\mathrm{H}_{-(n+2r)}^{!,G}(\mathcal{Z})\left(\simeq\mathrm{H}_{-(n+2r)}^{!,G}(\mathcal{Z})\right),$
(142) $\displaystyle\mathbf{A}_{\mathcal{G}}^{n}$
$\displaystyle\simeq\operatorname{{gr}}_{n}^{\bm{w}}\mathrm{H}_{-n}^{!,G}(\mathcal{Z}^{\prime}).$

It remains to compare the Borel–Moore homologies of $\mathcal{Z}$ and
$\mathcal{Z}^{\prime}$.

Let us outline the strategy. We will lift the Bruhat strata of
$\mathcal{B}\times\mathcal{B}$ to strata $\mathcal{Z}_{w}\subseteq\mathcal{Z}$
and $\mathcal{Z}^{\prime}_{w}\subseteq\mathcal{Z}^{\prime}$. To compare the
Borel–Moore homologies of $\mathcal{Z}_{w}$ and $\mathcal{Z}^{\prime}_{w}$, we
will further replace these strata with affine-space bundles $\mathcal{Y}_{w}$
and $\mathcal{Y}^{\prime}_{w}$ such that there is an explicit map
$\mathcal{Y}_{w}\times T\to\mathcal{Y}^{\prime}_{w}$, where $T$ is a maximal
torus of $G$. Finally, we will use the long exact sequence (131) to patch the
stratum-wise comparisons together.

### 4.5.

Let $\tilde{i}:\tilde{\mathcal{U}}_{0}\to\tilde{G}_{0}$ be the lift of the
inclusion $i:\mathcal{U}_{0}\to G_{0}$. Let
$\tilde{p}_{1},\tilde{p}_{2}:\mathcal{Z}_{0}\to\tilde{\mathcal{U}}_{0}$ be the
two projection maps, and let
$\tilde{p}^{\prime}_{1},\tilde{p}^{\prime}_{2}:\mathcal{Z}^{\prime}_{0}\to\tilde{G}_{0}$
be defined similarly. Below, we will continually refer to the commutative
diagram:

(143)
${\mathcal{Z}_{0}}$${\mathcal{Z}^{\prime}_{0}}$${\tilde{\mathcal{U}}_{0}}$${\tilde{G}_{0}}$${\mathcal{U}_{0}}$${G_{0}}$$i_{\mathcal{Z}}$$\tilde{p}_{j}$$\tilde{p}^{\prime}_{j}$$\tilde{i}$$p$$p^{\prime}$$i$

Note that both squares in the diagram are cartesian.

######  Lemma 4.5.1.

We have a commutative diagram

(144)
${\mathbf{A}_{\mathcal{G}}^{n}}$${\mathbf{A}_{\mathcal{S}}^{n}}$${\operatorname{{gr}}_{n}^{\bm{w}}\mathrm{H}_{-n}^{!,G}(\mathcal{Z}^{\prime})}$${\mathrm{H}_{-(n+2r)}^{!,G}(\mathcal{Z})}$$i^{\ast}$$\sim$$\sim$$i_{\mathcal{Z}}^{\ast}$

where $i^{\ast}$ is defined by (68), the vertical isomorphisms are (141) and
(142), and $i_{\mathcal{Z}}^{\ast}$ is the pullback with restricted support
induced by $\tilde{i}$ via the top square of (143).

###### Chứng minh.

The morphism $i_{\mathcal{Z}}^{\ast}$ is induced by the morphism

(145)
$\displaystyle(\tilde{p}^{\prime})_{1}^{!}\to(\tilde{p}^{\prime})_{1}^{!}\tilde{i}_{\ast}\tilde{i}^{\ast}\xrightarrow{\sim}i_{\mathcal{Z},\ast}{\tilde{p}}_{1}^{!}\tilde{i}^{\ast},$

while the morphism $i^{\ast}$ is induced by the composition

(146)
$\displaystyle\begin{split}\tilde{p}^{\prime}_{2,\ast}(\tilde{p}^{\prime})_{1}^{!}&\xrightarrow{\sim}{p^{\prime}}^{!}p^{\prime}_{\ast}\\\
&\to{p^{\prime}}^{!}i_{\ast}i^{\ast}p^{\prime}_{\ast}\\\
&\xrightarrow{\sim}\tilde{i}_{\ast}p^{!}p_{!}\tilde{i}^{\ast}\\\
&\xrightarrow{\sim}\tilde{i}_{\ast}\tilde{p}_{2,!}\tilde{p}_{1}^{!}i^{\ast}\\\
&\xrightarrow{\sim}\tilde{p}^{\prime}_{2,\ast}i_{\mathcal{Z},\ast}\tilde{p}_{1}^{!}i^{\ast}.\end{split}$

By further diagram-chasing, the latter is obtained by applying
$\tilde{p}^{\prime}_{2,\ast}$ to the former. ∎

To analyze $i_{\mathcal{Z}}^{\ast}$, we decompose $\mathcal{Z}_{0}$ and
$\mathcal{Z}^{\prime}_{0}$ into strata. For each $w\in W$, we define
subschemes $\mathcal{Z}_{w,0}\subseteq\mathcal{Z}_{0}$ and
$\mathcal{Z}^{\prime}_{w,0}\subseteq\mathcal{Z}^{\prime}_{0}$ by stipulating
that both squares in the commutative diagram below are cartesian:

(147)
${\mathcal{Z}_{w,0}}$${\mathcal{Z}^{\prime}_{w,0}}$${O_{w,0}}$${\mathcal{Z}_{0}}$${\mathcal{Z}^{\prime}_{0}}$${\mathcal{B}_{0}\times\mathcal{B}_{0}}$$i_{\mathcal{Z},w}$$i_{\mathcal{Z}}$$j_{w}$

Here, the map
$\mathcal{Z}^{\prime}_{0}\to\mathcal{B}_{0}\times\mathcal{B}_{0}$ is induced
by the map
$\mathcal{Z}^{\prime}_{0}\xrightarrow{\tilde{p}^{\prime}_{1}\times\tilde{p}^{\prime}_{2}}\tilde{G}_{0}\times\tilde{G}_{0}$.

To analyze $i_{\mathcal{Z},w}:\mathcal{Z}_{w,0}\to\mathcal{Z}^{\prime}_{w,0}$,
we rewrite these strata in a way that expresses their structure as fiber
bundles over $O_{w,0}$. Choose a split maximal torus $T_{0}\subseteq G_{0}$.
Choose a Borel $B_{0}\supseteq T_{0}$ with unipotent radical $U_{0}\subseteq
B_{0}$, so that $B_{0}\simeq T_{0}\ltimes U_{0}$. Let $N_{0}$ be the
normalizer of $T_{0}$ in $G_{0}$. For each $w\in W$, fix a lift $\dot{w}\in
N_{0}$. Then $\dot{w}B_{0}\dot{w}^{-1}$ contains
$\dot{w}T_{0}\dot{w}^{-1}=T_{0}$ and only depends on $w$. If we write

(148) $\displaystyle B_{w,0}$
$\displaystyle=B_{0}\cap\dot{w}B_{0}\dot{w}^{-1},$ (149) $\displaystyle
U_{w,0}$ $\displaystyle=U_{0}\cap\dot{w}U_{0}\dot{w}^{-1},$

then $B_{w,0}\simeq T_{0}\ltimes U_{w,0}$. In what follows, let $B_{w,0}$ act
on $G_{0}$ by right multiplication and on $U_{w,0}$ and itself by right
conjugation.

######  Lemma 4.5.2.

We have a commutative diagram

(150) ${(G_{0}\times U_{w,0})/B_{w,0}}$${(G_{0}\times
B_{w,0})/B_{w,0}}$${G_{0}/B_{w,0}}$${\mathcal{Z}_{w,0}}$${\mathcal{Z}^{\prime}_{w,0}}$${O_{w,0}}$$\sim$$\sim$$\sim$$i_{\mathcal{Z},w}$

where the middle vertical map sends
$[g,b]\mapsto[gbg^{-1},gBg^{-1},g\dot{w}B\dot{w}^{-1}g^{-1}]$.

######  Remark 4.5.3.

The vertical isomorphisms respectively transport the $G$-actions on
$\mathcal{Z}_{w}$, $\mathcal{Z}^{\prime}_{w}$, $O_{w}$ to the $G$-actions on
$(G\times U_{w})/B_{w}$, $(G\times B_{w})/B_{w}$, $G/B_{w}$ given by left
multiplication on the factor of $G$.

###### Chứng minh.

The $w=1$ case, entailing isomorphisms $(G\times
U)/B\simeq\tilde{\mathcal{U}}$ and $(G\times B)/B\simeq\tilde{G}$, is
explained in [Sl, Ch. 4], and the general case is similar. ∎

Again, let $T_{0}$ act on $G_{0}$ by right multiplication and on $U_{w,0}$ and
$B_{w,0}$ by right conjugation. Let $\mathcal{Y}_{w,0}$ and
$\mathcal{Y}^{\prime}_{w,0}$ be defined by the cartesian squares:

(151) ${\mathcal{Y}_{w,0}}$${(G_{0}\times
U_{w,0})/T_{0}}$${\mathcal{Z}_{w,0}}$${(G_{0}\times
U_{w,0})/B_{w,0}}$$\sim$$\sim$ ${\mathcal{Y}^{\prime}_{w,0}}$${(G_{0}\times
B_{w,0})/T_{0}}$${\mathcal{Z}^{\prime}_{w,0}}$${(G_{0}\times
B_{w,0})/B_{w,0}}$$\sim$$\sim$

Let $i_{\mathcal{Y},w}:\mathcal{Y}_{w,0}\to\mathcal{Y}^{\prime}_{w,0}$ be the
pullback of
$i_{\mathcal{Z},w}:\mathcal{Z}_{w,0}\to\mathcal{Z}^{\prime}_{w,0}$. The
advantage of $i_{\mathcal{Y},w}$ over $i_{\mathcal{Z},w}$ is that the former
can be factored as

(152)
$\displaystyle\mathcal{Y}_{w,0}\xrightarrow{\sim}\mathcal{Y}_{w,0}\times\\{1\\}\to\mathcal{Y}_{w,0}\times
T_{0}\xrightarrow{\sim}\mathcal{Y}^{\prime}_{w,0}.$

If we let $G_{0}$ act trivially on $T_{0}$, then these maps are all
$G_{0}$-equivariant.

### 4.6.

In what follows, if $M=\bigoplus_{n}M^{n}$ is a $\mathbf{Z}$-graded
$\bar{\mathbf{Q}}_{\ell}[F]$-module, then we write

(153)
$\displaystyle\begin{split}M^{\dagger}&\simeq\bigoplus_{n}{\operatorname{{gr}}_{n}^{\bm{w}}M^{n}}.\end{split}$

Let
$i_{\mathcal{Z},w}^{\ast}:\mathrm{H}_{-\ast}^{!,G}(\mathcal{Z}^{\prime}_{w})\to\mathrm{H}_{-\ast}^{!,G}(\mathcal{Z}_{w})\langle
2r\rangle$ be the pullback with restricted support induced by $\tilde{i}$ via
the top square of (143) and the left square of (147), and let

(154) $\displaystyle
i_{\mathcal{Z},w}^{\dagger}:\mathrm{H}_{-\ast}^{!,G}(\mathcal{Z}^{\prime}_{w})^{\dagger}\to\mathrm{H}_{-\ast}^{!,G}(\mathcal{Z}_{w})\langle
2r\rangle$

be its restriction to
$\mathrm{H}_{-\ast}^{!,G}(\mathcal{Z}^{\prime}_{w})^{\dagger}$. Let
$i_{\mathcal{Y},w}^{\dagger}$, $i_{\mathcal{Y},w}^{\dagger}$ be defined
similarly.

######  Lemma 4.6.1.

The map $i_{\mathcal{Y},w}^{\dagger}$ is an isomorphism.

###### Chứng minh.

Writing $N_{w}=\dim U_{w}$, we have

(155)
$\displaystyle\mathrm{H}_{-\ast}^{!,G}(\mathcal{Y}_{w})\simeq\mathrm{H}_{-\ast}^{!,G}(G/T)\langle
2N_{w}\rangle\simeq\mathrm{H}_{-\ast}^{!,T}(\mathit{pt})\langle
2N_{w}\rangle.$

By Poincaré duality,
$\mathrm{H}_{-i}^{!,T}(\mathit{pt})\simeq\mathrm{H}_{!,T}^{-i}(\mathit{pt})\simeq\mathrm{H}_{T}^{i-2r}(\mathit{pt})$,
where $\mathrm{H}_{!,T}^{\ast}$ denotes $T$-equivariant, compactly-supported
cohomology. So Frobenius acts on $\mathrm{H}_{-i}^{!,T}(\mathit{pt})$ by
$q^{\frac{i}{2}-r}$, which means it is concentrated in weight $i-2r$.

Again by Poincaré duality,
$\mathrm{H}_{-j}^{!}(T)\simeq\mathrm{H}_{!}^{-j}(T)^{\vee}\simeq\mathrm{H}^{j+2r}(T)$,
where $\mathrm{H}_{\ast}^{!}$, _resp._ $\mathrm{H}_{!}^{\ast}$, denotes non-
equivariant Borel–Moore homology, _resp._ compactly-supported cohomology. So
Frobenius acts on $\mathrm{H}_{-j}^{!}(T)$ by $q^{j+2r}$, which means it is
concentrated in weight $2j+4r$.

Altogether, the Künneth isomorphism

(156)
$\displaystyle\mathrm{H}_{-n}^{!,G}(\mathcal{Y}^{\prime}_{w})\xrightarrow{\sim}\bigoplus_{i+j=n}\mathrm{H}_{-i}^{!,G}(\mathcal{Y}_{w})\otimes\mathrm{H}_{-j}^{!}(T)$

restricts to an isomorphism

(157)
$\displaystyle\mathrm{H}_{-n}^{!,G}(\mathcal{Y}^{\prime}_{w})^{\dagger}\xrightarrow{\sim}\mathrm{H}_{-(n+2r)}^{!,G}(\mathcal{Y}_{w})\otimes
H_{2r}^{!}(T).$

But $i_{\mathcal{Y},w}^{\ast}$ is the composition of the Künneth isomorphism
with

(158)
$\displaystyle\begin{split}\mathrm{H}_{-\ast}^{!,G}(\mathcal{Y}_{w})\otimes\mathrm{H}_{-\ast}^{!}(T)&\to\mathrm{H}_{-\ast}^{!,G}(\mathcal{Y}_{w})\otimes\mathrm{H}_{2r}^{!}(T)\\\
&\xrightarrow{\sim}\mathrm{H}_{-\ast}^{!,G}(\mathcal{Y}_{w})\otimes\mathrm{H}_{0}^{!}(\mathit{pt})\langle
2r\rangle\\\
&\xrightarrow{\sim}\mathrm{H}_{-\ast}^{!,G}(\mathcal{Y}_{w})\langle
2r\rangle,\end{split}$

so we win. ∎

######  Lemma 4.6.2.

The map $i_{\mathcal{Z},w}^{\dagger}$ is an isomorphism.

###### Chứng minh.

The maps $\mathcal{Y}_{w,0}\to\mathcal{Z}_{w,0}$ and
$\mathcal{Y}^{\prime}_{w,0}\to\mathcal{Z}^{\prime}_{w,0}$ define affine-space
bundles of rank $N_{w}$, so they induce pullback isomorphisms

(159) $\displaystyle\mathrm{H}_{-\ast}^{!,G}(\mathcal{Z}_{w})$
$\displaystyle\xrightarrow{\sim}\mathrm{H}_{-\ast}^{!,G}(\mathcal{Y}_{w})\langle-2N_{w}\rangle,$
(160) $\displaystyle\mathrm{H}_{-\ast}^{!,G}(\mathcal{Z}^{\prime}_{w})$
$\displaystyle\xrightarrow{\sim}\mathrm{H}_{-\ast}^{!,G}(\mathcal{Y}^{\prime}_{w})\langle-2N_{w}\rangle$

that preserve weights. Now we are done by the previous lemma. ∎

###### Proof of Theorem 1.

Fix a total order $\preceq$ on $W$ in which $w_{1}\preceq w_{2}$ implies
$|w_{1}|\leq|w_{2}|$, where $|{-}|$ denotes Bruhat length. Such a total order
is necessarily a refinement of the Bruhat partial order. For any $w\in W$, let

(161) $\displaystyle\mathcal{Z}_{\preceq w,0}$
$\displaystyle=\coprod_{x\preceq w}\mathcal{Z}_{x,0}\subseteq\mathcal{Z}_{0},$
(162) $\displaystyle\mathcal{Z}_{\prec w,0}$ $\displaystyle=\coprod_{x\prec
w}\mathcal{Z}_{x,0}\subseteq\mathcal{Z}_{0}.$

Then $\mathcal{Z}_{w,0}\to\mathcal{Z}_{\preceq w,0}$ is a closed immersion
with complement $\mathcal{Z}_{\prec w,0}\to\mathcal{Z}_{\preceq w,0}$. Let
$\mathcal{Z}^{\prime}_{\preceq w,0},\mathcal{Z}^{\prime}_{\prec w,0}$ be
defined similarly. The map
$i_{\mathcal{Z}}:\mathcal{Z}_{0}\to\mathcal{Z}^{\prime}_{0}$ gives rise to the
following commutative diagram whose rows are long exact sequences and whose
vertical arrows are pullbacks with restricted support:

(163)
${\cdots}$${\mathrm{H}_{n}^{!,G}(\mathcal{Z}^{\prime}_{w})}$${\mathrm{H}_{n}^{!,G}(\mathcal{Z}^{\prime}_{\preceq
w})}$${\mathrm{H}_{n}^{!,G}(\mathcal{Z}^{\prime}_{\prec
w})}$${\cdots}$${\cdots}$${\mathrm{H}_{n}^{!,G}(\mathcal{Z}_{w})\langle
2r\rangle}$${\mathrm{H}_{n}^{!,G}(\mathcal{Z}_{\preceq w})\langle
2r\rangle}$${\mathrm{H}_{n}^{!,G}(\mathcal{Z}_{\prec w})\langle
2r\rangle}$${\cdots}$$i_{\mathcal{Z},w}^{\ast}$$i_{\mathcal{Z},\preceq
w}^{\ast}$$i_{\mathcal{Z},\prec w}^{\ast}$

Since $(-)^{\dagger}$ is exact, we obtain another commutative diagram

(164)
${\cdots}$${\mathrm{H}_{n}^{!,G}(\mathcal{Z}^{\prime}_{w})^{\dagger}}$${\mathrm{H}_{n}^{!,G}(\mathcal{Z}^{\prime}_{\preceq
w})^{\dagger}}$${\mathrm{H}_{n}^{!,G}(\mathcal{Z}^{\prime}_{\prec
w})^{\dagger}}$${\cdots}$${\cdots}$${\mathrm{H}_{n}^{!,G}(\mathcal{Z}_{w})\langle
2r\rangle}$${\mathrm{H}_{n}^{!,G}(\mathcal{Z}_{\preceq w})\langle
2r\rangle}$${\mathrm{H}_{n}^{!,G}(\mathcal{Z}_{\prec w})\langle
2r\rangle}$${\cdots}$$i_{\mathcal{Z},w}^{\dagger}$$i_{\mathcal{Z},\preceq
w}^{\dagger}$$i_{\mathcal{Z},\prec w}^{\dagger}$

in which the top row remains exact.

We induct along $\preceq$. If $w=1$, then $i_{\mathcal{Z},\prec
w}^{\dagger}=i_{\mathcal{Z},w}^{\dagger}$. By Lemma 4.6.2, the latter is an
isomorphism in every homological degree $n$. Now let $w$ be arbitrary. If $x$
is the immediate predecessor of $w$ in the total order $\preceq$, then
$\mathcal{Z}_{\prec w}=\mathcal{Z}_{\preceq x}$ and $i_{\mathcal{Z},\prec
w}^{\dagger}=i_{\mathcal{Z},\preceq x}^{\dagger}$. Lemma 4.6.2 and the
inductive hypothesis respectively imply that $i_{\mathcal{Z},w}^{\dagger}$ and
$i_{\mathcal{Z},\prec w}^{\dagger}$ are isomorphisms in every degree. So, by
the five lemma, $i_{\mathcal{Z},\preceq w}^{\dagger}$ is an isomorphism in
every degree as well, completing the induction. ∎

##  5\. Steinberg Schemes of Braids

### 5.1.

Let $S\subseteq W$ be the set of simple reflections in the chosen Coxeter
presentation of $W$. Let $\mathit{Br}_{W}$ be the Artin braid group of $W$,
and let $\mathit{Br}_{W}^{+}\subseteq\mathit{Br}_{W}$ be the monoid of
positive braids determined by $S$. Each element $s\in S$ lifts to an element
$\sigma_{s}\in\mathit{Br}_{W}^{+}$, such that $\\{\sigma_{s}\\}_{s\in S}$ is a
generating set for $\mathit{Br}_{W}^{+}$. The following result is [D97,
§1.11].

######  Theorem 5.1.1 (Deligne).

There is a map from elements $\beta\in\mathit{Br}_{W}^{+}$ to
$G_{0}$-varieties $O(\beta)_{0}$ over $\mathcal{B}_{0}\times\mathcal{B}_{0}$
such that:

  1. (1)

$O(\sigma_{s})_{0}=O_{s,0}$ for all $s\in S$.

  2. (2)

If $\beta=\beta^{\prime}\beta^{\prime\prime}$, then there is a fixed
isomorphism

(165) $\displaystyle
O(\beta)_{0}\xrightarrow{\sim}O(\beta^{\prime})_{0}\times_{\mathcal{B}_{0}}O(\beta^{\prime\prime})_{0},$

where the fiber product is formed with respect to the right projection
$O(\beta^{\prime})_{0}\to\mathcal{B}_{0}$ and the left projection
$O(\beta^{\prime\prime})_{0}\to\mathcal{B}_{0}$. These isomorphisms are
associative in the sense that the diagram

(166)
${O(\beta^{\prime}\beta^{\prime\prime}\beta^{\prime\prime\prime})_{0}}$${O(\beta^{\prime}\beta^{\prime\prime})_{0}\times_{\mathcal{B}_{0}}O(\beta^{\prime\prime\prime})_{0}}$${O(\beta^{\prime})_{0}\times_{\mathcal{B}_{0}}O(\beta^{\prime\prime}\beta^{\prime\prime\prime})_{0}}$${O(\beta^{\prime})_{0}\times_{\mathcal{B}_{0}}O(\beta^{\prime\prime})_{0}\times_{\mathcal{B}_{0}}O(\beta^{\prime\prime\prime})_{0}}$$\sim$$\sim$$\sim$$\sim$

is commutative.

In particular, if $w=s_{1}\cdots s_{\ell}$ is a reduced expression, then

(167) $\displaystyle O(\sigma_{w})_{0}\simeq
O(\sigma_{s_{1}})_{0}\times_{\mathcal{B}_{0}}\cdots\times_{\mathcal{B}_{0}}O(\sigma_{s_{\ell}})_{0}\simeq
O_{w,0}$

by way of these fixed isomorphisms.

Deligne attributes the varieties $O(\beta)_{0}$ to Broué–Michel [BM97], though
they are implicit in Lusztig’s earlier papers on character sheaves. Note that
$\dim O(\beta)=|\beta|+N$, where $|\beta|$ is the writhe of $\beta$ (see
§A.2).

### 5.2.

Rouquier introduced an extension of Deligne’s result in which the full group
$\mathit{Br}_{W}$ replaces the monoid $\mathit{Br}_{W}^{+}$, and in which
objects of $\mathsf{H}_{W}$ replace varieties over
$\mathcal{B}_{0}\times\mathcal{B}_{0}$. The fiber product over
$\mathcal{B}_{0}$ is replaced by the convolution on $\mathsf{H}_{W}$.

Each element $s\in S$ gives rise to objects
$\mathcal{R}_{s}^{+},\mathcal{R}_{s}^{-}\in\mathsf{H}_{W}$ that can be
represented by complexes of the following form:

(170)
$\displaystyle\begin{array}[]{rcllllll}\mathcal{R}_{s}^{+}&=&0&\to&\underline{{\mathit{IC}_{s}}}&\to&\mathit{IC}_{1}\langle
1\rangle,\\\\[4.30554pt]
\mathcal{R}_{s}^{-}&=&\mathit{IC}_{1}\langle-1\rangle&\to&\underline{{\mathit{IC}_{s}}}&\to&0.\end{array}$

Above, the underlined terms are the terms in degree zero. To describe the
morphisms, factor $j_{1}:O_{1,0}\to\mathcal{B}_{0}\times\mathcal{B}_{0}$ as
the composition

(171) $\displaystyle