$\mathcal{L}_{\lambda}$ on $\hat{G}/\hat{B}$, and we may form the associated
$\hat{G}$-equivariant coherent complex
$A_{\lambda}=\pi_{\ast}\eta^{\ast}\mathcal{L}_{\lambda}\in\mathrm{Coh}(\mathcal{N}/\hat{G})$.
The $A_{\lambda}$’s are sometimes called _Andersen-Jantzen sheaves_ , and they
are very interesting objects in geometric representation theory. We refer the
reader to [Ach15] for a beautiful overview of this topic. Among the highlights
of this theory, we note that Bezrukavnikov [Bez03] proved that the
$A_{\lambda}$’s are _perverse_ with respected to a suitable perverse coherent
t-structure on $\mathrm{Coh}(\mathcal{N}/\hat{G})$, whose heart we denote
$\mathrm{PCoh}(\mathcal{N}/\hat{G})$. We also note that
$A_{0}=\mathcal{O}_{\mathcal{N}}$, and that for any $\lambda\in
X^{\ast}(\hat{T})^{+}$, $A_{\lambda}$ is an honest coherent sheaf, i.e. is
concentrated in cohomological degree zero. When $\lambda$ is dominant, it is
conventional to write $\overline{\nabla}_{\lambda}=A_{\lambda}$ and
$\overline{\Delta}_{\lambda}=A_{w_{0}(\lambda)}$. We also note that for
$\lambda$ dominant, there is a unique (up to scalar) nonzero map
$\overline{\Delta}_{\lambda}\to\overline{\nabla}_{\lambda}$ whose image in
$\mathrm{PCoh}$ is an irreducible object $IC_{\lambda}$, and that this recipe
gives all the irreducible objects in the heart of the perverse coherent
t-structure.

######  Conjecture 2.2.1.

For any $\lambda\in X_{\ast}(T)^{+}$, there are isomorphisms

$c_{\psi}(i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda})\simeq\nu_{\ast}\overline{\nabla}_{\lambda}\;\;\;\mathrm{and}\;\;\;c_{\psi}(i_{b_{\lambda}\sharp}^{\mathrm{ren}}\pi_{\lambda})\simeq\nu_{\ast}\overline{\Delta}_{\lambda}.$

Moreover, the functor $c_{\psi}$ sends the canonical map

$i_{b_{\lambda}\sharp}^{\mathrm{ren}}\pi_{\lambda}\to
i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}$

to the map obtained by applying $\nu_{\ast}$ to the canonical map
$\overline{\Delta}_{\lambda}\to\overline{\nabla}_{\lambda}$.

######  Remark 2.2.2.

It is not hard to see that $\nu_{\ast}\overline{\nabla}_{\lambda}$ and
$\nu_{\ast}\overline{\Delta}_{\lambda}$ are perfect complexes on
$\mathrm{Par}_{G}$, so we could also formulate an obvious variant of this
conjecture using the more humane functor $a_{\psi}$ instead of $c_{\psi}$.

This conjecture follows from the expected compatibilities of the categorical
conjecture with Eisenstein series and duality. To see this, recall that the
unramified component of $\mathrm{Par}_{T}$ is canonically
$\mathrm{Par}_{T}^{\mathrm{nr}}\cong\hat{T}\times B\hat{T}$. We have a line
bundle $\mathcal{O}_{\lambda}$ on $B\hat{T}$ corresponding to $\lambda$.
Pushing forward along the evident closed immersion $e:B\hat{T}\to\hat{T}\times
B\hat{T}\cong\mathrm{Par}_{T}^{\mathrm{nr}}\subset\mathrm{Par}_{T}$ gives a
coherent sheaf $e_{\ast}\mathcal{O}_{\lambda}$ on $\mathrm{Par}_{T}$. Under
the known categorical equivalence for $T$ [Zou21], it corresponds to the sheaf
$i_{\lambda!}\mathbf{1}$ on
$\mathrm{Bun}_{T}=\coprod_{X_{\ast}(T)}[\ast/T(E)]$ given by the $!$-extension
of the trivial representation from the component labelled by $\lambda$.

######  Proposition 2.2.3.

Notation and assumptions as above, we have an isomorphism
$\mathrm{Eis}_{B}^{\mathrm{spec}}(e_{\ast}\mathcal{O}_{\lambda})\cong\nu_{\ast}A_{\lambda}$
in $\mathrm{Coh}(\mathrm{Par}_{G})$ for any $\lambda$.

###### Proof sketch.

The key point is that there is commutative diagram of (derived) Artin stacks

$\textstyle{B\hat{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\textstyle{\hat{\mathfrak{u}}/\hat{B}\cong\tilde{\mathcal{N}}/\hat{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{q^{\mathrm{nil}}}$$\scriptstyle{\;\;\;\pi}$$\textstyle{\mathcal{N}/\hat{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{\mathrm{Par}_{T}}$$\textstyle{\mathrm{Par}_{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{\mathrm{spec}}}$$\scriptstyle{p^{\mathrm{spec}}}$$\textstyle{\mathrm{Par}_{G}}$

where the vertical maps are closed immersions, and moreover the lefthand
square is Cartesian and $q^{\mathrm{spec}}$ is flat in a neighborbood of
$\mathrm{im}e$.121212The hard part here is showing that the left-hand square
is actually Cartesian, and not just Cartesian on classical truncations, since
$\mathrm{Par}_{B}$ is genuinely a derived Artin stack. To verify this, one
needs to check that the derived structure of $\mathrm{Par}_{B}$ is trivial in
a neighborhood of $(q^{\mathrm{spec}})^{-1}(\mathrm{im}e)$. See Proposition
2.3.3 for a much more general statement. It is then straightforward to compute

$\displaystyle\mathrm{Eis}_{B}^{\mathrm{spec}}(e_{\ast}\mathcal{O}_{\lambda})$
$\displaystyle=p_{\ast}^{\mathrm{spec}}q^{\mathrm{spec}\ast}e_{\ast}\mathcal{O}_{\lambda}$
$\displaystyle\simeq
p_{\ast}^{\mathrm{spec}}f_{\ast}q^{\mathrm{nil}\ast}\mathcal{O}_{\lambda}$
$\displaystyle\simeq\nu_{\ast}\pi_{\ast}q^{\mathrm{nil}\ast}\mathcal{O}_{\lambda}$
$\displaystyle\simeq\nu_{\ast}A_{\lambda}$

where the second line follows from flat base change, the third line is
trivial, and the fourth line follows from the definition of $A_{\lambda}$ plus
the simple observation that
$q^{\mathrm{nil}\ast}\mathcal{O}_{\lambda}=\eta^{\ast}\mathcal{L}_{\lambda}$
as line bundles on $\tilde{\mathcal{N}}/\hat{G}$. ∎

######  Proposition 2.2.4.

Notation as above, we have an isomorphism
$\mathrm{Eis}_{B}(i_{\lambda!}\mathbf{1})\simeq
i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}$ for $\lambda$ dominant.

###### Proof.

This follows from Remark 1.4.8 plus a little thought. See also [Ham22,
Proposition 9.4] and the discussion immediately afterwards. ∎

Now we can put things together: since $e_{\ast}\mathcal{O}_{\lambda}$ matches
$i_{\lambda!}\mathbf{1}$ under the (known) categorical equivalence for $T$,
compatibility of the categorical equivalence with Eisenstein functors on both
sides _forces_ us to expect that for any dominant $\lambda$,
$\nu_{\ast}\overline{\nabla}_{\lambda}\simeq\mathrm{Eis}_{B}^{\mathrm{spec}}(e_{\ast}\mathcal{O}_{\lambda})$
should match
$i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}\simeq\mathrm{Eis}_{B}(i_{\lambda!}\mathbf{1})$
under the categorical conjecture for $G$. This gives the first isomorphism in
Conjecture 2.2.1. The second isomorphism follows from the expected
compatibility with duality. Indeed, one checks directly that
$\mathbf{D}_{\mathrm{twGS}}(\nu_{\ast}A_{\lambda})\cong\nu_{\ast}A_{w_{0}(\lambda)}[-\dim
T]$ for any $\lambda$ (this follows, for instance, from the arguments in
section 4 of [AH19]), and also that
$\mathbf{D}_{\mathrm{BZ}}(i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda})\cong
i_{b_{\lambda}\sharp}^{\mathrm{ren}}\pi_{\lambda}[-\dim T]$, which is an easy
consequence of Proposition 1.1.4. Conjecture 1.7.5 then implies the desired
isomorphism.

We can use Conjecture 2.2.1 to do some actual calculations. Let us illustrate
this in the simplest case $G=\mathrm{PGL}_{2}$. Here we identify
$X^{\ast}(\hat{T})^{+}=\mathbf{Z}_{\geq 0}$ in the usual way, and write
$\overline{\nabla}_{n}$ and $\overline{\Delta}_{n}$ for the associated
perverse coherent sheaves as defined above. Let
$b_{n}\in|\mathrm{Bun}_{\mathrm{PGL}_{2}}|$ be the point labelled by the
highest weight $n$, so $b_{n}$ corresponds to the image of
$\mathrm{diag}(\varpi^{n},1)$ in $B(\mathrm{PGL}_{2})$. Note that one
component of $\mathrm{Bun}_{\mathrm{PGL}_{2}}$ consists of the chain of
specializations $b_{0}\rightsquigarrow b_{2}\rightsquigarrow b_{4}\cdots$,
while the other component consists of the chain $b_{1/2}\rightsquigarrow
b_{1}\rightsquigarrow b_{3}\cdots$. Here $b_{1/2}$ is the “missing” point,
which won’t play any role in our discussion (since no sheaf supported at this
point can have trivial _L_ -parameter). When $n=0$, $G_{b_{0}}=G$ and
$\pi_{0}=i_{B}^{G}(\mathbf{1})$ is an irreducible principal series
representation. For $n\geq 1$, $G_{b_{n}}(E)=E^{\times}$ and the
representation $\pi_{n}$ is just the trivial representation. In particular,
the sheaf $i_{b_{n}!}^{\mathrm{ren}}\mathbf{1}$ corresponding to
$\nu_{\ast}\overline{\nabla}_{n}$ is very simple and explicit, and is
supported at one point. But how to calculate the stalks of the sheaf
$i_{b_{n}\sharp}^{\mathrm{ren}}\mathbf{1}$? It turns out that the categorical
conjecture lets us do this!

For this, recall that (as we already mentioned) there is a unique nonzero map
$\overline{\Delta}_{\lambda}\to\overline{\nabla}_{\lambda}$ whose image is an
irreducible perverse coherent sheaf $IC_{\lambda}$. Translating the diagram
$\nu_{\ast}\overline{\Delta}_{\lambda}\to\nu_{\ast}IC_{\lambda}\to\nu_{\ast}\overline{\nabla}_{\lambda}$
to the other side of the categorical conjecture, we now predict the existence
of a canonical indecomposable sheaf $\mathscr{F}_{\lambda}$ on
$\mathrm{Bun}_{G}$ admitting maps
$i_{b_{\lambda}\sharp}^{\mathrm{ren}}\pi_{\lambda}\to\mathscr{F}_{\lambda}\to
i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}$ whose composite is the canonical
map, and such that
$\mathbf{D}_{\mathrm{BZ}}\mathscr{F}_{\lambda}\simeq\mathscr{F}_{\lambda}[-\dim
T]$. In general, these sheaves are hard to calculate. However, for
$\mathrm{PGL}_{2}$, everything can be made very explicit. In particular, the
following results are known [Ach15, Section 5]:

a) there are isomorphisms
$\overline{\Delta_{0}}\cong\overline{\nabla}_{0}\cong
IC_{0}\cong\mathcal{O}_{\mathcal{N}/\hat{G}}$ and
$\overline{\Delta_{1}}\cong\overline{\nabla}_{1}\cong IC_{1}$

b) for $n\geq 2$ there are short exact sequences of perverse coherent sheaves

$0\to IC_{n}\to\overline{\nabla}_{n}\to\overline{\nabla}_{n-2}\to 0$

and

$0\to\overline{\Delta}_{n-2}\to\overline{\Delta}_{n}\to IC_{n}\to 0,$

and isomorphisms $IC_{n}\simeq i_{0\ast}V_{n-2}[-1]$, where
$i_{0}:B\hat{G}\to\mathcal{N}/\hat{G}$ is the inclusion of the closed orbit.

Note that the information in a) translates into isomorphisms
$i_{b_{0}\sharp}^{\mathrm{ren}}\pi_{0}\simeq i_{b_{0}!}^{\mathrm{ren}}\pi_{0}$
and $i_{b_{1}\sharp}^{\mathrm{ren}}\mathbf{1}\simeq
i_{b_{1}!}^{\mathrm{ren}}\mathbf{1}\simeq\mathscr{F}_{1}$. The first of these
is completely tautological, while the second is not tautological but follows a
priori from the fact that no sheaf supported at $b_{1/2}$ (which is the unique
generization of $b_{1}$) can have trivial _L-_ parameter.

Using the first sequence in b) we can now calculate the stalks of
$\mathscr{F}_{n}$ inductively via the distinguished triangles
$\mathscr{F}_{n}\to i_{b_{n}!}^{\mathrm{ren}}\pi_{n}\to
i_{b_{n-2}!}^{\mathrm{ren}}\pi_{n-2}\to$ which map to this sequence under the
categorical equivalence. The result is easy to discern:

######  Proposition 2.2.5.

For $n\geq 2$, the stalks of $\mathscr{F}_{n}$ vanish outside the points
$b_{n}$ and $b_{n-2}$.

For $n=2$ we have $i_{b_{2}}^{\ast}\mathscr{F}_{2}=\delta^{1/2}[-2]$ and
$i_{b_{0}}^{\ast}\mathscr{F}_{2}=\pi_{0}[-1]$.

For $n>2$ we have $i_{b_{n}}^{\ast}\mathscr{F}_{n}=\delta^{1/2}[-n]$ and
$i_{b_{n-2}}^{\ast}\mathscr{F}_{n}=\delta^{1/2}[1-n]$.

We can now use this information together with the triangles
$i_{b_{n-2}\sharp}^{\mathrm{ren}}\pi_{n-2}\to
i_{b_{n}\sharp}^{\mathrm{ren}}\pi_{n}\to\mathscr{F}_{n}\to$ corresponding to
the second sequence in b) to inductively calculate the stalks of
$i_{b_{n}\sharp}^{\mathrm{ren}}\mathbf{1}$. For even $n$, the outcome is the
following:

######  Proposition 2.2.6.

Assume $n\geq 2$ is even. The stalks of
$i_{b_{n}\sharp}^{\mathrm{ren}}\mathbf{1}$ vanish outside the points
$b_{n},b_{n-2},\dots,b_{2},b_{0}$. At these points, all stalk cohomologies are
zero except the following:

_i._
$H^{n}(i_{b_{n}}^{\ast}i_{b_{n}\sharp}^{\mathrm{ren}}\mathbf{1})=\delta^{1/2}$

_ii._ For $0<2j<n$,
$H^{2j}(i_{b_{2j}}^{\ast}i_{b_{n}\sharp}^{\mathrm{ren}}\mathbf{1})=H^{2j+1}(i_{b_{2j}}^{\ast}i_{b_{n}\sharp}^{\mathrm{ren}}\mathbf{1})=\delta^{1/2}$

_iii.
$H^{0}(i_{b_{0}}^{\ast}i_{b_{n}\sharp}^{\mathrm{ren}}\mathbf{1})=H^{1}(i_{b_{0}}^{\ast}i_{b_{n}\sharp}^{\mathrm{ren}}\mathbf{1})=i_{B}^{G}(\mathbf{1})$._

We encourage the reader to check this for themselves, and to formulate and
prove a similar result for $n\geq 3$ odd. As a notable consequence of these
calculations, we get the following suggestive result.

######  Proposition 2.2.7.

For all $n\geq 0$ the sheaves $i_{b_{n}\sharp}^{\mathrm{ren}}\pi_{n}$,
$i_{b_{n}!}^{\mathrm{ren}}\pi_{n}$, and $\mathscr{F}_{n}$ are hadal sheaves,
and the distinguished triangles

$\mathscr{F}_{n}\to i_{b_{n}!}^{\mathrm{ren}}\pi_{n}\to
i_{b_{n-2}!}^{\mathrm{ren}}\pi_{n-2}\to$

and

$i_{b_{n-2}\sharp}^{\mathrm{ren}}\pi_{n-2}\to
i_{b_{n}\sharp}^{\mathrm{ren}}\pi_{n}\to\mathscr{F}_{n}\to$

are short exact sequences of hadal sheaves. The hadal sheaf $\mathscr{F}_{n}$
is irreducible.

###### Proof.

For $n=0,1$ it is clear that these sheaves are hadal. For $n\geq 2$, the
calculation in the previous proposition shows that
$i_{b_{n}\sharp}^{\mathrm{ren}}\pi_{n}$ is coconnective for the hadal
t-structure, but we also know it is connective by definition. Thus
$i_{b_{n}\sharp}^{\mathrm{ren}}\pi_{n}$ is a hadal sheaf. Now from the first
triangle we get that all $\mathscr{F}_{n}$’s are coconnective for the hadal
t-structure, so then the map $i_{b_{n-2}\sharp}^{\mathrm{ren}}\pi_{n-2}\to
i_{b_{n}\sharp}^{\mathrm{ren}}\pi_{n}$ in the second triangle is an injection
of hadal sheaves, which then implies that $\mathscr{F}_{n}$ is hadal. Now the
first triangle implies that $i_{b_{n}!}^{\mathrm{ren}}\pi_{n}$ is hadal by an
easy induction on $n$. ∎

In [FS21, Remark I.10.3] one finds the suggestion that the categorical
conjecture might match the perverse t-structure on $\mathrm{Bun}_{G}$ with
some perverse coherent t-structure on $\mathrm{Coh}(\mathrm{Par}_{G})$. Since
the $IC_{\lambda}$’s are perverse coherent, this might naively lead one to
guess that the $\mathscr{F}_{\lambda}$’s should be perverse, but already for
$\mathrm{PGL}_{2}$ this is false. Indeed, we calculated above that
$i_{b_{0}}^{\ast}\mathscr{F}_{2}$ has nonzero cohomology in degree $1$, which
could not happen if $\mathscr{F}_{2}$ were perverse. However, this numerology
matches the fact that $IC_{2}\simeq i_{0\ast}V_{0}[-1]$ has a nonzero
cohomology sheaf in degree one only. We will argue later in these notes that
after localizing over a large (and explicit) open substack of
$\mathrm{Par}_{G}$, the categorical equivalence should be t-exact for the
perverse t-structure on $\mathrm{Bun}_{G}$ and the _naive_ t-structure on
$\mathrm{Coh}(\mathrm{Par}_{G})$. On the other hand, we also saw above that
$\mathscr{F}_{n}$ is a hadal sheaf. This suggests that perverse coherent
t-structures on $\mathrm{Coh}(\mathrm{Par}_{G}$) should be relevant, but that
they will be related to the _hadal_ t-structure on $\mathrm{Bun}_{G}$ instead.

We now make these ideas precise for the trivial parameter, again in the
context of a general split $G$.

######  Theorem 2.2.8.

Suppose that for all $\lambda\in X^{\ast}(\hat{T})^{+}$ we have isomorphisms

$a_{\psi}(\nu_{\ast}\overline{\Delta}_{\lambda})\simeq
i_{b_{\lambda}\sharp}^{\mathrm{ren}}\pi_{\lambda}\;\;\;\mathrm{and}\;\;\;a_{\psi}(\nu_{\ast}\overline{\nabla}_{\lambda})\simeq
i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}$

as in Remark 2.2.2. Then the following results are true.

_i._ The triangulated functor

$a_{\psi}\circ\nu_{\ast}:\mathrm{Coh}(\mathcal{N}/\hat{G})\to
D(\mathrm{Bun}_{G})_{\mathrm{fin}}$

is t-exact with respect to the perverse coherent t-structure on the left and
the hadal t-structure on the right. In particular, it induces an exact functor

$a_{\psi}\circ\nu_{\ast}:\mathrm{PCoh}(\mathcal{N}/\hat{G})\to\mathrm{Had}(\mathrm{Bun}_{G})_{\mathrm{fin}}.$

_ii._ The sheaves $i_{b_{\lambda}\sharp}^{\mathrm{ren}}\pi_{\lambda}$ and
$i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}$ are hadal.

_iii._ The image of the (unique up to scalar) nonzero map
$\overline{\Delta}_{\lambda}\to\overline{\nabla}_{\lambda}$ under the functor
$a_{\psi}\circ\nu_{\ast}$ is nonzero, and we have an isomorphism

$a_{\psi}(\nu_{\ast}IC_{\lambda})\simeq
i_{b_{\lambda}\sharp!}^{\mathrm{ren}}\pi_{\lambda}.$

In particular, the exact functor

$a_{\psi}\circ\nu_{\ast}:\mathrm{PCoh}(\mathcal{N}/\hat{G})\to\mathrm{Had}(\mathrm{Bun}_{G})_{\mathrm{fin}}$

is faithful, and sends irreducible objects to irreducible objects.

_iv._ Assume that the pairs $(b_{\lambda},\pi_{\lambda})$ exhaust all pairs
$(b,\pi)$ for which $i_{b!}^{\mathrm{ren}}\pi$ has trivial Fargues-Scholze
parameter. Then the sheaves $i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}$ are
perverse.

Note that the sheaves $\mathscr{F}_{\lambda}$ which we predicted earlier, and
calculated by hand for $\mathrm{PGL}_{2}$, are now revealed more clearly: they
are exactly the irreducible hadal sheaves
$i_{b_{\lambda}\sharp!}^{\mathrm{ren}}\pi_{\lambda}$.

We emphasize again that the _only_ assumption in this theorem is a set-
theoretic matching of objects, and this matching is _forced_ on us by the most
basic expectations regarding the categorical local Langlands conjecture. The
fact that this matching implies so much more is really some magic, related to
the remarkable properties of the hadal and perverse coherent t-structures.

###### Proof.

By [Bez03], the left half, resp. right half of the perverse coherent
t-structure is generated under extensions by objects of the form
$\overline{\Delta}_{\lambda}[n],\,n\geq 0$, resp. objects of the form
$\overline{\nabla}_{\lambda}[n],\,n\leq 0$. Our assumption now guarantees that
$a_{\psi}\circ\nu_{\ast}$ sends the left resp. right half into the left resp.
right half of the hadal t-structure. This gives i, and then ii. is an
immediate consequence of the fact that $\overline{\Delta}_{\lambda}$ and
$\overline{\nabla}_{\lambda}$ are perverse coherent.

For iii., let
$\gamma_{\lambda}:\overline{\Delta}_{\lambda}\to\overline{\nabla}_{\lambda}$
be a nonzero map. Pick a total ordering $<$ on $X^{\ast}(\hat{T})^{+}$
refining the usual partial ordering. By [Bez03], the cone
$\mathcal{K}_{\lambda}$ of the map $\gamma_{\lambda}$ lies in the triangulated
subcategory of $\mathrm{Coh}(\mathcal{N}/\hat{G})$ generated by
$\overline{\nabla}_{\mu}[m]$ for $m\in\mathbf{Z}$ and $\mu<\lambda$. This
implies that the $\ast$-stalks of the sheaf
$a_{\psi}(\nu_{\ast}\mathcal{K}_{\lambda})$ can only be nonzero at points
$b_{\mu}$ with $\mu<\lambda$, and in particular the $\ast$-stalk at
$b_{\lambda}$ vanishes. This implies that the first map in the distinguished
triangle

$a_{\psi}(\nu_{\ast}\overline{\Delta}_{\lambda})\simeq
i_{b_{\lambda}\sharp}^{\mathrm{ren}}\pi_{\lambda}\overset{(a_{\psi}\circ\nu_{\ast})(\gamma_{\lambda})}{\longrightarrow}a_{\psi}(\nu_{\ast}\overline{\nabla}_{\lambda})\simeq
i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}\to
a_{\psi}(\nu_{\ast}\mathcal{K}_{\lambda})\to$

cannot be zero, because otherwise we would get an isomorphism

$a_{\psi}(\nu_{\ast}\mathcal{K}_{\lambda})\simeq
i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}\oplus
i_{b_{\lambda}\sharp}^{\mathrm{ren}}\pi_{\lambda}[1],$

and the right-hand side here clearly has nonzero $\ast$-stalk at
$b_{\lambda}$. Therefore, the image of $\gamma_{\lambda}$ under
$a_{\psi}\circ\nu_{\ast}$ is the (unique up to scalar) nonzero map

$i_{b_{\lambda}\sharp}^{\mathrm{ren}}\pi_{\lambda}\overset{\delta_{\lambda}}{\to}i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}.$

Since we have already seen that the source and target of $\delta_{\lambda}$
are hadal sheaves, it is clear that
$\mathrm{im}\delta_{\lambda}=i_{b_{\lambda}\sharp!}^{\mathrm{ren}}\pi_{\lambda}$
by the definition of the latter as in Theorem 1.2.7. On the other hand,
$a_{\psi}\circ\nu_{\ast}$ is an exact functor of abelian categories, so it
preserves images, and therefore

$i_{b_{\lambda}\sharp!}^{\mathrm{ren}}\pi_{\lambda}=\mathrm{im}\delta_{\lambda}\simeq
a_{\psi}\circ\nu_{\ast}(\mathrm{im}\gamma_{\lambda})\simeq
a_{\psi}\circ\nu_{\ast}(IC_{\lambda}).$

This also implies that $a_{\psi}\circ\nu_{\ast}$ is conservative (since it
doesn’t send any irreducible object to the zero object), and thus faithful.

For iv., we already know that $i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}$
lies in the left half of the of the perverse t-structure. To see that it lies
in the right half, it is enough (by the exhaustion assumption in iv.) to check
that

$\mathrm{Hom}(i_{b_{\mu}!}^{\mathrm{ren}}\pi_{\mu}[n],i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda})=0$

for all $\mu$ and all $n>0$. We already know that
$i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}\in\phantom{}^{h}D^{\geq 0}$ in
general, but from ii. we also know that $i_{b_{\mu}!}^{\mathrm{ren}}\pi_{\mu}$
is hadal for all $\mu$, and therefore
$i_{b_{\mu}!}^{\mathrm{ren}}\pi_{\mu}[n]\in\phantom{}^{h}D^{\leq-1}$ for all
$n>0$. The desired $\mathrm{Hom}$-vanishing then follows from basic properties
of t-structures. ∎

######  Exercise 2.2.9.

1\. Show that for $\lambda,\mu$ dominant with $\lambda\neq\mu$, we have
$\mathrm{Hom}(i_{b_{\lambda}\sharp}^{\mathrm{ren}}\pi_{\lambda},i_{b_{\mu}!}^{\mathrm{ren}}\pi_{\mu})=0$.
(Hint: Use Proposition 1.1.2.iii.) Match this under the categorical conjecture
with Proposition 4.b) from [Bez03].

2\. Show that for $\lambda,\mu$ dominant with $\lambda\ngeq\mu$, we have
$\mathrm{Hom}(i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda},i_{b_{\mu}!}^{\mathrm{ren}}\pi_{\mu})=0$.
(Hint: Use Proposition 1.1.2.ii.) Match this under the categorical conjecture
with the vanishing portion of [Ach15, Theorem 4.12]. How does the “other part”
of [Ach15, Theorem 4.12] translate under the categorical conjecture?

3\. Assume that the pairs $(b_{\lambda},\pi_{\lambda})$ exhaust all pairs
$(b,\pi)$ for which $i_{b!}^{\mathrm{ren}}\pi$ has trivial Fargues-Scholze
parameter. Pick a total ordering $<$ on $X^{\ast}(\hat{T})^{+}$ refining the
usual partial ordering. Prove by hand that the collection
$\left\\{i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}\right\\}_{\lambda\in
X^{\ast}(\hat{T})^{+}}$ generates the triangulated category
$D(\mathrm{Bun}_{G})_{\mathrm{fin},\phi_{\mathrm{triv}}}$ and defines a
dualizable quasi-exceptional set within it, whose associated t-structure is
the hadal t-structure, and with dual quasi-exceptional set
$\left\\{i_{b_{\lambda}\sharp}^{\mathrm{ren}}\pi_{\lambda}\right\\}_{\lambda\in
X^{\ast}(\hat{T})^{+}}$.

4\. Assume that $G=\mathrm{GL}_{n}$. If $\lambda$ is minuscule, show that the
natural map $i_{b_{\lambda}\sharp}^{\mathrm{ren}}\pi_{\lambda}\to
i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}$ is an isomorphism. Observe that
this is _predicted_ by the categorical conjecture for all $G$: for $\lambda$
minuscule, the natural map
$\overline{\Delta}_{\lambda}\to\overline{\nabla}_{\lambda}$ is always an
isomorphism (see e.g. [Ach15, Proof of Prop. 3.8]). (Hint: If $b$ is a proper
generization of $b_{\lambda}$, show that $G_{b}$ is not quasisplit. Deduce
that no sheaf supported at $b$ can have trivial _L-_ parameter, using the
known compatibility of the FS correspondence with the classical LLC for inner
forms of $\mathrm{GL}_{n}$ [HKW22].)

######  Remark 2.2.10.

Let $i_{0}:B\hat{G}\to\mathcal{N}/\hat{G}$ be the inclusion of the closed
orbit. It is a general fact [Ach15, Proposition 3.9] that for any dominant
$\lambda$, $IC_{\lambda+2\rho}\simeq i_{0\ast}V_{\lambda}[-\dim U]$.131313This
is more or less the only part of the Lusztig-Vogan bijection which can be
explicitly understood. In the particular case of $G=\mathrm{PGL}_{2}$, this
reduces to isomorphisms $IC_{n}\simeq i_{0\ast}V_{n-2}[-1]$ for all $n\geq 2$,
as we’ve already noted. Writing $\mathcal{O}(\hat{G})\simeq\oplus_{n\geq
2}V_{n-2}^{\oplus n-1}$, we formally see that
$\nu_{\ast}i_{0\ast}\mathcal{O}(\hat{G})\simeq\oplus_{n\geq
2}\nu_{\ast}IC_{n}^{\oplus n-1}[1]$, so translating to the other side we get
that

$\displaystyle a_{\psi}(\nu_{\ast}i_{0\ast}\mathcal{O}(\hat{G}))$
$\displaystyle\simeq\oplus_{n\geq 2}a_{\psi}(IC_{n})^{\oplus n-1}[1]$
$\displaystyle\simeq\oplus_{n\geq 2}\mathscr{F}_{n}^{\oplus n-1}[1]$

should be a Hecke eigensheaf with trivial eigenvalue. For general $G$, similar
reasoning leads formally to the expectation that
$\mathscr{G}=\oplus_{\lambda\in
X^{\ast}(\hat{T})^{+}}\mathscr{F}_{\lambda+2\rho}^{\oplus\mathrm{dim}V_{\lambda}}$
should be a Hecke eigensheaf with trivial eigenvalue.

It seems that much more can be said about the sheaves $\mathscr{F}_{\lambda}$.
We already noted above that there are isomorphisms
$\mathbf{D}_{\mathrm{BZ}}\mathscr{F}_{\lambda}\simeq\mathscr{F}_{\lambda}[-\dim
T]$. However, a heuristic argument with compactified Eisenstein series
suggests that for $\lambda\in 2\rho+X^{\ast}(\hat{T})^{+}$, we should also
have isomorphisms
$\mathbf{D}_{\mathrm{Verd}}\mathscr{F}_{\lambda}\simeq\mathscr{F}_{\lambda}[2\mathrm{dim}U]$,
and $\mathscr{F}_{\lambda}[\dim U]$ should be an irreducible perverse sheaf.
We will come back to this later in the notes. For now, we note that the
special case of $\mathrm{PGL}_{2}$ can be understood directly using our
previous calculations. Indeed, recall from earlier the natural map
$i_{b_{n}!}^{\mathrm{ren}}\pi_{n}\to i_{b_{n-2}!}^{\mathrm{ren}}\pi_{n-2}$,
which is a surjection of hadal sheaves with kernel $\mathscr{F}_{n}$. However,
the source and target of this map are also perverse (by Theorem 2.2.8.iv), and
by some general nonsense this same map must be an _injection_ of perverse
sheaves, with cokernel in the perverse category given by $\mathscr{F}_{n}[1]$!
This immediately implies that $\mathscr{F}_{n}[1]$ is perverse, and a small
additional argument identifies it with the intermediate extension
$i_{b_{n-2}!\ast}^{\mathrm{ren}}\pi_{n-2}$. In other words, we have short
exact sequences of perverse sheaves

$0\to i_{b_{n+2}!}^{\mathrm{ren}}\pi_{n+2}\to
i_{b_{n}!}^{\mathrm{ren}}\pi_{n}\to i_{b_{n}!\ast}^{\mathrm{ren}}\pi_{n}\to 0$

for all $n\geq 0$. This implies that the perverse sheaf
$i_{b_{n}!}^{\mathrm{ren}}\pi_{n}$ is uniserial of infinite length, with
Jordan-Holder series

$\cdots
i_{b_{n+6}!\ast}^{\mathrm{ren}}\pi_{n+6}-i_{b_{n+4}!\ast}^{\mathrm{ren}}\pi_{n+4}-i_{b_{n+2}!\ast}^{\mathrm{ren}}\pi_{n+2}-i_{b_{n}!\ast}^{\mathrm{ren}}\pi_{n}.$

Note also that $i_{b_{n}!}^{\mathrm{ren}}\pi_{n}$ has infinite length as a
perverse sheaf, despite clearly being an object of
$D(\mathrm{Bun}_{G})_{\mathrm{fin}}$. Finally, taking the Verdier dual of the
previous short exact sequence and using the self-duality of intermediate
extensions, we get short exact seqences of perverse sheaves

$0\to i_{b_{n}!\ast}^{\mathrm{ren}}\pi_{n}\to
i_{b_{n}\ast}^{\mathrm{ren}}\pi_{n}\to
i_{b_{n+2}\ast}^{\mathrm{ren}}\pi_{n+2}\to 0$

for all $n\geq 0$. This immediately implies that
$i_{b_{n}\ast}^{\mathrm{ren}}\pi_{n}$ is uniserial of infinite length, with
the same perverse Jordan-Holder factors as $i_{b_{n}!}^{\mathrm{ren}}\pi_{n}$
but arranged in the opposite order. By a simple induction, this triangle also
lets us easily compute the $\ast$-stalks of the sheaves
$i_{b_{n}\ast}^{\mathrm{ren}}\pi_{n}$.

######  Proposition 2.2.11.

For all $n\geq 0$ and $j>0$, we have

$i_{b_{n+2j}}^{\ast\mathrm{ren}}i_{b_{n}\ast}^{\mathrm{ren}}\pi_{n}\simeq\pi_{n+2j}\oplus\pi_{n+2j}[1].$

###  2.3 Semisimple generic parameters

In this section we sketch our expectations regarding the maximal common
generalizations of the discussion in section 2.1 and 2.2. We again allow any
quasisplit $G$. It will be convenient to adopt the following notation: If $H$
is any linear algebraic group over $\overline{\mathbf{Q}_{\ell}}$, we have the
associated Artin stack $\mathcal{N}_{H}=\mathrm{Lie}(H)^{\mathrm{nil}}/H$
where $\mathrm{Lie}(H)^{\mathrm{nil}}$ is the nilpotent cone. If the connected
component $H^{\circ}$ is reductive, the obvious map
$\mathcal{N}_{H^{\circ}}\to\mathcal{N}_{H}$ is finite etale, and it is not
hard to see that that we can define a perverse coherent t-structure on
$\mathrm{Coh}(\mathcal{N}_{H})$ whose heart is exactly the objects in
$\mathrm{Coh}(\mathcal{N}_{H})$ whose pullback to
$\mathrm{Coh}(\mathcal{N}_{H^{\circ}})$ is perverse coherent.

Let $\phi$ be a semisimple parameter, $x_{\phi}\in X_{G}^{\mathrm{spec}}$ the
associated closed point. To begin, observe that there is a canonical closed
immersion $\nu_{\phi}:\mathcal{N}_{S_{\phi}}\to\mathrm{Par}_{G}$ factoring
over the closed substack $q^{-1}(x_{\phi})$, which parametrizes exactly the
_L_ -parameters with open kernel whose Frobenius-semisimplification is $\phi$.
We omit the proof, which is an easy exercise with the Jordan-Chevalley
decomposition.

######  Proposition 2.3.1.

The following conditions on a semisimple parameter $\phi$ are equivalent.

1) The morphism $\nu_{\phi}$ factors through an isomorphism
$\mathcal{N}_{S_{\phi}}\overset{\sim}{\to}q^{-1}(x_{\phi})$.

2) The parameter $\phi$ is _generic_ , i.e. the adjoint L-function
$L(s,\mathrm{ad}\circ\phi)$ is regular at $s=1$.

3) The composite map
$BS_{\phi}\to\mathcal{N}_{S_{\phi}}\overset{\nu_{\phi}}{\to}\mathrm{Par}_{G}$
defines a smooth point of $\mathrm{Par}_{G}$.

4) The stack $\mathrm{Par}_{G}$ is smooth in a Zariski neighborhood of
$q^{-1}(x_{\phi})$.

###### Proof.

Omitted. We note that condition 2) may seem to depend a priori on a choice of
isomorphism $\mathbf{C}\simeq\overline{\mathbf{Q}_{\ell}}$, and we encourage
the reader to convince themselves that it is actually independent of any such
choice. ∎

######  Example 2.3.2.

Suppose $G=\mathrm{GL}_{n}$. Then an _L-_ parameter $\phi$ is semisimple
generic iff it is a direct sum $\phi\simeq\phi_{1}\oplus\cdots\oplus\phi_{d}$
where the $\phi_{i}$’s are supercuspidal parameters with
$\sum_{i}\mathrm{dim}\phi_{i}=n$, and with $\phi_{i}\not\simeq\phi_{j}(1)$ for
any $i\neq j$. However, unlike in the generous case, we allow the $\phi_{i}$’s
to occur with arbitrary multiplicities. Note that a semisimple _L-_ parameter
$\phi$ for $\mathrm{GL}_{n}$ is generic _if and only if_ the associated
$\mathrm{GL}_{n}(E)$-representation $\pi_{\phi}$ is an irreducible parabolic
induction of a supercuspidal representation on a Levi subgroup.

It is not hard to see that the locus
$X_{G}^{\mathrm{spec},\mathrm{gen}}\subset X_{G}^{\mathrm{spec}}$
parametrizing semisimple generic parameters is open and dense, and contains
all generous parameters. Here is another key simplifying feature associated
with these parameters.

######  Proposition 2.3.3.

Choose any parabolic $P=MU\subset G$, so we have the natural composite map
$\mathrm{Par}_{P}\overset{p^{\mathrm{spec}}}{\to}\mathrm{Par}_{G}\overset{q}{\to}X_{G}^{\mathrm{spec}}$.
Let

$\mathrm{Par}_{P}^{G-\mathrm{gen}}=\mathrm{Par}_{P}\times_{X_{G}^{\mathrm{spec}}}X_{G}^{\mathrm{spec,gen}}\subset\mathrm{Par}_{P}$

be the evident open substack. Then the derived structure on
$\mathrm{Par}_{P}^{G-\mathrm{gen}}$ is trivial.

###### Proof.

By the calculations in [Zhu21, Section 2.3], we know that
$q^{\mathrm{spec}}:\mathrm{Par}_{P}\to\mathrm{Par}_{M}$ is a quasismooth
morphism of quasismooth derived stacks with classical target, and where the
source and target both have virtual (equi)dimension zero. By Proposition
B.0.1, it is then enough to prove the following: for every closed point
$x_{\phi}\in X_{M}^{\mathrm{spec}}$ whose image in $X_{G}^{\mathrm{spec}}$
lies in $X_{G}^{\mathrm{spec,gen}}$, with associated closed immersion
$\nu_{\phi}^{M}:\mathcal{N}_{S_{\phi}}\simeq(q^{M})^{-1}(x_{\phi})\hookrightarrow\mathrm{Par}_{M}$,
the natural map on classical truncations
$\mathrm{Par}_{P}^{\mathrm{cl}}\times_{\mathrm{Par}_{M},\nu_{\phi}^{M}}\mathcal{N}_{S_{\phi}}\to\mathcal{N}_{S_{\phi}}$
is smooth of relative dimension zero. (Here
$S_{\phi}=\mathrm{Cent}_{\hat{M}}(\phi)$ is computed relative to the Levi.)
This follows in turn by explicitly identifying this fiber product with
$\mathcal{N}_{S_{\phi}^{P}}$ where
$S_{\phi}^{P}=\mathrm{Cent}_{\hat{P}}(\phi)$, and then using the fact that for
any surjection $H\to G$ of linear algebraic groups with unipotent kernel, the
associated map $\mathcal{N}_{H}\to\mathcal{N}_{G}$ is smooth of relative
dimension zero. ∎

######  Remark 2.3.4.

It is probably true that for most groups, $X_{G}^{\mathrm{spec,gen}}$ is the
maximal open subvariety of $X_{G}^{\mathrm{spec}}$ whose preimage in
$\mathrm{Par}_{P}$ has trivial derived structure for all parabolics $P\subset
G$.

Let $\phi$ be a semisimple generic parameter, so we have maps
$BS_{\phi}\overset{i_{0}}{\to}\mathcal{N}_{S_{\phi}}\overset{\nu_{\phi}}{\to}\mathrm{Par}_{G}$.
Here we write $i_{0}$ for the inclusion of the zero orbit, so
$i_{\phi}=\nu_{\phi}\circ i_{0}:BS_{\phi}\to\mathrm{Par}_{G}$ is the closed
orbit of the semisimple parameter $\phi$ just as in the discussion of generous
parameters. For any $\rho\in\mathrm{Irr}(S_{\phi})$, we set
$\delta_{\rho}=i_{0\ast}\rho\in\mathrm{Coh}(\mathcal{N}_{S_{\phi}})$, so the
pushforward $\nu_{\phi\ast}\delta_{\rho}=i_{\phi\ast}\rho$ is still a perfect
complex on $\mathrm{Par}_{G}$. However, there should be several other sheaves
in the picture, reflecting the more complicated geometry of $q^{-1}(x_{\phi})$
and our experience with the trivial _L_ -parameter. More precisely, we expect
that there is a clean and explicit recipe which assigns to any
$\rho\in\mathrm{Irr}(S_{\phi})$ three canonical perverse-coherent objects
$\overline{\Delta}_{\rho},IC_{\rho},\overline{\nabla}_{\rho}\in\mathrm{PCoh}(\mathcal{N}_{S_{\phi}})$
together with maps

$\overline{\Delta}_{\rho}\twoheadrightarrow
IC_{\rho}\hookrightarrow\overline{\nabla}_{\rho}$

realizing $IC_{\rho}$ as the socle of $\overline{\nabla}_{\rho}$ and the
cosocle of $\overline{\Delta}_{\rho}$. Moreover, $IC_{\rho}$ should be
irreducible, and $\overline{\nabla}_{\rho}$ should be a genuine coherent
sheaf. These sheaves should interpolate the following properties.

1\. When $\phi$ is generous, $\mathcal{N}_{S_{\phi}}\cong BS_{\phi}$ and
$\overline{\Delta}_{\rho}=IC_{\rho}=\overline{\nabla}_{\rho}=\delta_{\rho}=\rho$.

2\. When $S_{\phi}$ is connected, $\mathrm{Irr}(S_{\phi})$ is parametrized by
highest weights and
$\overline{\Delta}_{\rho},IC_{\rho},\overline{\nabla}_{\rho}$ should be the
standard, resp. irreducible, resp. costandard perverse-coherent objects as in
the discussion preceding Conjecture 2.2.1.

3\. When $\rho$ factors over the quotient $S_{\phi}\to S_{\phi}^{\natural}$,
$\overline{\Delta}_{\rho}=IC_{\rho}=\overline{\nabla}_{\rho}$ should be the
pullback of $\rho$ along the tautological map $\mathcal{N}_{S_{\phi}}\to
BS_{\phi}$.

4\. For any $\rho$, there should be a uniquely determined $\rho^{\prime}$ such
that $\delta_{\rho}\simeq IC_{\rho^{\prime}}[d]$, where
$d=\mathrm{dim}U_{S_{\phi}^{\circ}}$ for $U_{S_{\phi}^{\circ}}$ the unipotent
radical of a(ny) Borel subgroup of $S_{\phi}^{\circ}$.

5\. There should be a surjection of coherent sheaves
$\overline{\nabla}_{\rho}\twoheadrightarrow\delta_{\rho}$.

Finally, these sheaves should have the property that if $(b,\pi)$ is the pair
matching $(\phi,\rho)$ under the BM-O bijection, then the categorical
equivalence induces the following matchings of sheaves ($c_{\psi}$ in one
direction, $a_{\psi}$ in the other)

$\displaystyle\nu_{\phi\ast}\overline{\Delta}_{\rho}$
$\displaystyle\longleftrightarrow i_{b\sharp}^{\mathrm{ren}}\pi$
$\displaystyle\nu_{\phi\ast}IC_{\rho}$ $\displaystyle\longleftrightarrow
i_{b\sharp!}^{\mathrm{ren}}\pi$
$\displaystyle\nu_{\phi\ast}\overline{\nabla}_{\rho}$
$\displaystyle\longleftrightarrow i_{b!}^{\mathrm{ren}}\pi$
$\displaystyle\nu_{\phi\ast}\delta_{\rho}=i_{\phi\ast}\rho$
$\displaystyle\longleftrightarrow i_{b!\ast}^{\mathrm{ren}}\pi$

compatible with the evident maps. The first three sheaves on the right should
be hadal, and the last two should be perverse. The surjection
$\nu_{\phi\ast}\overline{\nabla}_{\rho}\twoheadrightarrow\nu_{\phi\ast}\delta_{\rho}$
from 5. above should correspond to the natural surjection
$i_{b!}^{\mathrm{ren}}\pi\twoheadrightarrow i_{b!\ast}^{\mathrm{ren}}\pi$ of
perverse sheaves. The isomorphism $\delta_{\rho}\simeq IC_{\rho^{\prime}}[d]$
from 4. should correspond to an isomorphism
$i_{b!\ast}^{\mathrm{ren}}\pi\simeq
i_{b^{\prime}\sharp!}^{\mathrm{ren}}\pi^{\prime}[d]$, where
$(b^{\prime},\pi^{\prime})$ is the pair matching $(\phi,\rho^{\prime})$ under
the BM-O bijection. In particular, we see that irreducible perverse sheaves
over a semisimple generic parameter should be finite sheaves.

If we believe in this matching, it is also not hard to see that Hecke
operators should be t-exact for the hadal t-structure on
$D(\mathrm{Bun}_{G})_{\mathrm{fin},\phi}$. Indeed,
$T_{V}i_{b\sharp}^{\mathrm{ren}}\pi$ matches with
$\nu_{\phi\ast}(V\otimes\overline{\Delta}_{\rho})$, and
$V\otimes\overline{\Delta}_{\rho}$ is still perverse coherent, so by general
nonsense about t-structures associated with quasiexceptional collections it
will have a finite filtration with graded pieces of the form
$\overline{\Delta}_{\rho_{i}}[n_{i}]$ for some $n_{i}\geq 0$. Passing back to
the other side, we see that $T_{V}i_{b\sharp}^{\mathrm{ren}}\pi$ should have a
finite filtration with graded pieces of the form
$i_{b_{i}\sharp}^{\mathrm{ren}}\pi_{i}[n_{i}]$, which are connective for the
hadal t-structure. This shows that $T_{V}$ is right t-exact, and a similar
argument gives left t-exactness.

######  Remark 2.3.5.

It is natural to wonder how Bernstein-Zelevinsky duality interacts with the
hadal t-structure. Let us say that a semisimple _L_ -parameter $\phi$ is
_cohomologically inert_ if there is a fixed (nonnegative) integer $d_{\phi}$
such that for all $b\in B(G)$ and all $\pi\in\Pi(G_{b})$ with Fargues-Scholze
parameter $\phi$,
$\mathbf{D}_{\mathrm{coh}}(\pi)\simeq\mathrm{Zel}(\pi)[-d_{\phi}]$. Here
$\mathrm{Zel}(-)$ denotes the Aubert-Zelevinsky involution on $\Pi(G_{b})$ as
in Remark 1.1.5. It is easy to see that if $\phi$ is cohomologically inert,
then also $\phi^{\vee}$ is cohomologically inert with
$d_{\phi}=d_{\phi^{\vee}}$, and one can also show that
$\mathbf{D}_{\mathrm{BZ}}(-)[d_{\phi}]$ restricts to an exact anti-equivalence
of abelian categories

$\displaystyle\mathrm{Had}(\mathrm{Bun}_{G})_{\mathrm{fin},\phi}$
$\displaystyle\overset{\sim}{\to}\mathrm{Had}(\mathrm{Bun}_{G})_{\mathrm{fin},\phi^{\vee}}$
$\displaystyle A$ $\displaystyle\mapsto\mathbf{D}_{\mathrm{BZ}}(A)[d_{\phi}]$

sending the irreducible hadal sheaf $\mathscr{G}_{b,\pi}$ to the irreducible
hadal sheaf $\mathscr{G}_{b,\mathrm{Zel}(\pi)}$. One can also show that this
result is best possible: if $\phi$ is not cohomologically inert, then no fixed
shift of $\mathbf{D}_{\mathrm{BZ}}(-)$ can induce such an equivalence.
Finally, all evidence and examples point to the speculation that a semisimple
parameter is cohomologically inert if and only if it is generic.

###  2.4 Perverse t-exactness of Hecke operators

In the previous section we pointed out that Hecke operators should be hadal
t-exact on $\phi$-local finite sheaves when $\phi$ is semisimple generic. It
is also natural to wonder how the Hecke action interacts with the perverse
t-structure. Our main goal in this section is to justify the following
conjecture.

######  Conjecture 2.4.1.

If $\phi$ is a semisimple generic parameter, the Hecke action of
$\mathrm{Rep}(\hat{G})$ on $D(\mathrm{Bun}_{G})_{\phi}^{\mathrm{ULA}}$ is
perverse t-exact.

When $\phi$ is supercuspidal, this conjecture was essentially formulated in
Fargues-Scholze. For generous parameters, this conjecture is a straightforward
consequence of Conjecture 2.1.8, as discussed in the proof of Proposition
2.1.14. For general semisimple generic parameters, however, there is no easy
evidence in its favor. Nevertheless, we will rigorously show that when $\phi$
is the trivial parameter, Conjecture 2.2.1 actually implies Conjecture 2.4.1!
A very similar argument will apply to all semisimple generic parameters, once
the ideas sketched in section 2.3 are more thoroughly developed.

The essential point on the spectral side is the following result. Here we
return to the notation and setup of section 2.2; in particular we assume $G$
is split.

######  Proposition 2.4.2.

For any $\lambda\in X^{\ast}(\hat{T})^{+}$ and any
$V\in\mathrm{Rep}(\hat{G})$, the sheaf $V\otimes
A_{\lambda}\in\mathrm{Coh}(\mathcal{N}/\hat{G})$ admits a finite filtration
with graded pieces of the form $A_{\nu}[n]$ for some $\nu\in
X^{\ast}(\hat{T})^{+}$ and $n\geq 0$.

Despite its innocent nature, I was only able to prove this by using the full
power of the Arkhipov-Bezrukavnikov equivalence.141414It is quite striking
that in order to prove something about the Hecke action on
$D(\mathrm{Bun}_{G})$, we will first go to the spectral side via the
categorical equivalence, and then pass through the Langlands mirror _again_
via the AB equivalence. To prepare for the argument, recall that we have a
diagram of functors between triangulated categories

$\textstyle{\mathrm{Coh}(\tilde{\mathcal{N}}/\hat{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\ast}}$$\textstyle{D^{b}\mathrm{ExCoh}(\tilde{\mathcal{N}}/\hat{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\ast}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{F_{\mathcal{IW}}}$$\textstyle{D^{b}\mathrm{Perv}_{\mathcal{IW}}(\mathrm{Fl}_{\mathbf{G}},\overline{\mathbf{Q}_{\ell}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{D_{\mathcal{IW}}^{b}(\mathrm{Fl}_{\mathbf{G}},\overline{\mathbf{Q}_{\ell}})}$$\textstyle{\mathrm{Coh}(\mathcal{N}/\hat{G})}$$\textstyle{D^{b}\mathrm{PCoh}(\mathcal{N}/\hat{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{D^{b}\mathrm{Perv}_{I}(\mathrm{Fl}_{\mathbf{G}},\overline{\mathbf{Q}_{\ell}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Av}_{\mathcal{IW}}}$$\textstyle{D_{I}^{b}(\mathrm{Fl}_{\mathbf{G}},\overline{\mathbf{Q}_{\ell}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Av}_{\mathcal{IW}}}$

where all but one horizontal arrow is an equivalence of categories. Here
$\mathbf{G}/\overline{\mathbf{F}_{p}}$ is a split reductive group equipped
with a fixed identification of dual groups $\hat{\mathbf{G}}\cong\hat{G}$ over
$\overline{\mathbf{Q}_{\ell}}$, and all our remaining notation essentially
follows the book [AR23], which we will refer to heavily. Aside from notation
we have already seen in section 2.2, we recall that
$\mathrm{ExCoh}(\tilde{\mathcal{N}}/\hat{G})$ denotes the heart of the
“exotic” t-structure on equivariant coherent sheaves on the Springer
resolution, and $F_{\mathcal{IW}}$ denotes the Arkhipov-Bezrukavnikov
equivalence. All unlabelled horizontal arrows are induced by the usual
realization functors, and we will elide them in our notation; this should
cause no confusion. We also recall that $F_{\mathcal{IW}}$ is t-exact for the
exotic and perverse t-structures, and defines an exact equivalence of abelian
categories between $\mathrm{Ex}\mathrm{Coh}$ and
$\mathrm{Perv}_{\mathcal{IW}}$. Moreover, $\pi_{\ast}$ is t-exact for the
exotic and perverse coherent t-structures. Finally,
$\mathrm{Av}_{\mathcal{IW}}$ is t-exact for the evident perverse t-structures.

Each of the abelian categories $\mathrm{PCoh}$, $\mathrm{ExCoh}$,
$\mathrm{Perv}_{\mathcal{IW}}$, $\mathrm{Perv}_{I}$ is equipped with a
canonical collection of costandard objects. For $\mathrm{PCoh}$ these are
exactly the sheaves $A_{\lambda}=\overline{\nabla}_{\lambda}$, $\lambda\in
X^{\ast}(\hat{T})^{+}$, which we have seen in section 2.2. For
$\mathrm{ExCoh}$ and $\mathrm{Perv}_{\mathcal{IW}}$ the indexing set for the
costandard objects is the set of all $\mu\in X^{\ast}(\hat{T})$, and we write
$\nabla_{\mu}^{\mathrm{ex}}\in\mathrm{ExCoh}$ resp.
$\nabla_{\mu}^{\mathcal{IW}}\in\mathrm{Perv}_{\mathcal{IW}}$ for the
associated costandard objects. For $\mathrm{Perv}_{I}$ the indexing set for
costandard objects is the extended affine Weyl group $\widetilde{W}=W\ltimes
X^{\ast}(\hat{T})$, and we write $\nabla_{w}^{I}$ for the costandard object
corresponding to an element $w\in\widetilde{W}$.

For $\mathcal{C}$ any one of these four abelian categories, we write
$D_{+}^{b}\mathcal{C}$ for the full (but not triangulated!) subcategory of
$D^{b}\mathcal{C}$ spanned by objects which admit a finite filtration with
graded pieces of the form $A[n]$ where $A$ is costandard and $n\geq 0$. We
also write
$D_{\mathcal{IW}}^{b}(\mathrm{Fl}_{\mathbf{G}},\overline{\mathbf{Q}_{\ell}})_{+}$
and $D_{I}^{b}(\mathrm{Fl}_{\mathbf{G}},\overline{\mathbf{Q}_{\ell}})_{+}$ for
the full subcategories spanned by objects admitting finite filtrations whose
graded pieces are nonnegative shifts of costandard objects.

######  Proposition 2.4.3.

The functors $\pi_{\ast}$, $F_{\mathcal{IW}}$, and
$\mathrm{Av}_{\mathcal{IW}}$, together with the realization functors, induce a
commutative diagram of functors

$\textstyle{D_{+}^{b}\mathrm{ExCoh}(\tilde{\mathcal{N}}/\hat{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\ast}}$$\scriptstyle{\sim}$$\scriptstyle{F_{\mathcal{IW}}}$$\textstyle{D_{+}^{b}\mathrm{Perv}_{\mathcal{IW}}(\mathrm{Fl}_{\mathbf{G}},\overline{\mathbf{Q}_{\ell}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{D_{\mathcal{IW}}^{b}(\mathrm{Fl}_{\mathbf{G}},\overline{\mathbf{Q}_{\ell}})_{+}}$$\textstyle{D_{+}^{b}\mathrm{PCoh}(\mathcal{N}/\hat{G})}$$\textstyle{D_{+}^{b}\mathrm{Perv}_{I}(\mathrm{Fl}_{\mathbf{G}},\overline{\mathbf{Q}_{\ell}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Av}_{\mathcal{IW}}}$$\textstyle{D_{I}^{b}(\mathrm{Fl}_{\mathbf{G}},\overline{\mathbf{Q}_{\ell}})_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Av}_{\mathcal{IW}}}$

where both upper horizontal arrows are equivalences.

###### Proof.

Recall that all three functors in the left half of the diagram come from exact
functors on the evident abelian categories. For $\pi_{\ast}$ the claim follows
from the fact that
$\pi_{\ast}\nabla_{\mu}^{\mathrm{ex}}\simeq\overline{\nabla}_{\mathrm{dom}(\mu)}$
for all $\mu$, where $\mathrm{dom}(\mu)$ is the unique dominant weight in the
$W$-orbit of $\mu$ [AR23, Lemma 7.3.10]. For $F_{\mathcal{IW}}$ the claim
follows from the fact that
$F_{\mathcal{IW}}(\nabla_{\mu}^{\mathrm{ex}})\simeq\nabla_{\mu}^{\mathcal{IW}}$
for all $\mu$ [AR23, Proposition 7.1.5]. Finally, for
$\mathrm{Av}_{\mathcal{IW}}$ the claim follows from the fact that
$\mathrm{Av}_{\mathcal{IW}}(\nabla_{w}^{I})\simeq\nabla_{\mu}^{\mathcal{IW}}$,
where $\mu\in X^{\ast}(\hat{T})$ is the unique element with
$W\cdot(1,\mu)=W\cdot w$ [AR23, Lemma 6.4.5]. ∎

We now return to the task of proving Proposition 2.4.2. Note that in the “$+$”
notation introduced above, we are simply trying to prove that for all
$\lambda\in X^{\ast}(\hat{T})^{+}$ and $V\in\mathrm{Rep}(\hat{G})$,
$\overline{\nabla}_{\lambda}\otimes V$ lies in
$D_{+}^{b}\mathrm{PCoh}(\mathcal{N}/\hat{G})$. A trivial projection formula
gives an isomorphism $\overline{\nabla}_{\lambda}\otimes
V\simeq\pi_{\ast}(\nabla_{\lambda}^{\mathrm{ex}}\otimes V)$, so by Proposition
2.4.3 it’s enough to prove that $\nabla_{\lambda}^{\mathrm{ex}}\otimes V$ lies
in $D_{+}^{b}\mathrm{ExCoh}(\tilde{\mathcal{N}}/\hat{G})$. Going to the other
side of the Arkhipov-Bezrukavnikov equivalence and using Proposition 2.4.3
again, it’s enough in turn to prove that
$F_{\mathcal{IW}}(\nabla_{\lambda}^{\mathrm{ex}}\otimes V)$ lies in
$D_{+}^{b}\mathrm{Perv}_{\mathcal{IW}}(\mathrm{Fl}_{\mathbf{G}},\overline{\mathbf{Q}_{\ell}})$.

Now the magic happens. Recall that $D_{\mathcal{IW}}^{b}$ is a right module
over $D_{I}^{b}$ via the convolution action of $D_{I}^{b}$ on itself,
compatibly with the functor $\mathrm{Av}_{\mathcal{IW}}$. It is then true that

$F_{\mathcal{IW}}(\mathcal{G}\otimes V)\simeq
F_{\mathcal{IW}}(\mathcal{G})\star^{I}\mathscr{Z}(V)$

for any $\mathcal{G}\in D^{b}\mathrm{ExCoh}$ and any
$V\in\mathrm{Rep}(\hat{G})$, where $\mathscr{Z}(V)$ denotes the central sheaf
in $\mathrm{Perv}_{I}$ associated with $V$. Applying this property with
$\mathcal{G}=\nabla_{\lambda}^{\mathrm{ex}}$, using the identification of
costandard objects under $F_{\mathcal{IW}}$ and $\mathrm{Av}_{\mathcal{IW}}$
stated previously, and rearranging using the centrality of $\mathscr{Z}(V)$,
we get that

$\displaystyle F_{\mathcal{IW}}(\nabla_{\lambda}^{\mathrm{ex}}\otimes V)$
$\displaystyle\simeq
F_{\mathcal{IW}}(\nabla_{\lambda}^{\mathrm{ex}})\star^{I}\mathscr{Z}(V)$
$\displaystyle\simeq\nabla_{\lambda}^{\mathcal{IW}}\star^{I}\mathscr{Z}(V)$
$\displaystyle\simeq\mathrm{Av}_{\mathcal{IW}}\left(\nabla_{w_{\lambda}}^{I}\star^{I}\mathscr{Z}(V)\right)$
$\displaystyle\simeq\mathrm{Av}_{\mathcal{IW}}\left(\mathscr{Z}(V)\star^{I}\nabla_{w_{\lambda}}^{I}\right)$
$\displaystyle\simeq\mathrm{Av}_{\mathrm{\mathcal{IW}}}(\mathscr{Z}(V))\star^{I}\nabla_{w_{\lambda}}^{I}$

where $w_{\lambda}$ is the evident lift. By [AR23, Theorem 6.5.2],
$\mathrm{Av}_{\mathcal{IW}}(\mathscr{Z}(V))$ admits a finite filtration in
$\mathrm{Perv}_{\mathcal{IW}}$ with costandard graded pieces. This immediately
reduces us to proving that any complex of the form
$\nabla_{\nu}^{\mathcal{IW}}\star^{I}\nabla_{w}^{I}$ lies in
$D_{+}^{b}\mathrm{Perv}_{\mathcal{IW}}$. But

$\nabla_{\nu}^{\mathcal{IW}}\star^{I}\nabla_{w}^{I}\simeq\mathrm{Av}_{\mathcal{IW}}(\nabla_{w_{\nu}}^{I}\star^{I}\nabla_{w}^{I}),$

so using Proposition 2.4.3 one more time, we’re now reduced to showing that
any complex of the form $\nabla_{w^{\prime}}^{I}\star^{I}\nabla_{w}^{I}$ lies
in $D_{I}^{b}(\mathrm{Fl}_{\mathbf{G}},\overline{\mathbf{Q}_{\ell}})_{+}$. But
this is exactly the second half of [AR23, Lemma 6.5.8].

######  Theorem 2.4.4.

Assume Conjecture 2.2.1, and also that the pairs $(b_{\lambda},\pi_{\lambda})$
exhaust all pairs $(b,\pi)$ for which $i_{b!}^{\mathrm{ren}}\pi$ has trivial
Fargues-Scholze parameter. Then the Hecke action of $\mathrm{Rep}(\hat{G})$ on
$D(\mathrm{Bun}_{G})_{\phi_{\mathrm{triv}}}^{\mathrm{ULA}}$ is perverse
t-exact.

The exhaustion hypothesis is known unconditionally for many groups, including
$\mathrm{GL}_{n}$, $\mathrm{SL}_{n}$, and $\mathrm{GSp}_{4}$.

###### Proof.

It suffices to prove that any Hecke operator $T_{V}$ acting on
$D(\mathrm{Bun}_{G})_{\phi_{\mathrm{triv}}}^{\mathrm{ULA}}$ is perverse right
t-exact. Indeed, since the right adjoint of $T_{V}$ is $T_{V^{\vee}}$, this
automatically implies that $T_{V^{\vee}}$ is perverse left t-exact. Varying
over all $V$, we get the claimed reduction.

Next, using the exhaustion hypothesis, one checks that
$D(\mathrm{Bun}_{G})_{\phi_{\mathrm{triv}}}^{\mathrm{ULA}}\cap\,^{p}\\!D^{\leq
0}$ is generated under extensions and colimits by objects of the form
$i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}[n]$ for $\lambda\in
X^{\ast}(\hat{T})^{+}$ and $n\geq 0$. This reduces us to checking that any
sheaf of the form $T_{V}i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}$ is
perverse connective. Under the categorical equivalence, Conjecture 2.2.1,
$T_{V}i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}$ corresponds to the
coherent complex $\nu_{\ast}(V\otimes A_{\lambda})$. By Proposition 2.4.2,
this has a finite filtration with graded pieces of the form
$\nu_{\ast}A_{\mu}[n]$ for some $\mu\in X^{\ast}(\hat{T})^{+}$ and $n\geq 0$.
Translating back to the other side, we deduce that
$T_{V}i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}$ has a finite filtration
with graded pieces of the form $i_{b_{\mu}!}^{\mathrm{ren}}\pi_{\mu}[n]$ for
some $\mu\in X^{\ast}(\hat{T})^{+}$ and $n\geq 0$. Since these graded pieces
are perverse connective, we get the desired result. ∎

###  2.5 Two t-exactness conjectures

At this point, we are ready to confidently pose some precise t-exactness
conjectures for the categorical equivalence with restricted variation,
localized over semisimple generic parameters.

Set
$\mathrm{Par}_{G}^{\mathrm{gen}}=\mathrm{Par}_{G}\times_{X_{G}^{\mathrm{spec}}}X_{G}^{\mathrm{spec,gen}}$,
so $\mathrm{Par}_{G}^{\mathrm{gen}}$ is a smooth algebraic stack. In fact,
$X_{G}^{\mathrm{spec,gen}}$ is the _maximal_ open subscheme of
$X_{G}^{\mathrm{spec}}$ with the property that its preimage in
$\mathrm{Par}_{G}$ is a smooth algebraic stack.

Let $D(\mathrm{Bun}_{G})_{\mathrm{fin}}^{\mathrm{gen}}$ be the full
subcategory of finite sheaves $A$ whose $\phi$-local summand $A_{\phi}$
vanishes for every semisimple parameter $\phi$ which is not semisimple
generic. It is clear that Conjecture 1.7.8 localizes to a conjectural
equivalence

$c_{\psi}:D(\mathrm{Bun}_{G})_{\mathrm{fin}}^{\mathrm{gen}}\overset{\sim}{\to}\mathrm{Coh}(\mathrm{Par}_{G}^{\mathrm{gen}})_{\mathrm{fin}}.$

######  Conjecture 2.5.1.

_i._ The category $D(\mathrm{Bun}_{G})_{\mathrm{fin}}^{\mathrm{gen}}$ is
stable under the perverse truncation functors.

_ii._ The equivalence

$D(\mathrm{Bun}_{G})_{\mathrm{fin}}^{\mathrm{gen}}\overset{\sim}{\to}\mathrm{Coh}(\mathrm{Par}_{G}^{\mathrm{gen}})_{\mathrm{fin}}$

is t-exact with respect to the _perverse_ t-structure on the left-hand side
and the _standard_ t-structure on the right-hand side.

Note that ii. is not well-posed unless one assumes i. is true. However, we
warn the reader that i. definitely fails before passing to the localization
around semisimple generic parameters. On the other hand, part ii. of this
conjecture is not a wild guess:

######  Proposition 2.5.2.

If Conjectures 1.7.8 and 2.4.1 are true, then

$c_{\psi}:D(\mathrm{Bun}_{G})_{\mathrm{fin}}^{\mathrm{gen}}\to\mathrm{Coh}(\mathrm{Par}_{G}^{\mathrm{gen}})_{\mathrm{fin}}$

satisfies the t-exactness property of Conjecture 2.5.1.

###### Sketch..

If $A\in D(\mathrm{Bun}_{G})_{\mathrm{fin}}^{\mathrm{gen}}$ is perverse, then
$T_{V}A$ is perverse for all $V\in\mathrm{Rep}(\hat{G})$ by Conjecture 2.4.1,
so $i_{1}^{\ast}T_{V}A$ is concentrated in degree zero and then

$R\mathrm{Hom}(i_{1!}W_{\psi},T_{V}A)\simeq
R\mathrm{Hom}(W_{\psi},i_{1}^{\ast}T_{V}A)$

is also concentrated in degree zero by the projectivity of $W_{\psi}$ (see the
discussion around Theorem A.0.1). On the other side of Conjecture 1.7.8, this
translates into the knowledge that

$R\Gamma(\mathrm{Par}_{G},V\otimes c_{\psi}(A))\simeq
R\mathrm{Hom}(i_{1!}W_{\psi},T_{V}A)$

is concentrated in degree zero for all $V\in\mathrm{Rep}(\hat{G})$. But this
implies that $c_{\psi}(A)$ is concentrated in degree zero for the standard
t-structure. ∎

Finally we formulate a t-exactness conjecture for the hadal t-structure. This
is somewhat implicit in the discussion from section 2.3, but for completeness
we spell it out fully. Here we freely reuse the notations introduced in
section 2.3. For a fixed semisimple generic parameter $\phi$, define
${}^{pcoh}D^{\leq 0}$ resp. ${}^{pcoh}D^{\geq 0}$ inside
$\mathrm{Coh}(\mathrm{Par}_{G})_{\phi}$ as the full subcategory generated
under extensions by $\nu_{\phi\ast}\overline{\Delta}_{\rho}[n]$ for $n\geq 0$,
resp. by $\nu_{\phi\ast}\overline{\nabla}_{\rho}[n]$ for $n\leq 0$.

######  Conjecture 2.5.3.

_i._ The pair $(^{pcoh}D^{\leq 0},{}^{pcoh}D^{\geq 0})$ defines a perverse
coherent t-structure on $\mathrm{Coh}(\mathrm{Par}_{G})_{\phi}$. Writing
$\mathrm{PCoh}(\mathrm{Par}_{G})_{\phi}$ for the heart, the functor
$\nu_{\phi\ast}:\mathrm{Coh}(\mathcal{N}_{S_{\phi}})\to\mathrm{Coh}(\mathrm{Par}_{G})_{\phi}$
should induce a faithful exact functor

$\mathrm{PCoh}(\mathcal{N}_{S_{\phi}})\to\mathrm{PCoh}(\mathrm{Par}_{G})_{\phi}$

which is bijective on isomorphism classes of irreducible objects.

_ii._ The equivalence

$c_{\psi}:D(\mathrm{Bun}_{G})_{\mathrm{fin},\phi}\overset{\sim}{\to}\mathrm{Coh}(\mathrm{Par}_{G})_{\phi}$

induced by $\phi$-localizing the equivalence of Conjecture 1.7.8 should be
t-exact with respect to the _hadal_ t-structure on the left-hand side and the
_perverse coherent_ t-structure defined above on the right-hand side. In
particular, it should restrict to an exact equivalence of abelian categories

$\mathrm{Had}(\mathrm{Bun}_{G})_{\mathrm{fin},\phi}\overset{\sim}{\to}\mathrm{PCoh}(\mathrm{Par}_{G})_{\phi}.$

Part i. of this conjecture should be quite easy to verify, conditionally on
working out the speculations from section 2.3.

##  3 Additional conjectures

###  3.1 ULA sheaves and generic parameters

The reader may have noticed that general (i.e. non-finite) ULA sheaves have
been largely absent from our discussion. This is due to a psychological
complication, which we have tried to avoid confronting until now: when
translating general ULA sheaves to the spectral side, we are forced
unavoidably to reckon with $\mathrm{IndCoh}$. More precisely, we have the
following definition.

######  Definition 3.1.1.

An object $A\in\mathrm{IndCoh}(\mathrm{Par}_{G})$ is _admissible_ if for all
$B\in\mathrm{Coh}(\mathrm{Par}_{G})$, $R\mathrm{Hom}(B,A)$ lies in
$\mathrm{Perf}(\overline{\mathbf{Q}_{\ell}})$. We write
$\mathrm{Adm}(\mathrm{Par}_{G})\subset\mathrm{IndCoh}(\mathrm{Par}_{G})$ for
the evident stable $\infty$-category of admissible ind-coherent sheaves.

If we believe the categorical conjecture, then these are exactly the sheaves
on the spectral side which should match with ULA sheaves on the automorphic
side.

######  Proposition 3.1.2.

If Conjecture 1.7.3 is true, then the ind-extension of $c_{\psi}$ to an
equivalence
$c_{\psi}:D(\mathrm{Bun}_{G})\overset{\sim}{\to}\mathrm{IndCoh}(\mathrm{Par}_{G})$
restricts to an equivalence of categories

$c_{\psi}:D(\mathrm{Bun}_{G})^{\mathrm{ULA}}\overset{\sim}{\to}\mathrm{Adm}(\mathrm{Par}_{G})$

linear over the $\otimes$-action of $\mathrm{Perf}(\mathrm{Par}_{G})$.

###### Proof.

This is immediate from the fact that ULA sheaves $A$ on $\mathrm{Bun}_{G}$ are
characterized by the condition that $R\mathrm{Hom}(B,A)$ lies in
$\mathrm{Perf}(\overline{\mathbf{Q}_{\ell}})$ for all $B\in
D(\mathrm{Bun}_{G})^{\omega}$, which follows from [FS21, Prop. VII.7.4 and
Prop. VII.7.9]. ∎

######  Exercise 3.1.3.

1\. Check that there is an inclusion
$\mathrm{Coh}(\mathrm{Par}_{G})_{\mathrm{fin}}\subset\mathrm{Adm}(\mathrm{Par}_{G})$.

2\. (Slightly harder.) Check that there is an equality
$\mathrm{Coh}(\mathrm{Par}_{G})_{\mathrm{fin}}=\mathrm{Adm}(\mathrm{Par}_{G})\cap\mathrm{Coh}(\mathrm{Par}_{G})$.

We warn the reader that most admissible sheaves are not coherent. For
instance, let $x\in\mathrm{Par}_{G}(\overline{\mathbf{Q}_{\ell}})$ be any
point in the smooth locus of the stack $\mathrm{Par}_{G}$, with associated
residual gerbe $BS_{x}$, and let $i_{x}:BS_{x}\to\mathrm{Par}_{G}$ be the
evident immersion. If $\rho$ is any irreducible algebraic representation of
$S_{x}$, one can show that $i_{x\ast}^{\mathrm{IndCoh}}\rho$ is always an
admissible sheaf (we will prove a more general result below). However, if $x$
is not a closed point, this sheaf will not be coherent.

Next we formulate a duality conjecture. If $X$ is any quasismooth algebraic
stack over $\mathrm{Spec}\,\overline{\mathbf{Q}_{\ell}}$ with structure map
$f_{X}$, and $A\in\mathrm{IndCoh}(X)$ is a given object, then we get a
contravariant functor

$\displaystyle\mathrm{IndCoh}(X)$
$\displaystyle\to\mathrm{IndCoh}(\mathrm{Spec}\,\overline{\mathbf{Q}_{\ell}})=\mathrm{Vect}_{\overline{\mathbf{Q}_{\ell}}}$
$\displaystyle B$ $\displaystyle\mapsto
R\mathrm{Hom}(f_{X\ast}^{\mathrm{IndCoh}}(A\otimes^{!}B),\overline{\mathbf{Q}_{\ell}})$

where the notation for pushforward and tensor product of ind-coherent sheaves
follows Gaitsgory-Rozenblyum’s book. By some general nonsense with the
$\infty$-categorical adjoint functor theorem, this functor is representable by
$R\mathrm{Hom}(-,\mathbf{D}_{\mathrm{adm}}A)$ for a uniquely determined object
$\mathbf{D}_{\mathrm{adm}}A\in\mathrm{IndCoh}(X)$. The association
$A\mapsto\mathbf{D}_{\mathrm{adm}}A$ is a contravariant endofunctor of
$\mathrm{IndCoh}(X)$. For general stacks and general sheaves, this functor of
“admissible dual” will not be well-behaved. However, for the stack of _L_
-parameters, we expect the following.151515A proof of Conjecture 3.1.4.1 and
substantial evidence for Conjecture 3.1.4.2 will be given in future joint work
with Lucas Mann [HM23].

######  Conjecture 3.1.4.

_1._ The functor $\mathbf{D}_{\mathrm{adm}}(-)$ defines an involutive anti-
equivalence from $\mathrm{Adm}(\mathrm{Par}_{G})$ to itself.

_2._ The equivalence
$c_{\psi}:D(\mathrm{Bun}_{G})^{\mathrm{ULA}}\overset{\sim}{\to}\mathrm{Adm}(\mathrm{Par}_{G})$
conjectured in Proposition 3.1.2 should satisfy the duality compatibility

$\mathbf{D}_{\mathrm{tw.adm}}\circ c_{\psi}\simeq
c_{\psi^{-1}}\circ\mathbf{D}_{\mathrm{Verd}},$

where $\mathbf{D}_{\mathrm{tw.adm}}=c^{\ast}\mathbf{D}_{\mathrm{adm}}$ is the
composition of admissible duality with pullback along the Chevalley
involution.

This suggests another perspective on the “best hope” discussed at the
beginning of section 2. Specifically, if we believe in Proposition 3.1.2 and
Conjecture 3.1.4, we are forced to believe in the existence of an exotic
t-structure on $\mathrm{Adm}(\mathrm{Par}_{G})$ matching the perverse
t-structure on $D(\mathrm{Bun}_{G})^{\mathrm{ULA}}$, whose left and right
halves are swapped by $\mathbf{D}_{\mathrm{tw.adm}}$. Over the smooth locus of
$\mathrm{Par}_{G}$, this should just be the standard t-structure, but it seems
extremely subtle to extend the desired t-structure over the singularities of
the stack of $L$-parameters. For instance, one can show that for most
$V\in\mathrm{Rep}(\hat{G})$, the endofunctor $V\otimes-$ corresponding to the
Hecke operator $T_{V}$ is neither left nor right t-exact for this t-structure,
and its failure of t-exactness around a given point $x$ seems to correlate
with “how singular” the stack is at $x$. This seems to suggest that any direct
definition of this t-structure will need to use specific features of the stack
of singularities of $\mathrm{Par}_{G}$. We will discuss these ideas in more
detail elsewhere.

Finally, we formulate some conjectures attaching ULA sheaves on
$\mathrm{Bun}_{G}$ to generic _L_ -parameters. In some sense this brings us
back to the very origins of the entire subject in Fargues’s 2014 MSRI lecture
[Far14]. To simplify the discussion, we will consider _L_ -parameters
satisfying the following condition.

($\dagger$) $\phi$ is Frobenius-semisimple and generic, and $S_{\phi}^{\circ}$
is reductive.

The condition $(\dagger)$ is very mild. For instance, it holds for all
discrete parameters, all parameters which are $\iota$-essentially tempered for
some isomorphism
$\iota:\overline{\mathbf{Q}_{\ell}}\overset{\sim}{\to}\mathbf{C}$, all
semisimple generic parameters, and all generic parameters such that
$\phi(\mathrm{Fr})$ is regular semisimple. The last two claims here are easy,
and the first two (which require some work) are proved in [BMIY22, Section 3].
Given any such _L_ -parameter, we get a canonical immersion
$i_{\phi}:BS_{\phi}\to\mathrm{Par}_{G}$ which factors through the smooth locus
in the stack of _L_ -parameters by our genericity assumption.

######  Proposition 3.1.5.

Let $M\in\mathrm{IndCoh}(BS_{\phi})$ be any object with the property that for
each $\rho\in\mathrm{Irr}(S_{\phi})$, the total multiplicity
$\sum_{n}\mathrm{dim}\mathrm{Hom}_{S_{\phi}}(\rho,H^{n}(M))$ is finite. Then
$i_{\phi\ast}^{\mathrm{IndCoh}}M$ is an admissible sheaf.

###### Proof.

If $B$ lies in $\mathrm{Coh}(\mathrm{Par}_{G})$, then
$i_{\phi}^{\ast\mathrm{IndCoh}}B$ lies in $\mathrm{Coh}(BS_{\phi})$, using
that $i_{\phi}$ has finite tor-dimension since it is a regular immersion of
stacks. We then get that

$R\mathrm{Hom}(B,i_{\phi\ast}^{\mathrm{IndCoh}}M)\simeq
R\mathrm{Hom}(i_{\phi}^{\ast\mathrm{IndCoh}}B,M)$

by adjunction, and this lies in $\mathrm{Perf}(\overline{\mathbf{Q}_{\ell}})$
by the coherence of $i_{\phi}^{\ast\mathrm{IndCoh}}B$, the reductivity of
$S_{\phi}^{\circ}$, and our assumption on $M$. ∎

Note that the condition in this proposition is trivially satisfied if $M$ is
an irreducible algebraic $S_{\phi}$-representation, but it is also satisfied
if $M=\mathcal{O}(S_{\phi})$ is the regular representation, even though the
latter is not coherent! In particular, we get an admissible sheaf
$i_{\phi\ast}^{\mathrm{IndCoh}}\mathcal{O}(S_{\phi})$ which (by our
reductivity assumption) admits a canonical direct sum decomposition with
admissible pieces

$i_{\phi\ast}^{\mathrm{IndCoh}}\mathcal{O}(S_{\phi})\simeq\oplus_{\rho\in\mathrm{Irr}(S_{\phi})}i_{\phi\ast}^{\mathrm{IndCoh}}\rho^{\oplus\mathrm{dim}\rho}.$

Translating to the other side of the putative equivalence in Proposition
3.1.2, we conjecture the existence of a canonical ULA sheaf
$\mathscr{F}_{\phi}$ together with a direct sum decomposition

$\mathscr{F}_{\phi}\simeq\oplus_{\rho\in\mathrm{Irr}(S_{\phi})}\mathscr{F}_{\phi,\rho}^{\oplus\mathrm{dim}\rho}$

satisfying many remarkable properties.

1\. By the usual formal argument, $\mathscr{F}_{\phi}$ should be a Hecke
eigensheaf with eigenvalue $\phi$. A more refined argument leads to the
prediction that for any $V\in\mathrm{Rep}(^{L}G)$, there is an isomorphism

$T_{V}\mathscr{F}_{\phi,\rho}\simeq\bigoplus_{\rho^{\prime}\in\mathrm{Irr}(S_{\phi})}\mathscr{F}_{\phi,\rho^{\prime}}\boxtimes\mathrm{Hom}_{S_{\phi}}(\rho^{\vee}\otimes\rho^{\prime},V\circ\phi)$

as objects of $D(\mathrm{Bun}_{G})^{BW_{E}}$.

2\. We expect that each $\mathscr{F}_{\phi,\rho}$ is indecomposable and
perverse. Moreover, the “easy part” of the BMO algorithm begins by
(unconditionally!) attaching to the pair $(\phi,\rho)$ an element
$b=b_{\phi,\rho}\in B(G)$, and we expect that
$\mathrm{supp}\mathscr{F}_{\phi,\rho}\subseteq\overline{\\{b\\}}$. Moreover,
we expect that $i_{b}^{\ast\mathrm{ren}}\mathscr{F}_{\phi,\rho}$ is a single
irreducible $G_{b}(E)$-representation $\pi$ concentrated in degree zero, and
that this is _the_ $G_{b}(E)$-representation attached to $(\phi,\rho)$ by the
$B(G)$ local Langlands correspondence. In other words, we expect that this
stalk realizes the “hard part” of the BMO algorithm. We also expect that the
induced map

$i_{b!}^{\mathrm{ren}}\pi\to\mathscr{F}_{\phi,\rho}$

induces a surjection (in the perverse category) from
${}^{p}H^{0}(i_{b!}^{\mathrm{ren}}\pi)$ onto the socle of
$\mathscr{F}_{\phi,\rho}$. In general, $\mathscr{F}_{\phi,\rho}$ should _not_
be the intermediate extension $i_{b!\ast}^{\mathrm{ren}}\pi$ \- we believe
that this happens exactly when $\phi$ is semisimple generic. When $\phi$ is
discrete, $\mathscr{F}_{\phi,\rho}$ seems to be a _tilting_ perverse sheaf.

3\. We expect that
$\mathbf{D}_{\mathrm{Verd}}(\mathscr{F}_{\phi,\rho})\simeq\mathscr{F}_{\phi^{\vee},\rho^{\vee}}$,
where $\mathscr{F}_{\phi^{\vee},\rho^{\vee}}$ is the sheaf matching
$i_{\phi^{\vee}\ast}^{\mathrm{IndCoh}}\rho^{\vee}$ under the functor
$c_{\psi^{-1}}$. Indeed, one can prove unconditionally that
$\mathbf{D}_{\mathrm{tw.adm}}$ exchanges $i_{\phi\ast}^{\mathrm{IndCoh}}\rho$
and $i_{\phi^{\vee}\ast}^{\mathrm{IndCoh}}\rho^{\vee}$, so this expectation is
forced upon us by Conjecture 3.1.4.2.

4\. We expect that suitable linear combinations (over varying $\rho$) of the
virtual stalks $[i_{b}^{\ast\mathrm{ren}}\mathscr{F}_{\phi,\rho}]$ can be
described explicitly. For instance, if $\phi$ is discrete and $b$ has trivial
Kottwitz invariant, we expect that for every elliptic endoscopic datum
$\mathcal{H}=(H,s,\eta)$ such that $\phi$ admits a factorization
$\phi=\phantom{}^{L}\eta\circ\phi^{H}$ there should be an equality of the form

$\sum_{\rho\in\mathrm{Irr}(\pi_{0}(\overline{S}_{\phi}))}\mathrm{tr}\rho(s)\cdot[i_{b}^{\ast\mathrm{ren}}\mathscr{F}_{\phi,\rho}]=\mathrm{Red}_{b}^{\mathcal{H}}(S\Theta_{\phi^{H}}).$

Here $\mathrm{Red}_{b}^{\mathcal{H}}$ is the map from stable virtual
representations of $H(E)$ towards virtual representations of $G_{b}(E)$
defined in [BM21, Definition 5.6], and $S\Theta_{\phi^{H}}$ is the stable
virtual representation of $H(E)$ attached to $\phi^{H}$. In general, the
stalks of the individual sheaves $\mathscr{F}_{\phi,\rho}$ seem extremely hard
to describe, even as virtual representations.

Some of these phenomena would be neatly explained by yet another
conjecture.161616Conjecture 3.1.6 and Proposition 3.1.7 below were
independently discovered by Koshikawa, who also independently noticed the
utility of the condition $(\dagger)$. I would also like to acknowledge that
the tilting property for Hecke eigensheaves at discrete parameters was
suggested by a beautiful example explained to me by Koshikawa.  To formulate
this, note that for any $b$ and any open compact $K\subset G_{b}(E)$,
$i_{b!}^{\mathrm{ren}}\mathrm{ind}_{K}^{G_{b}(E)}\overline{\mathbf{Q}_{\ell}}$
is compact, so Conjecture 1.7.3 predicts that

$\mathscr{C}_{b,K}\overset{\mathrm{def}}{=}c_{\psi}\left(i_{b!}^{\mathrm{ren}}\mathrm{ind}_{K}^{G_{b}(E)}\overline{\mathbf{Q}_{\ell}}\right)$

is a bounded coherent complex with quasicompact support. (Strictly speaking,
$\mathscr{C}_{b,K}$ depends also on the Whittaker datum, but we suppress this
from the notation.)

######  Conjecture 3.1.6.

For all $b$ and $K$ as above (and all Whittaker data), $\mathscr{C}_{b,K}$ is
a genuine coherent sheaf, i.e. it lies in the heart
$\mathrm{Coh}(\mathrm{Par}_{G})^{\heartsuit}$ of the standard t-structure on
$\mathrm{Coh}$.

When $b=1$, $G$ is unramified, and $K=I$ is an Iwahori, $\mathscr{C}_{1,I}$ is
the “coherent Springer sheaf” studied by Ben–Zvi-Chen-Helm-Nadler, Hellmann
and Zhu. For non-basic $b$, Conjecture 3.1.6 is a natural $\mathrm{Bun}_{G}$
variant of the conjecture formulated in [Zhu21, Remark 4.6.5].171717I do not
believe Zhu’s formulation is quite correct. This conjecture turns out to have
many consequences.

######  Proposition 3.1.7.

Assume Conjectures 1.7.3 and 1.7.5, and also assume that Conjecture 3.1.6 is
true. Then the following hold.

_i._ For all $b$ and $K$, the coherent complex
$c_{\psi}\left(i_{b\sharp}^{\mathrm{ren}}\mathrm{ind}_{K}^{G_{b}(E)}\overline{\mathbf{Q}_{\ell}}\right)$
is concentrated in nonnegative degrees.

_ii._ The functor $c_{\psi}$ is perverse right semi-exact, i.e. it carries
$\phantom{}{}^{p}D^{\leq 0}(\mathrm{Bun}_{G})$ into $\mathrm{IndCoh}^{\leq
0}(\mathrm{Par}_{G})$. In particular, for any $b$ and any $\pi\in\Pi(G_{b})$,
$c_{\psi}(i_{b!}^{\mathrm{ren}}\pi)$ is concentrated in nonpositive
cohomological degrees.

_iii._ For any $L$-parameter $\phi$ satisfying $(\dagger)$ as above and any
$\rho\in\mathrm{Irr}(S_{\phi})$, the sheaf $\mathscr{F}_{\phi,\rho}$ defined
above is perverse. Moreover, if $\phi$ is a discrete parameter then
$\mathscr{F}_{\phi,\rho}$ is a _tilting_ perverse sheaf: for all $b$,
$i_{b}^{\ast\mathrm{ren}}\mathscr{F}_{\phi,\rho}$ and
$i_{b}^{!\mathrm{ren}}\mathscr{F}_{\phi,\rho}$ are concentrated in degree
zero.

_iv._ For any local shtuka datum $(G,\mu,b)$ and any open compact subgroup
$K\subset G(E)$, the compactly supported intersection cohomology

$R\Gamma_{c}(\mathrm{Sht}(G,\mu,b)_{K},IC_{\mu})$

of the local shtuka space $\mathrm{Sht}(G,\mu,b)_{K}$ at level $K$ is
concentrated in degrees $[\left\langle
2\rho_{G},\nu_{b}\right\rangle,\left\langle 2\rho_{G},\mu\right\rangle]$.

Here $R\Gamma_{c}(\mathrm{Sht}(G,\mu,b)_{K},IC_{\mu})\in
D(G_{b}(E),\overline{\mathbf{Q}_{\ell}})^{\omega}$ is defined as in [HKW22,
Lemma 6.4.4], taking $\Lambda=\overline{\mathbf{Q}_{\ell}}$ in the notation
there. A priori, the complex $R\Gamma_{c}(\mathrm{Sht}(G,\mu,b)_{K},IC_{\mu})$
lives in degrees $[-\left\langle 2\rho_{G},\mu\right\rangle,\left\langle
2\rho_{G},\mu\right\rangle]$. When $\mu$ is minuscule, the Stein property
formulated in [Han23a, Conjecture 1.10] would imply concentration in degrees
$[0,\left\langle 2\rho_{G},\mu\right\rangle]$. Thus for non-basic $b$, the
vanishing result in iv. above goes strictly beyond what is predicted by
geometry. We also observe that i. is best possible: it is certainly not true
that
$c_{\psi}\left(i_{b\sharp}^{\mathrm{ren}}\mathrm{ind}_{K}^{G_{b}(E)}\overline{\mathbf{Q}_{\ell}}\right)$
is concentrated in degree zero.

###### Proof.

For i., we have a duality isomorphism

$\mathbf{D}_{\mathrm{BZ}}\left(i_{b!}^{\mathrm{ren}}\mathrm{ind}_{K}^{G_{b}(E)}\overline{\mathbf{Q}_{\ell}}\right)\simeq
i_{b\sharp}^{\mathrm{ren}}\mathrm{ind}_{K}^{G_{b}(E)}\overline{\mathbf{Q}_{\ell}}.$

Combining this with the duality compatibility gives

$c_{\psi^{-1}}\left(i_{b\sharp}^{\mathrm{ren}}\mathrm{ind}_{K}^{G_{b}(E)}\overline{\mathbf{Q}_{\ell}}\right)\simeq\mathbf{D}_{\mathrm{twGS}}\left(\mathscr{C}_{b,K}\right),$

and the right-hand side is clearly concentrated in nonnegative degrees. For
ii., the claim follows from observing that $\phantom{}{}^{p}D^{\leq
0}(\mathrm{Bun}_{G})$ is generated under extensions and colimits by objects of
the form
$i_{b!}^{\mathrm{ren}}\mathrm{ind}_{K}^{G_{b}(E)}\overline{\mathbf{Q}_{\ell}}[n]$
for arbitrary $b,K$ and arbitrary $n\geq 0$.

For iii. it is enough to prove that $\mathscr{F}_{\phi,\rho}$ is perverse
coconnective, and that $i_{b}^{!\mathrm{ren}}\mathscr{F}_{\phi,\rho}$ is
concentrated in degree zero when $\phi$ is discrete. These reductions follow
from the Verdier duality property of $\mathscr{F}_{\phi,\rho}$ implied by
Conjecture 3.1.4.2 as in the discussion above. In turn, Conjecture 3.1.4.2 is
an _unconditional_ consequence of Conjectures 1.7.3 and 1.7.5; the argument
for this will appear in [HM23]. Now by design, Conjecture 1.7.3 implies that

$R\mathrm{Hom}(\mathscr{C}_{b,K},c_{\psi}(-)):D(\mathrm{Bun}_{G})\to
D(\overline{\mathbf{Q}_{\ell}})$

is exactly the functor $A\mapsto(i_{b}^{!\mathrm{ren}}A)^{K}$, so we compute
that

$\displaystyle(i_{b}^{!\mathrm{ren}}\mathscr{F}_{\phi,\rho})^{K}$
$\displaystyle\simeq
R\mathrm{Hom}(\mathscr{C}_{b,K},c_{\psi}(\mathscr{F}_{\phi,\rho}))$
$\displaystyle\simeq
R\mathrm{Hom}(\mathscr{C}_{b,K},i_{\phi\ast}^{\mathrm{IndCoh}}\rho)$
$\displaystyle\simeq
R\mathrm{Hom}(i_{\phi}^{\ast\mathrm{IndCoh}}\mathscr{C}_{b,K},\rho).$

Since $\mathscr{C}_{b,K}$ is in degree zero by assumption,
$i_{\phi}^{\ast\mathrm{IndCoh}}\mathscr{C}_{b,K}$ is concentrated in
nonpositive degrees, so we immediately get that the above expression is
concentrated in nonnegative degrees. Moreover, if $\phi$ is discrete, one can
check that $i_{\phi}^{\ast\mathrm{IndCoh}}\mathscr{C}_{b,K}$ is necessarily
concentrated in degree zero, which implies that the above expression is
concentrated also in degree zero. If $G$ is semisimple this is obvious since
the morphism $i_{\phi}$ is then an open immersion; for general groups one
needs a small extra argument, using that the orbit of any discrete parameter
$\phi$ under unramified twisting is open in $\mathrm{Par}_{G}$ together with
the equivariance of $\mathscr{C}_{b,K}$ under unramified twisting.

For iv., concentration in degrees $\leq\left\langle
2\rho_{G},\mu\right\rangle$ follows from the general étale cohomology
formalism. For the bound in the other direction, the key point is the formula

$R\Gamma_{c}(\mathrm{Sht}(G,\mu,b)_{K},IC_{\mu})^{K^{\prime}}\simeq
R\mathrm{Hom}\left(\mathscr{C}_{1,K}\otimes
V_{\mu},\mathscr{C}_{b,K^{\prime}}\right)[-\left\langle
2\rho_{G},\nu_{b}\right\rangle]$

where $K^{\prime}\subset G_{b}(E)$ is any open compact subgroup.181818This is
a variant of [Zhu21, Conjecture 4.7.18], but again I do not think Zhu’s
formulation is correct. This formula is an unconditional consequence of
Conjecture 1.7.3. Indeed, [HKW22, Lemma 6.4.4] gives an isomorphism

$R\Gamma_{c}(\mathrm{Sht}(G,\mu,b)_{K},IC_{\mu})\simeq
i_{b}^{\ast}T_{V_{\mu}}i_{1!}\mathrm{ind}_{K}^{G(E)}\overline{\mathbf{Q}_{\ell}},$

which easily implies an isomorphism

$R\Gamma_{c}(\mathrm{Sht}(G,\mu,b)_{K},IC_{\mu})^{K^{\prime}}\simeq
R\mathrm{Hom}(i_{b\sharp}\mathrm{ind}_{K^{\prime}}^{G_{b}(E)}\overline{\mathbf{Q}_{\ell}},T_{V_{\mu}}i_{1!}\mathrm{ind}_{K}^{G(E)}\overline{\mathbf{Q}_{\ell}}).$

Applying BZ duality on the right-hand side, we can rewrite it as

$R\mathrm{Hom}(T_{V_{\mu}}i_{1!}\mathrm{ind}_{K}^{G(E)}\overline{\mathbf{Q}_{\ell}},i_{b!}\mathrm{ind}_{K^{\prime}}^{G_{b}(E)}\overline{\mathbf{Q}_{\ell}}[-2\left\langle
2\rho_{G},\nu_{b}\right\rangle]).$

Now we have isomorphisms

$c_{\psi}\left(T_{V_{\mu}}i_{1!}\mathrm{ind}_{K}^{G(E)}\overline{\mathbf{Q}_{\ell}}\right)\simeq
V_{\mu}\otimes\mathscr{C}_{1,K}$

and

$c_{\psi}\left(i_{b!}\mathrm{ind}_{K^{\prime}}^{G_{b}(E)}\overline{\mathbf{Q}_{\ell}}[-2\left\langle
2\rho_{G},\nu_{b}\right\rangle]\right)\simeq\mathscr{C}_{b,K^{\prime}}[-\left\langle
2\rho_{G},\nu_{b}\right\rangle]$

by unwinding the definitions and using the linearity of $c_{\psi}$ over the
spectral action, so passing to the other side via Conjecture 1.7.3 we arrive
at the desired formula. Then by Conjecture 3.1.6,
$R\mathrm{Hom}\left(\mathscr{C}_{1,K}\otimes
V_{\mu},\mathscr{C}_{b,K^{\prime}}\right)$ is concentrated in nonnegative
degrees, so accounting for the shift and shrinking $K^{\prime}$ arbitrarily
gives the claim. ∎

###  3.2 More examples

In this section we collect some miscellaneous remarks and examples. We also
try to illustrate how complicated the situation becomes once we allow _L_
-parameters with nontrivial monodromy into the discussion.

######  Example 3.2.1.

It is natural to expect that $c_{\psi}\circ i_{1!}$ should coincide with
Hellmann’s functor $R_{\psi}$. Taking this for granted, we can use the
calculations in [Hel23] as a source of examples. In particular, take
$G=\mathrm{GL}_{2}$, and let $\mathrm{St}$ be the Steinberg representation, so
we have a short exact sequence

$0\to\mathrm{St}\to\mathrm{Ind}_{B}^{G}\delta_{B}\to\mathbf{1}\to 0.$

Using [Hel23, Theorem 4.34 and Remark 4.43] it is not difficult to see that
$R_{\psi}$ should send this sequence to the distinguished triangle

$i_{Z\ast}\mathcal{O}_{Z}\to i_{y\ast}\mathcal{O}_{y}\to
i_{Z\ast}\mathcal{L}[1]\to.$

Here $Z\simeq\mathbf{A}^{1}/\mathbf{G}_{m}^{2}\subset\mathrm{Par}_{G}$ is the
closure of the orbit of the Steinberg parameter, $y\simeq
B(\mathbf{G}_{m}^{2})\in Z$ is the unique closed orbit, $i_{Z}$ and $i_{y}$
are the evident closed immersions, and $\mathcal{L}$ is the line bundle on $Z$
of functions which vanish at $y$. Note that this triangle is a rotation of the
more natural triangle

$i_{Z\ast}\mathcal{L}\to i_{Z\ast}\mathcal{O}_{Z}\to
i_{y\ast}\mathcal{O}_{y}\to$

which is an honest short exact sequence in $\mathrm{Coh}^{\heartsuit}$.

It is very instructive to see how this sequence interacts with duality. On the
$\mathrm{Bun}_{G}$ side, it is easy to see that $\mathbf{D}_{\mathrm{coh}}$
sends $\mathbf{1}$ to $\mathrm{St}[-2]$ and $\mathrm{St}$ to $\mathbf{1}[-2]$.
It is also well-known that $\mathbf{D}_{\mathrm{coh}}$ intertwines
$\mathrm{Ind}_{B}^{G}$ and $\mathrm{Ind}_{\overline{B}}^{G}$. Putting these
observations together appropriately, we see that $\mathbf{D}_{\mathrm{coh}}$
sends our initial short exact sequence to the distinguished triangle

$\mathrm{St}[-2]\to\mathrm{Ind}_{\overline{B}}^{G}\delta_{\overline{B}}[-2]\to\mathbf{1}[-2]\to,$

which is isomorphic to the sequence we began with shifted by $-2$. In order
for this to be compatible with duality on the _L-_ parameter side, we must
have $\mathbf{D}_{\mathrm{twGS}}i_{Z\ast}\mathcal{O}_{Z}\simeq
i_{Z\ast}\mathcal{L}[-1]$, or equivalently we must have

$\omega_{Z}=i_{Z}^{!}\mathcal{O}_{\mathrm{Par}_{G}}\simeq\mathcal{L}[-1].$

This is indeed true, and follows from an explicit calculation via the
isomorphism $Z\simeq\mathbf{A}^{1}/\mathbf{G}_{m}^{2}$ noted above.

Examination of this example and various other examples in [Hel23], together
with some optimism, leads to the following speculation. Let $G$ be quasisplit
with a fixed Whittaker datum as usual. Let $\pi\in\mathrm{Rep}(G(E))$ be
irreducible and $\psi$-generic, with Fargues-Scholze parameter $\phi$. Assume
that $\phi(\mathrm{Fr})$ is _regular semisimple_. Write $Z=q^{-1}(x_{\phi})$,
so this is an Artin stack which comes with a tautological closed immersion
$i:Z\hookrightarrow\mathrm{Par}_{G}$. Our assumption on $\phi$ guarantees that
$Z$ is extremely nice, and is nothing more than the Vogan variety for the
infinitesimal character $\phi$. Explicitly, there is a presentation

$Z\cong\left\\{N\in\mathfrak{g}^{\mathrm{ad}\phi(I_{E})}\mid\mathrm{ad}\phi(\mathrm{Fr})\cdot
N=q^{-1}N\right\\}/S_{\phi}.$

Question. Is it true that $c_{\psi}(i_{1!}\pi)\simeq i_{\ast}\mathcal{O}_{Z}$?

######  Example 3.2.2.

In this example we illustrate how complicated the functor
$i_{b\sharp}^{\mathrm{ren}}$ can be in general. Take $E=\mathbf{Q}_{p}$ with
$p>2$, and $G=\mathrm{GSp}_{4}$, so we have an exceptional isomorphism
$\hat{G}\simeq\mathrm{GSp}_{4}\subset\mathrm{GL}_{4}$. Let $b^{\prime}$ be the
element such that
$\mathcal{E}_{b^{\prime}}\simeq\mathcal{O}(\tfrac{1}{2})\oplus\mathcal{O}(-\tfrac{1}{2})$.
Note that $b^{\prime}$ is an immediate specialization of the point $b=1$.
There is a natural identification $G_{b^{\prime}}(\mathbf{Q}_{p})\simeq
D^{\times}\times\mathbf{Q}_{p}^{\times}$, where $D/\mathbf{Q}_{p}$ is the
usual quaternion algebra. Let $\rho$ be an irreducible
$D^{\times}$-representation of dimension $>1$ with trivial central character,
with _L_ -parameter
$\phi_{\rho}:W_{\mathbf{Q}_{p}}\to\mathrm{GL}_{2}(\overline{\mathbf{Q}_{\ell}}$).
There is then a unique supercuspidal representation $\pi$ of
$\mathrm{GSp}_{4}(\mathbf{Q}_{p})$ with semisimple _L-_ parameter
$\phi_{\rho}\oplus|\cdot|^{\frac{1}{2}}\oplus|\cdot|^{-\frac{1}{2}}$. Note
that this parameter is _not_ semisimple generic in the sense of section 2.3:
it is the semisimplification of a discrete parameter with nontrivial
monodromy, which is the true _L_ -parameter of $\pi$.

Claim. For $n\in\\{0,1\\}$,
$H^{n}(i_{1}^{\ast}i_{b^{\prime}\sharp}^{\mathrm{ren}}(\rho|\mathrm{Nm}|^{-\frac{1}{2}}\boxtimes|\cdot|^{-\frac{1}{2}}))$
contains $\pi$ with multiplicity one. Moreover, this is the entire
supercuspidal part of
$H^{\ast}(i_{1}^{\ast}i_{b^{\prime}\sharp}^{\mathrm{ren}}(\rho|\mathrm{Nm}|^{-\frac{1}{2}}\boxtimes|\cdot|^{-\frac{1}{2}}))$.

The proof of this is very indirect. The idea is to compute

$A\overset{\mathrm{def}}{=}i_{1}^{\ast}T_{V}i_{b!}i_{P_{b}}^{G_{b}}(\rho|\mathrm{Nm}|^{-\frac{1}{2}}\boxtimes|\cdot|^{-\frac{1}{2}})$

in two different ways. Here $P=MU\subset G$ is the Siegel parabolic, $b\in
B(G)$ is (a representative in $M(\mathbf{Q}_{p})$ of) the basic element such
that $\mathcal{E}_{b}\simeq\mathcal{O}(\tfrac{1}{2})^{2}$,
$P_{b}=M_{b}U_{b}\subset G_{b}$ is the evident twist, and $V$ is the dual
standard representation of $\hat{G}$. Note that $M_{b}\cong G_{b^{\prime}}$ as
inner forms of $M\simeq\mathrm{GL}_{2}\times\mathbf{G}_{m}$. On one hand,
there is a short exact sequence

$0\to\mathrm{St}(\rho,1)\to
i_{P_{b}}^{G_{b}}(\rho|\mathrm{Nm}|^{-\frac{1}{2}}\boxtimes|\cdot|^{-\frac{1}{2}})\to\mathrm{Sp}(\rho,1)\to
0$

(cf. Proposition 5.3.(i) of [GT14]), and one can compute the supercuspidal
part of $i_{1}^{\ast}T_{V}i_{b!}(-)$ applied to the outer terms of this
sequence using a result of Ito-Mieda (Theorem 3.1 of [Mie21]; note that
$\mathrm{St}(\rho,1)=\rho_{\mathrm{disc}}$ and
$\mathrm{Sp}(\rho,1)=\rho_{\mathrm{nt}}$ in Mieda’s notation). This yields a
two-term filtration on the supercuspidal part of $A$ with comprehensible
graded pieces.

On the other hand, we can rewrite $A$ as
$i_{1}^{\ast}T_{V}\mathrm{Eis}_{P}^{G}i_{b!}^{M}(\rho|\mathrm{Nm}|^{-\frac{1}{2}}\boxtimes|\cdot|^{-\frac{1}{2}})$.
Using the filtered commutation of $\mathrm{Eis}$ with Hecke operators
(Conjecture 1.5.2), the latter expression acquires a three-term filtration
with graded pieces
$i_{1}^{\ast}\mathrm{Eis}_{P}^{G}T_{W_{k}}i_{b!}^{M}(\rho|\mathrm{Nm}|^{-\frac{1}{2}}\boxtimes|\cdot|^{-\frac{1}{2}})$.
Here the $W_{k}$ are the irreducible algebraic $\hat{M}$-representations whose
inflations to the Klingen parabolic191919We remind the reader that under the
exceptional identification of $\mathrm{GSp}_{4}$ with its own dual group, the
Klingen and Siegel parabolics correspond. $\hat{P}$ yield the three
irreducible subquotients of $V|\hat{P}$. These graded pieces can be computed
explicitly, and after passing to the supercuspidal part only one of them
survives and yields exactly the supercuspidal part of
$i_{1}^{\ast}i_{b^{\prime}\sharp}^{\mathrm{ren}}(\rho|\mathrm{Nm}|^{-\frac{1}{2}}\boxtimes|\cdot|^{-\frac{1}{2}})$.
A careful comparison of these two calculations now yields the claim.

Note that by adjunction, the claim easily implies that
$i_{b^{\prime}}^{\ast\mathrm{ren}}i_{1\ast}\pi$ is nonzero, with
$\rho|\mathrm{Nm}|^{-\frac{1}{2}}\boxtimes|\cdot|^{-\frac{1}{2}}$ occurring as
a subquotient. In the early days of the Fargues-Scholze program, many people
(including the present author) hoped the functors $i_{b}^{\ast}i_{1\ast}$
would have a reasonably simple explicit description in terms of Jacquet
modules and other concrete representation-theoretic operations. Examples of
this sort suggest that such hopes are woefully misguided.

######  Exercise 3.2.3.

Give a similar analysis of the supercuspidal part of
$H^{n}(i_{1}^{\ast}i_{b^{\prime}\sharp}^{\mathrm{ren}}(\rho|\mathrm{Nm}|^{\frac{1}{2}}\boxtimes|\cdot|^{-\frac{1}{2}}))$
\- note the change in the twisting exponent on $\rho$. (Hint: There is a
relevant variant of the short exact sequence used above where Sp and St are
swapped.)

###  3.3 Grothendieck groups and vanishing conjectures

Since $D(\mathrm{Bun}_{G})$ is cocomplete, its $K_{0}$ is zero by the usual
trick. On the other hand, $D(\mathrm{Bun}_{G})_{\mathrm{fin}}$ is much
smaller, and its $K_{0}$ is very interesting.

For any $b$, $i_{b}^{\ast\mathrm{ren}}$ induces a map
$[i_{b}^{\ast\mathrm{ren}}]:K_{0}D(\mathrm{Bun}_{G})_{\mathrm{fin}}\to
K_{0}\mathrm{Rep}(G_{b}(E))_{\mathrm{fin}}$, as well as maps
$[i_{b!}^{\mathrm{ren}}],[i_{b\sharp}^{\mathrm{ren}}]:K_{0}\mathrm{Rep}(G_{b}(E))_{\mathrm{fin}}\to
K_{0}D(\mathrm{Bun}_{G})_{\mathrm{fin}}$. Note that
$[i_{b\sharp}^{\mathrm{ren}}]$ is well-defined, since we know from the proof
of Theorem 1.6.3 that $i_{b\sharp}^{\mathrm{ren}}\pi$ is a finite sheaf for
$\pi$ of finite length. These clearly assemble into maps

$\gamma_{!}:\bigoplus_{b\in
B(G)}K_{0}\mathrm{Rep}(G_{b}(E))_{\mathrm{fin}}\overset{\sum_{b}[i_{b!}^{\mathrm{ren}}]}{\longrightarrow}K_{0}D(\mathrm{Bun}_{G})_{\mathrm{fin}},$
$\gamma_{\sharp}:\bigoplus_{b\in
B(G)}K_{0}\mathrm{Rep}(G_{b}(E))_{\mathrm{fin}}\overset{\sum_{b}[i_{b\sharp}^{\mathrm{ren}}]}{\longrightarrow}K_{0}D(\mathrm{Bun}_{G})_{\mathrm{fin}}$

and

$\gamma^{\ast}:K_{0}D(\mathrm{Bun}_{G})_{\mathrm{fin}}\overset{\\{[i_{b}^{\ast\mathrm{ren}}]\\}_{b}}{\longrightarrow}\bigoplus_{b\in
B(G)}K_{0}\mathrm{Rep}(G_{b}(E))_{\mathrm{fin}}.$

######  Proposition 3.3.1.

The maps $\gamma_{!}$, $\gamma_{\sharp}$ and $\gamma^{\ast}$ are isomorphisms
of abelian groups. The map $\gamma^{\ast}$ is a left inverse of $\gamma_{!}$.

###### Proof.

The last part is clear, since
$i_{b}^{\ast\mathrm{ren}}i_{b!}^{\mathrm{ren}}=\mathrm{id}$ and
$i_{b^{\prime}}^{\ast\mathrm{ren}}i_{b!}^{\mathrm{ren}}=0$ for all
$b^{\prime}\neq b$. The first part is an easy consequence of the
semiorthogonal decomposition together with the fact that
$i_{b\sharp}^{\mathrm{ren}}$ preserves finite sheaves as in the proof of
Theorem 1.6.3. ∎

Of course, the unrenormalized variants of these statements are also true.

By Theorem 1.6.3, Bernstein-Zelevinsky duality induces a well-defined
involution on $K_{0}D(\mathrm{Bun}_{G})_{\mathrm{fin}}$, which we denote
$[\mathbf{D}_{\mathrm{BZ}}]$. Similarly, cohomological duality induces an
involution on $K_{0}\mathrm{Rep}(G_{b}(E))_{\mathrm{fin}}$ for each $b$;
taking the direct sum over $b$, we get an involution
$[\mathbf{D}_{\mathrm{coh}}]$ on $\bigoplus_{b\in
B(G)}K_{0}\mathrm{Rep}(G_{b}(E))_{\mathrm{fin}}$.

######  Proposition 3.3.2.

The following statements are equivalent.

_1)_ The maps $\gamma_{!}$ and $\gamma_{\sharp}$ are equal.

_2)_ For all $b^{\prime}\neq b$, the map
$[i_{b^{\prime}}^{\ast\mathrm{ren}}i_{b\sharp}^{\mathrm{ren}}]:K_{0}\mathrm{Rep}(G_{b}(E))_{\mathrm{fin}}\to
K_{0}\mathrm{Rep}(G_{b^{\prime}}(E))_{\mathrm{fin}}$ is zero.

_2’)_ For all $b^{\prime}\neq b$, the map
$[i_{b^{\prime}}^{\ast}i_{b\sharp}]:K_{0}\mathrm{Rep}(G_{b}(E))_{\mathrm{fin}}\to
K_{0}\mathrm{Rep}(G_{b^{\prime}}(E))_{\mathrm{fin}}$ is zero.

_3)_ There is an equality of maps
$\gamma_{!}\circ[\mathbf{D}_{\mathrm{coh}}]=[\mathbf{D}_{\mathrm{BZ}}]\circ\gamma_{!}$.

_4)_ There is an equality of maps
$\gamma^{\ast}\circ[\mathbf{D}_{\mathrm{BZ}}]=[\mathbf{D}_{\mathrm{coh}}]\circ\gamma^{\ast}$.

_5)_ The maps $\gamma^{\ast}$ and $\gamma_{!}$ are mutually inverse.

###### Proof.

The equivalence of 1) and 2) is immediate from the definitions together with
the fact that $\gamma^{\ast}$ is a left inverse of $\gamma_{!}$. The
equivalence of 2) and 2’) is trivial. The equivalence of 1) and 3) follows
from the equality
$\gamma_{\sharp}\circ[\mathbf{D}_{\mathrm{coh}}]=[\mathbf{D}_{\mathrm{BZ}}]\circ\gamma_{!}$
together with the fact that both dualities are isomorphisms on the relevant
$K_{0}$’s. The equivalence of 3) and 4) follows along similar lines. Finally,
we leave the implications 3) + 4) $\Rightarrow$5) $\Rightarrow$1) as
exercises. ∎

We now have the following key vanishing conjecture.

######  Conjecture 3.3.3.

The equivalent statements in Proposition 3.3.2 are true.

This conjecture seems to lie very deep, and it has significant consequences
for the cohomology of local Shimura varieties.

It is fruitful to study this conjecture one _L_ -parameter at a time. More
precisely, passing to $\phi$-localizations, one can introduce an additional
grading on the source and target of the $\gamma$’s, indexed by semisimple _L_
-parameters. Conjecture 3.3.3 is then equivalent, for instance, to the
statement that for all semisimple L-parameters $\phi$ and all $b^{\prime}\neq
b$, the map

$[i_{b^{\prime}}^{\ast\mathrm{ren}}i_{b\sharp}^{\mathrm{ren}}]:K_{0}\mathrm{Rep}(G_{b}(E))_{\mathrm{fin},\phi}\to
K_{0}\mathrm{Rep}(G_{b^{\prime}}(E))_{\mathrm{fin},\phi}$

is zero. Note that if $\phi$ is generous, the final part of Conjecture 2.1.8
predicts that

$i_{b^{\prime}}^{\ast\mathrm{ren}}i_{b\sharp}^{\mathrm{ren}}:D(G_{b}(E),\overline{\mathbf{Q}_{\ell}})_{\mathrm{fin},\phi}\to
D(G_{b^{\prime}}(E),\overline{\mathbf{Q}_{\ell}})_{\mathrm{fin},\phi}$

is _already_ identically zero before passing to $K_{0}$! However, in general
this map will not vanish before passing to $K_{0}$.

A good example of this phenomenon is given by the trivial _L_ -parameter.
Here, to see that $\gamma_{!}$ and $\gamma_{\sharp}$ are equal on the summand
indexed by the trivial parameter, we need to see that
$[i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}]=[i_{b_{\lambda}\sharp}^{\mathrm{ren}}\pi_{\lambda}]$
for all dominant $\lambda$. On the other side of the categorical conjecture,
this corresponds to the expectation that
$[\nu_{\ast}A_{\lambda}]=[\nu_{\ast}A_{w_{0}(\lambda)}]$ in
$K_{0}\mathrm{Coh}(\mathrm{Par}_{G})_{\mathrm{fin}}$. But this is true! In
fact, the equality $[A_{\lambda}]=[A_{w_{0}(\lambda)}]$ already holds in
$K_{0}\mathrm{Coh}(\mathcal{N}/\hat{G})$, which follows from the results in
section 4 of [AH19]. Indeed, Achar-Hardesty prove that twisted Grothendieck-
Serre duality induces the _identity_ on
$K_{0}\mathrm{Coh}(\mathcal{N}/\hat{G})$, but on the other hand it is easy to
check that $\mathbf{D}_{\mathrm{twGS}}A_{\lambda}\simeq A_{w_{0}(\lambda)}$.

There is another, closely related conjecture. Fix a Levi subgroup $M\subset G$
and a parabolic $P=MU$ containing it. Recall from our discussion of Eisenstein
series that $\mathrm{Eis}_{P}^{G}$ is expected to preserve compact objects,
and also ULA objects with quasicompact support. With
$\overline{\mathbf{Q}_{\ell}}$-coefficients, we thus expect that it will
preserve finite sheaves, and in particular it will induce a functor

$[\mathrm{Eis}_{P}^{G}]:K_{0}D(\mathrm{Bun}_{M})_{\mathrm{fin}}\to
K_{0}D(\mathrm{Bun}_{G})_{\mathrm{fin}}.$

######  Conjecture 3.3.4.

The functor $[\mathrm{Eis}_{P}^{G}]$ depends only on the Levi $M$.

This is a geometric analogue of the classical fact that for any parabolic
$P=MU\subset G$, the map
$[i_{P}^{G}]:K_{0}\mathrm{Rep}(M(E))_{\mathrm{fin}}\to
K_{0}\mathrm{Rep}(G(E))_{\mathrm{fin}}$ depends only on the Levi $M$, which is
an easy consequence of van Dijk’s formula for the Harish-Chandra character of
a parabolic induction [vD72].

######  Proposition 3.3.5.

Conjecture 3.3.3 and Conjecture 3.3.4 are equivalent.

###### Sketch.

One implication follows immediately from Remark 1.4.8. The other implication
is much less obvious, and a detailed proof will appear in [HHS]. ∎

It turns out that Conjecture 1.5.2 and Conjecture 3.3.4 together imply a vast
generalization of the “Harris-Viehmann conjecture” describing the
$K_{0}$-class of the cohomology of non-basic local Shimura varieties in terms
of parabolic inductions. A detailed discussion will appear in [HHS].

######  Exercise 3.3.6.

Return to the notation and setup of Section 2.2.

1) (Difficult.) For any $\lambda\in X^{*}(\hat{T})$ and any $w\in W$, prove
that $[A_{\lambda}]=[A_{w(\lambda)}]$ in
$K_{0}\mathrm{Coh}(\mathcal{N}/\hat{G})$.

2) Show that 1) is consistent with Conjecture 3.3.4 and the categorical
conjecture. Hint: Generalize the arguments in Section 2.2 to show that for
$\lambda$ dominant and $w$ arbitrary,
$\mathrm{Eis}_{B}(i_{w(\lambda)!}\mathbf{1})\simeq\mathrm{Eis}_{{}^{w}B}(i_{\lambda!}\mathbf{1})$
should match with $A_{w(\lambda)}$ under the categorical conjecture.

3) (Difficult.) Assume Conjecture 2.2.1. Prove that for any $\lambda\in
X^{\ast}(\hat{T})^{+}$,

$[\mathscr{F}_{\lambda+2\rho}]=\sum_{w\in
W}(-1)^{\ell(w)}[i_{b_{\mathrm{dom}(\lambda+\rho+w\rho)}!}^{\mathrm{ren}}\pi_{\mathrm{dom}(\lambda+\rho+w\rho)}]$

in $K_{0}D(\mathrm{Bun}_{G})_{\mathrm{fin}}$. Here $\mathrm{dom}(\mu)$ denotes
the unique dominant element in the $W$-orbit of $\mu$. For
$G=\mathrm{PGL}_{3}$ and $\lambda=0$, compute the right-hand side explicitly.

######  Remark 3.3.7.

According to Conjecture 3.3.3, the elements
$[i_{b!}^{\mathrm{ren}}\pi]=[i_{b\sharp}^{\mathrm{ren}}\pi]$ should coincide,
and should give a canonical $\mathbf{Z}$-basis for
$K_{0}D(\mathrm{Bun}_{G})_{\mathrm{fin}}$ parametrized by pairs $(b,\pi)$.
There is a second canonical $\mathbf{Z}$-basis parametrized by the same set,
given by the elements $[\mathscr{G}_{b,\pi}]$ where $\mathscr{G}_{b,\pi}$ is
the irreducible hadal sheaf constructed in Theorem 1.2.7. In particular, we
can write any $[\mathscr{G}_{b,\pi}]$ uniquely as a finite $\mathbf{Z}$-linear
combination of $[i_{b^{\prime}!}^{\mathrm{ren}}\pi^{\prime}]$, and vice versa.
Experiments (including the previous exercise) suggest that the coefficients in
the expansion
$[\mathscr{G}_{b,\pi}]=\sum_{(b^{\prime},\pi^{\prime})}c_{b^{\prime},\pi^{\prime}}[i_{b^{\prime}!}^{\mathrm{ren}}\pi^{\prime}]$
are somewhat complicated, with unpredictable signs. However, in every example
I have worked out, the coefficients in the expansion

$[i_{b!}^{\mathrm{ren}}\pi]=\sum_{(b^{\prime},\pi^{\prime})}\alpha_{b^{\prime},\pi^{\prime}}[\mathscr{G}_{b^{\prime},\pi^{\prime}}]$

are _nonnegative_ integers. At semisimple generic parameters, this is
predicted by the discussion in section 2.3, but this is also true in some
nontrivial cases around the Steinberg parameter for $\mathrm{GL}_{2}$. Does
this positivity phenomenon persist in general?

##  Appendix A Homological properties of Whittaker functions

Fix $E/\mathbf{Q}_{p}$ finite, $G$ (the $E$-points of) a quasisplit reductive
group, $B=TU\subset G$ as usual. If $R$ is any commutative ring, we write
$\mathrm{Mod}_{R}(G)$ for the category of smooth $R[G]$-modules.

Let $\Lambda$ be any $\mathbf{Z}[\tfrac{1}{p},\zeta_{p^{\infty}}]$ algebra,
and let $\psi:U\to\Lambda^{\times}$ be any nondegenerate additive character.
We are interested in the space

$W_{\psi}=\mathrm{ind}_{U}^{G}(\psi)$

of compactly supported Whittaker functions with coefficients in $\Lambda$. In
other words, $W_{\psi}\subset\mathcal{C}(G,\Lambda)$ is the space of functions
such that $f(ug)=\psi(u)f(g)$ for all $u\in U$ and $g\in G$, $f$ is right-
invariant by some open compact subgroup of $G$, and the support of $f$ has
compact image in $U\backslash G$.

In the most classical situation where $\Lambda=\mathbf{C}$, Chan-Savin proved
that $W_{\psi}$ is a projective object in $\mathrm{Mod}_{\mathbf{C}}(G)$
[CS19], and Bushnell-Henniart proved that the summand $W_{\psi,\mathfrak{s}}$
corresponding to an individual Bernstein component $\mathfrak{s}$ is finitely
generated [BH03]. The following theorem is a generalization of these results.

######  Theorem A.0.1.

_i._ For any $\Lambda$, $W_{\psi}$ is a projective object in
$\mathrm{Mod}_{\Lambda}(G)$.

_ii._ Let $e_{r}:\mathrm{Mod}_{\Lambda}(G)\to\mathrm{Mod}_{\Lambda}(G)_{r}$ be
the projector onto the depth $\leq r$ factor category. Then
$e_{r}W_{\psi}\in\mathrm{Mod}_{\Lambda}(G)$ is finitely generated projective,
and $\mathbf{D}_{\mathrm{coh}}(e_{r}W_{\psi})\simeq e_{r}W_{\psi^{-1}}$.

_iii._ Assume that $\Lambda=\mathbf{C}$. Then
$\mathbf{D}_{\mathrm{coh}}(W_{\psi,\mathfrak{s}})\simeq
W_{\psi^{-1},\mathfrak{s}^{\vee}}$ for any Bernstein component $\mathfrak{s}$.

Here as before, $\mathbf{D}_{\mathrm{coh}}(-)$ denotes the cohomological
duality functor $R\mathrm{Hom}(-,\mathcal{C}_{c}^{\infty}(G,\Lambda))$. We
note that a different proof of i. and of the finite generation statement in
ii. was also discovered by Dat-Helm-Kurinczuk-Moss.

The main tools in our analysis of $W_{\psi}$ are Bushnell-Henniart’s
finiteness results over $\mathbf{C}$, together with a beautiful approximation
technique due to Rodier [Rod75], which seems perhaps under-appreciated. To
explain this, fix $\Lambda$-valued Haar measures $dg$ on $G$ and $du$ on $U$.
We may assume that the volume of any pro-$p$ open subgroup is a unit in
$\Lambda$. Following Rodier, we may choose a sequence of pairs
$(K_{n},\psi_{n})_{n\geq 1}$, where $K_{n}\subset G$ is an open compact
pro-$p$ subgroup and $\psi_{n}:K_{n}\to\Lambda^{\times}$ is a smooth
character, with the following remarkable properties.

  * P1

$K_{n}\cap U$ is an increasing sequence of groups, with
$U=\bigcup_{n}K_{n}\cap U$.

  * P2

$K_{n}\cap\overline{B}$ is a decreasing sequence of groups, with
$\\{e\\}=\bigcap_{n}K_{n}\cap\overline{B}$.

  * P3

For all $n$, $K_{n}=(K_{n}\cap
U)\cdot(K_{n}\cap\overline{B})=(K_{n}\cap\overline{B})\cdot(K_{n}\cap U)$.

  * P4

$\psi_{n}|_{K_{n}\cap\overline{B}}=1$ and $\psi_{n}|_{K_{n}\cap
U}=\psi|_{K_{n}\cap U}$.

  * P5

Let $\varphi_{n}\in\mathcal{C}_{c}^{\infty}(G,\Lambda)$ be the function on $G$
obtained from $\psi_{n}$ via extension by zero. Then there is an explicit
integer $B$ such that for all $n\geq B$,

$\varphi_{n}\ast\varphi_{n+1}\ast\varphi_{n}=\mathrm{vol}(K_{n})\mathrm{vol}(K_{n+1}\cap
K_{n})\varphi_{n}$

where $\ast$ denotes the usual convolution structure on
$\mathcal{C}_{c}^{\infty}(G,\Lambda)$ relative to the chosen Haar measure.

The existence of a sequence of pairs satisfying P1-P4 is not particularly
hard, but P5 lies significantly deeper. Note that Rodier’s paper only treated
the case of $G$ split and $p$ sufficiently large, in which case P5 corresponds
to the key “Lemme 5”, whose proof occupies the entirety of [Rod75, Section 5].
In the present generality, the existence of a system satisfying P1-P5 follows
from Varma’s paper [Var14].202020I am very grateful to Sandeep Varma for
helpful communications regarding his paper [Var14]. (Note that we write
$K_{n}$ where Varma writes $G_{n}^{\prime}$.) The essential point is that P5
follows from Lemma 9 of [Var14], upon observing that in the present situation,
the element $Y$ is principal nilpotent and contained in
$\overline{\mathfrak{b}}$, so $G(Y)\subset\overline{B}$ and then (by P2)
$G^{\prime}_{n+1}\cap G(Y)\subset G^{\prime}_{n+1}\cap\overline{B}\subset
G^{\prime}_{n}$, so we may take $\mathcal{Y}_{n}=\\{e\\}$ in Lemma 9 of
[Var14]. Note that [Rod75] and [Var14] are written in the usual setting where
the coefficient ring is $\mathbf{C}$, but it is immediate that the arguments
work for any $\Lambda$ as above.

With these preparations in hand, consider the compact inductions
$\mathrm{ind}_{K_{n}}^{G}(\psi_{n})$. Then for all positive integers $m,n$, we
have the $G$-equivariant map

$\displaystyle A_{n}^{m}:\mathrm{ind}_{K_{n}}^{G}(\psi_{n})$
$\displaystyle\to\mathrm{ind}_{K_{m}}^{G}(\psi_{m})$ $\displaystyle f$
$\displaystyle\mapsto\frac{1}{\mathrm{vol}(K_{m})}\int_{K_{m}}\psi_{m}(k)^{-1}f(kg)dk.$

Note that $A_{n}^{m}$ is well-defined independently of the choice of Haar
measure $dk$ on $K_{m}$. Note also that $A_{m}^{\ell}\circ
A_{n}^{m}=A_{n}^{\ell}$ for all $\ell\geq m\geq n$, so the representations
$\mathrm{ind}_{K_{n}}^{G}(\psi_{n})$ form an inductive system with transition
maps given by the $A_{n}^{m}$’s. It is also easy to check that for $m\geq n$,
the formula

$A_{n}^{m}(f)=\frac{1}{\mathrm{vol}(K_{m}\cap U)}\int_{K_{m}\cap
U}\psi(u)^{-1}f(ug)du$

is true. On the other hand, for any $n\geq 1$, we have a $G$-equivariant map

$\displaystyle\phi_{n}:\mathrm{ind}_{K_{n}}^{G}(\psi_{n})$ $\displaystyle\to
W_{\psi}$ $\displaystyle f$ $\displaystyle\mapsto\int_{U}\psi(u)^{-1}f(ug)du.$

Using the preceding formula for $A_{n}^{m}$, it is easy to check that
$\phi_{n}=\phi_{m}\circ A_{n}^{m}$ for all $m\geq n$, so passing to the
colimit we get a $G$-equivariant map

$\phi_{\infty}:\mathrm{colim}_{n}\mathrm{ind}_{K_{n}}^{G}(\psi_{n})\to
W_{\psi}.$

It is then formal (Proposition 3 in Rodier’s paper) that $\phi_{\infty}$ is an
isomorphism. Note that all the arguments so far used only P1-P4.

The key remaining observation is that for all $m\geq n\geq B$, $A_{n}^{m}$ is
injective and $A_{m}^{n}$ is surjective. This follows immediately from the
claim that $A_{m}^{n}\circ A_{n}^{m}=\frac{\mathrm{vol}(K_{m}\cap
K_{n})}{\mathrm{vol}(K_{m})}\mathrm{id}$ for all $m\geq n\geq B$ (note that
the fraction here is a power of $p$, hence a unit in $\Lambda$). To prove this
claim, first note (using P1-P3) that $\frac{\mathrm{vol}(K_{m}\cap
K_{n})}{\mathrm{vol}(K_{m})}=\frac{\mathrm{vol}(K_{n}\cap
U)}{\mathrm{vol}(K_{m}\cap U)}$, whence the claim reduces by induction to the
special case $m=n+1$. The case $m=n+1$ in turn is a direct consequence of P5
above, upon noting that
$\mathrm{ind}_{K_{n}}^{G}(\psi_{n})\subset\mathcal{C}_{c}^{\infty}(G,\Lambda)$
is exactly the image of the map

$\displaystyle\mathcal{C}_{c}^{\infty}(G,\Lambda)$
$\displaystyle\to\mathcal{C}_{c}^{\infty}(G,\Lambda)$ $\displaystyle f$
$\displaystyle\mapsto\varphi_{n}\ast f,$

and that the map $A_{n+1}^{n}\circ A_{n}^{n+1}$ coincides with further left
convolution by $\varphi_{n}\ast\varphi_{n+1}$ up to an explicit scaling. For
details, see the proof of [Rod75, Proposition 4].

The equality $A_{m}^{n}\circ A_{n}^{m}=\frac{\mathrm{vol}(K_{m}\cap
K_{n})}{\mathrm{vol}(K_{m})}\mathrm{id}$ immediately implies that for all
$m\geq n\geq B$, $A_{n}^{m}$ is a split injection, whence also
$\phi_{n}=\phi_{\infty}\circ\mathrm{colim}_{m}A_{n}^{m}$ is a split injection.

###### Proof of Theorem A.1..

First we prove i. With the above preparations on Rodier approximation at hand,
this result proves itself. The only thing to observe is that each
$\mathrm{ind}_{K_{n}}^{G}(\psi_{n})$ is projective, so Rodier approximation
presents $W_{\psi}$ as the colimit of a directed system of projective objects
along split injective transition maps. But the colimit of any such directed
system is itself a projective object.

Next we prove iii. Note first that in the discussion of Rodier approximation,
we may replace $\psi$ resp. $\psi_{n}$ with $\psi^{-1}$ resp. $\psi_{n}^{-1}$
everywhere without changing the validity of any statements. In particular, we
may simultaneously write
$W_{\psi}\simeq\mathrm{colim}_{n}\mathrm{ind}_{K_{n}}^{G}(\psi_{n})$ and
$W_{\psi^{-1}}\simeq\mathrm{colim}_{n}\mathrm{ind}_{K_{n}}^{G}(\psi_{n}^{-1})$,
where for all sufficiently large $n$ the transition maps in both systems are
(split) injections. Now pick any Bernstein component $\mathfrak{s}$. Again, we
have
$W_{\psi,\mathfrak{s}}\simeq\mathrm{colim}_{n}\mathrm{ind}_{K_{n}}^{G}(\psi_{n})_{\mathfrak{s}}$
and
$W_{\psi^{-1},\mathfrak{s}^{\vee}}\simeq\mathrm{colim}_{n}\mathrm{ind}_{K_{n}}^{G}(\psi_{n}^{-1})_{\mathfrak{s}^{\vee}}$,
where for all sufficiently large $n$ the transition maps are injections. Since
$W_{\psi,\mathfrak{s}}$ is finitely generated, this immediately implies that
the map $\mathrm{ind}_{K_{n}}^{G}(\psi_{n})_{\mathfrak{s}}\to
W_{\psi,\mathfrak{s}}$ is an isomorphism for all sufficiently large $n$.
Repeating the same argument, we also get that the map
$\mathrm{ind}_{K_{n}}^{G}(\psi_{n}^{-1})_{\mathfrak{s}^{\vee}}\to
W_{\psi^{-1},\mathfrak{s}^{\vee}}$ is an isomorphism for all sufficiently
large $n$. But
$\mathbf{D}_{\mathrm{coh}}(\mathrm{ind}_{K_{n}}^{G}(\psi_{n})_{\mathfrak{s}})\simeq\mathrm{ind}_{K_{n}}^{G}(\psi_{n}^{-1})_{\mathfrak{s}^{\vee}}$,
which gives the result.

Finally we prove ii. Observe first that the formation of the representations
$W_{\psi}$ and $\mathrm{ind}_{K_{n}}^{G}(\psi_{n})$ and the maps $A_{n}^{m}$,
$\phi_{n}$, $\phi_{\infty}$ are compatible with extension of scalars along any
ring map $\Lambda\to\Lambda^{\prime}$. Since BZ duality and the depth $\leq r$
projector $e_{r}$ are also compatible with any extension of scalars, we are
immediately reduced to the universal case
$\Lambda=\mathbf{Z}[\tfrac{1}{p},\zeta_{p^{\infty}}]$. Pick an embedding
$\Lambda\to\mathbf{C}$; we will write $(-)_{\mathbf{C}}$ for objects obtained
by the evident extension of scalars along this map.

Since $\mathrm{Mod}_{\mathbf{C}}(G)_{r}$ is a product of finitely many
Bernstein components, the proof of iii. shows that the maps
$e_{r}\phi_{n,\mathbf{C}}:e_{r}\mathrm{ind}_{K_{n}}^{G}(\psi_{n})_{\mathbf{C}}\to
e_{r}W_{\psi,\mathbf{C}}$ and
$e_{r}\mathrm{ind}_{K_{n}}^{G}(\psi_{n}^{-1})_{\mathbf{C}}\to
e_{r}W_{\psi^{-1},\mathbf{C}}$ are isomorphisms for sufficiently large $n$.
Now consider the maps
$e_{r}\phi_{n}:e_{r}\mathrm{ind}_{K_{n}}^{G}(\psi_{n})_{\mathbf{}}\to
e_{r}W_{\psi}$. Since $\phi_{n}$ is a split injection for all sufficiently
large $n$, we get also that $e_{r}\phi_{n}$ is a split injection for all
sufficiently large $n$. Moreover, the source and target of $\phi_{n}$ are
projective $\Lambda$-modules,212121For $\mathrm{ind}_{K_{n}}^{G}(\psi_{n})$
this is clear, and for $W_{\psi}$ it follows from the proof of i. Is it clear
from first principles that $W_{\psi}$ is projective as a $\Lambda$-module? so
also the source and target of $e_{r}\phi_{n}$ are projective
$\Lambda$-modules. In particular, $\mathrm{coker}\,e_{r}\phi_{n}$ is a
projective $\Lambda$-module for sufficiently large $n$. But we’ve already
established that

$(\mathrm{coker}\,e_{r}\phi_{n})\otimes_{\Lambda}\mathbf{C}\simeq\mathrm{coker}\,e_{r}\phi_{n,\mathbf{C}}=0$

for all sufficiently large $n$. This implies that
$\mathrm{coker}\,e_{r}\phi_{n}=0$ for all sufficiently large $n$, so
$e_{r}\phi_{n}:e_{r}\mathrm{ind}_{K_{n}}^{G}(\psi_{n})\to e_{r}W_{\psi}$ is an
isomorphism for all sufficiently large $n$. Since
$\mathrm{ind}_{K_{n}}^{G}(\psi_{n})$ and its summand
$e_{r}\mathrm{ind}_{K_{n}}^{G}(\psi_{n})$ are finitely generated, we deduce
that $e_{r}W_{\psi}$ is finitely generated.

Repeating the same argument with inverses on all characters, we also get that
$e_{r}\mathrm{ind}_{K_{n}}^{G}(\psi_{n}^{-1})\to e_{r}W_{\psi^{-1}}$ is an
isomorphism for all sufficiently large $n$. Since
$\mathbf{D}_{\mathrm{coh}}(\mathrm{ind}_{K_{n}}^{G}(\psi_{n}))=\mathrm{ind}_{K_{n}}^{G}(\psi_{n}^{-1})$
and $e_{r}$ commutes with $\mathbf{D}_{\mathrm{coh}}$, we compute that

$\displaystyle\mathbf{D}_{\mathrm{coh}}(e_{r}W_{\psi})$
$\displaystyle\simeq\mathbf{D}_{\mathrm{coh}}(e_{r}\mathrm{ind}_{K_{n}}^{G}(\psi_{n}))$
$\displaystyle=e_{r}\mathbf{D}_{\mathrm{coh}}(\mathrm{ind}_{K_{n}}^{G}(\psi_{n}))$
$\displaystyle=e_{r}\mathrm{ind}_{K_{n}}^{G}(\psi_{n}^{-1})$
$\displaystyle\simeq e_{r}W_{\psi^{-1}}.$

This concludes the proof. ∎

###  A.1 A reasonable condition

In this section we again take our coefficients to be
$\Lambda=\overline{\mathbf{Q}_{\ell}}$. Let $G/E$ be any connected reductive
group.

######  Definition A.1.1.

i. The group $G$ is _reasonable_ if the Fargues-Scholze map
$\Psi_{G}^{\mathrm{geom}}:X_{G}\to X_{G}^{\mathrm{spec}}$ is finite, or
equivalently (by Lemma 1.6.4) if the associated map
$\pi_{0}X_{G}\to\pi_{0}X_{G}^{\mathrm{spec}}$ has finite fibers.

ii. The group $G$ is _very reasonable_ if $G_{b}$ is reasonable for all $b\in
B(G)$.

It is clear that if $G$ is very reasonable, then $G_{b}$ is very reasonable
for all $b\in B(G)$.

######  Exercise A.1.2.

Prove that if $G$ is reasonable, then any Levi subgroup $M\subset G$ is
reasonable.

Our main reason for considering these conditions is the following basic
finiteness result, which we leave to the reader as an exercise. (The key
ingredient for i. is Bushnell-Henniart’s finiteness theorem cited above.)

######  Proposition A.1.3.

_i._ If $\mathrm{char}E=0$ and $G$ is quasisplit and reasonable, the functor
$\mathcal{F}\mapsto\mathcal{F}\ast i_{1!}W_{\psi}$ sends
$\mathrm{Perf^{qc}(}\mathrm{Par}_{G})$ into $D(\mathrm{Bun}_{G})^{\omega}$.

_ii._ If $G$ is very reasonable, then for every $b$ and every semisimple
parameter $\phi:W_{E}\to\,^{L}G(\overline{\mathbf{Q}_{\ell}})$, the set of
irreducible smooth representations $\pi\in\Pi(G_{b})$ with Fargues-Scholze
parameter $\phi$ is finite.

_iii._ If $G$ is very reasonable, then every ULA sheaf in
$D(\mathrm{Bun}_{G})$ is a filtered colimit of finite sheaves.

Note that the condition in Proposition A.1.3.ii is actually equivalent to very
reasonableness. This follows from finiteness of the map
$X_{G_{b}}^{\mathrm{spec}}\to X_{G}^{\mathrm{spec}}$, Theorem 1.4.2, and basic
structure theory a la Bernstein.

It’s also quite plausible that Proposition A.1.3 iii. could be proved
unconditionally for any $G$, but I didn’t try very hard to check this. If $G$
is very reasonable, however, it is nearly trivial: if $A$ is ULA, then for any
semisimple parameter $\phi$ and any quasicompact open substack
$j:U\to\mathrm{Bun}_{G}$, the sheaf $j_{!}j^{\ast}A_{\phi}$ is finite by the
very reasonableness condition, and the natural map

$\mathrm{colim}_{U,S}\oplus_{\phi\in S}j_{!}j^{\ast}A_{\phi}\to A$

is an isomorphism (using the decomposition $A\cong\oplus_{\phi}A_{\phi}$
proved in [Han23b]), where $S$ runs over (the filtered collection of) finite
sets of semisimple _L_ -parameters.

Of course we expect that every group is very reasonable. Right now, we know
that $\mathrm{GL}_{n}$ is very reasonable [FS21, HKW22], as well as
$\mathrm{GSp}_{4}$ with $E/\mathbf{Q}_{p}$ unramified and $p>2$ [Ham21],
unramified $\mathrm{(G)U}_{2n+1}/\mathbf{Q}_{p}$ [BMHN22],
$\mathrm{SO}_{2n+1}$ with $E/\mathbf{Q}_{p}$ unramified and $p>2$ (H.,
unpublished), and groups obtained from these by passing to derived groups,
products, central isogenies, twisted Levis, etc. To my knowledge, there is no
group which is known to be reasonable but not known to be very reasonable.

######  Exercise A.1.4.

Show that if $G$ and $G^{\prime}$ have isomorphic adjoint groups and $G$ is
reasonable, then $G^{\prime}$ is reasonable. Can you formulate a similar
statement for “very reasonableness”?

##  Appendix B A dimensional classicality criterion for derived stacks, by
Adeel Khan

We define (co)dimension of derived schemes and stacks on classical truncations
(see [Sta21, Tag 04N3] or [GD71, 0_IV, 14.1.2, 14.2.4] for schemes, and
[Sta21, Tags 0AFL and 0DRL] for stacks).

######  Proposition B.0.1.

Let $f:\mathcal{X}\to\mathcal{Y}$ be a quasi-smooth morphism of derived
$1$-Artin stacks where $\mathcal{Y}_{\mathrm{cl}}$ is Cohen-
Macaulay.222222Equivalently, $\mathcal{Y}$ admits a smooth surjection
$Y\twoheadrightarrow\mathcal{Y}$ where $Y_{\mathrm{cl}}$ is a Cohen–Macaulay
scheme. If $x\in|\mathcal{X}|$ is a point at which the relative dimension of
$f$ is equal to the relative virtual dimension of $f$, then
$\mathcal{X}\operatorname*{\times}^{\mathbf{R}}_{\mathcal{Y}}\mathcal{Y}_{{\mathrm{cl}}}$
is classical in a Zariski neighbourhood of $x$.

We will make use of the following lemma from [KR18, 2.3.12].

######  Lemma B.0.2.

Let $Z\to X$ be a quasi-smooth closed immersion of derived schemes where
$X_{{\mathrm{cl}}}$ is Cohen–Macaulay. Then we have
$-\operatorname{vdim}(Z/X)\geqslant\operatorname{\operatorname{codim}}(Z,X)$,
with equality if and only if
$Z\operatorname*{\times}^{\mathbf{R}}_{X}X_{{\mathrm{cl}}}$ is classical in a
Zariski neighbourhood of $x$.

###### Proof of Proposition B.0.1.

The statement is invariant under replacing $\mathcal{Y}$ by
$\mathcal{Y}_{{\mathrm{cl}}}$ and $\mathcal{X}$ by
$\mathcal{X}\operatorname*{\times}^{\mathbf{R}}_{\mathcal{Y}}\mathcal{Y}_{{\mathrm{cl}}}$,
so we may assume $\mathcal{Y}$ classical.

Suppose first that $\mathcal{X}=X$ and $\mathcal{Y}=Y$ are schemes. Since
$f:X\to Y$ is quasi-smooth, there exists for every $x\in\lvert X\rvert$ over
$y$ a Zariski neighbourhood $U\subseteq X$ of $x$, a derived scheme $M$ which
is smooth over $Y$, and a quasi-smooth closed immersion $U\hookrightarrow M$
over $Y$ (see [KR18, Prop. 2.3.14]). We have

$\displaystyle\operatorname{vdim}_{x}(U_{y}/M_{y})$
$\displaystyle=\operatorname{vdim}_{x}(U/M),$
$\displaystyle\operatorname{\operatorname{codim}}_{x}(U_{y},M_{y})$
$\displaystyle\leqslant\operatorname{\operatorname{codim}}_{x}(U,M).$

Since $\operatorname{vdim}_{x}(U_{y}/\kappa(y))=\dim_{x}(U_{y})$ by
assumption, we also have

$-\operatorname{vdim}_{x}(U_{y}/M_{y})=\dim_{x}(M_{y})-\dim_{x}(U_{y})=\operatorname{\operatorname{codim}}_{x}(U_{y},M_{y})$

where the last equality holds because $M_{y}$ is catenary (see [GD71, 0_IV
Cor.16.5.12]; [GD71, IV_2 Prop. 5.1.9]). We conclude that

$-\operatorname{vdim}_{x}(U/M)\leqslant\operatorname{\operatorname{codim}}_{x}(U,M).$

Now Lemma B.0.2 implies that $U$ is classical in a Zariski neighbourhood of
$x$.

Next suppose that $\mathcal{X}=X$ and $\mathcal{Y}=Y$ are algebraic spaces.
Choose an Ã©tale surjection $X_{0}\twoheadrightarrow X$ where $X_{0}$ is a
derived scheme, and let $x_{0}\in\lvert X_{0}\rvert$ be a lift of the given
point $x\in\lvert X\rvert$. Choose also an Ã©tale surjection
$Y_{0}\twoheadrightarrow Y$ where $Y_{0}$ is a Cohen–Macaulay scheme and a
lift $y_{0}\in\lvert Y_{0}\rvert$ of $y$. Since $Y$ has schematic diagonal,
$X_{0}\operatorname*{\times}_{Y}Y_{0}$ is a derived scheme. Applying the case
above to the morphism $X_{0}\operatorname*{\times}_{Y}Y_{0}\to Y_{0}$, we
obtain a Zariski neighbourhood of $(x_{0},y_{0})\in
X_{0}\operatorname*{\times}_{Y}Y_{0}$ which is classical. Its image along the
Ã©tale morphism $X_{0}\operatorname*{\times}_{Y}Y_{0}\twoheadrightarrow
X_{0}\twoheadrightarrow X$ is then a Zariski neighbourhood of $x\in X$ which
is classical.

Finally we consider the general case. Choose a smooth surjection
$X\twoheadrightarrow\mathcal{X}$ where $X$ is a derived scheme, a lift
$x_{0}\in\lvert X_{0}\rvert$ of the given point $x\in\lvert X\rvert$, a smooth
surjection $Y_{0}\twoheadrightarrow Y$ where $Y_{0}$ is a Cohen–Macaulay
scheme, and a lift $y_{0}\in\lvert Y_{0}\rvert$ of $y$. Since $Y$ has
representable diagonal, $X\operatorname*{\times}_{\mathcal{Y}}Y$ is a derived
algebraic space. Hence the previous case applied to the morphism
$X\operatorname*{\times}_{\mathcal{Y}}Y\to Y$ yields a Zariski neighbourhood
of $(x_{0},y_{0})\in X\operatorname*{\times}_{\mathcal{Y}}Y$ which is
classical. Its image along the smooth morphism
$X\operatorname*{\times}_{\mathcal{Y}}Y\twoheadrightarrow
X\twoheadrightarrow\mathcal{X}$ is then a Zariski neighbourhood of
$x\in\mathcal{X}$ which is classical. ∎

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