# Beijing notes on the categorical local Langlands conjecture

David Hansen
with an appendix by Adeel Khan

_A fool can ask more questions in a minute_

_than a wise man can answer in an hour._

###### Contents

  1. 1 The categorical conjecture
    1. 1.1 The automorphic side
      1. 1.1.1 Finiteness and duality
    2. 1.2 t-structures
    3. 1.3 The spectral side
    4. 1.4 Categorical conjecture
    5. 1.5 More conjectures
    6. 1.6 Finiteness conditions and spectral decomposition of sheaves
    7. 1.7 The categorical conjecture over $\overline{\mathbf{Q}_{\ell}}$, unconditionally
      1. 1.7.1 A variant with restricted variation
  2. 2 Compatibility with classical local Langlands
    1. 2.1 Generous _L_ -parameters
      1. 2.1.1 Example: Supercuspidal parameters
      2. 2.1.2 Example: Generic toral parameters for $\mathrm{GL}_{n}$
    2. 2.2 The trivial _L_ -parameter
    3. 2.3 Semisimple generic parameters
    4. 2.4 Perverse t-exactness of Hecke operators
    5. 2.5 Two t-exactness conjectures
  3. 3 Additional conjectures
    1. 3.1 ULA sheaves and generic parameters
    2. 3.2 More examples
    3. 3.3 Grothendieck groups and vanishing conjectures
  4. A Homological properties of Whittaker functions
    1. A.1 A reasonable condition
  5. B A dimensional classicality criterion for derived stacks, by Adeel Khan

### Acknowledgements

These are expanded notes written for a lecture series at the Morningside
Center of Mathematics in July 2023. I’m very grateful for the opportunity to
give these lectures, which significantly pushed me to clarify a number of
vague ideas and write them out carefully. Hopefully this written account will
be helpful to others in the field. On a personal level, I’m especially
grateful to Shizhang Li for his extremely kind and generous hospitality during
my visit to Beijing.

It’s a pleasure to thank Pramod Achar, Alexander Bertoloni Meli, Jean-François
Dat, Laurent Fargues, Linus Hamann, Eugen Hellmann, Christian Johansson, Tasho
Kaletha, Adeel Khan, Teruhisa Koshikawa, Lucas Mann, Sam Raskin, Peter
Scholze, Sug Woo Shin, Sandeep Varma, Xinwen Zhu, Yihang Zhu, and Konrad Zou
for some very helpful and inspiring discussions. I’d particularly like to
thank Adeel Khan for proving Proposition B.0.1 and agreeing to include it as
an appendix; Linus Hamann for teaching me about Eisenstein series; and
Alexander Bertoloni Meli for many discussions and valuable encouragement as I
worked out these ideas. I also got crucial inspiration from beautiful lectures
by Ian Gleason, Teruhisa Koshikawa, João Lourenço, and Xinwen Zhu. Finally,
Christian Johansson pointed out back in January that the paper [Bez03] might
be worth looking at; as always, his advice was on the mark.

### What these notes are trying to do

Fix a prime $p$, and a nonarchimedean local field $E$ with uniformizer
$\varpi$ and residue field $\mathbf{F}_{q}$ of characteristic $p$. Let $G/E$
be a connected reductive group, which we assume for simplicity is quasisplit.
The _local Langlands conjecture,_ in its vaguest form, seeks to parametrize
the irreducible smooth representations of $G(E)$ in terms of Galois-theoretic
data. A huge amount of effort has been expended on making this conjecture
precise, and proving it in many cases. In its modern form, the conjecture
proposes that irreducible smooth representations of $G(E)$ can be parametrized
by (suitable) pairs $(\phi,\rho)$ where
$\phi:W_{E}\times\mathrm{SL}_{2}\to\phantom{}^{L}G$ is an _L-_ parameter and
$\rho$ is an irreducible algebraic representation of a group closely related
to the centralizer group $S_{\phi}=\mathrm{Cent}_{\hat{G}}(\phi)$. Moreover,
this parametrization should depend only on a choice of Whittaker datum for
$G$, and should satisfy many good properties. We refer to [Kal16] for a
precise formulation.

In recent years, the subject has undergone a major transformation, thanks to
Fargues’s amazing discovery that ideas from _geometric Langlands_ can be
imported into the study of local Langlands, by reinterpreting the basic
structures in local Langlands in terms of $G$-bundles on the Fargues–Fontaine
curve [Far16, Far14]. The foundations for this rebuilding of the field were
then laid in the revolutionary manuscripts [FS21, Sch17]. Among other things,
they succeed in constructing a canonical map $\pi\mapsto\varphi_{\pi}$ from
irreducible smooth representations towards semisimple _L_ -parameters, which
should be the semisimplification of the “true” local Langlands
parametrization.

In a parallel stream, the idea emerged that the derived category of all smooth
$G(E)$-representations should embed fully faithfully into ind-coherent sheaves
on the stack of _L_ -parameters. This was made precise (independently) by
Hellmann [Hel23], Zhu [Zhu21], and Ben-Zvi–Chen–Helm–Nadler. It is then
natural to conjecture that some form of the geometric Langlands conjecture
holds in this setting, namely that the category of all sheaves on
$\mathrm{Bun}_{G}$ should be equivalent to the category of ind-coherent
sheaves on the stack of _L_ -parameters. A formal conjecture along these lines
was proposed by Fargues–Scholze [FS21, Conjecture I.10.2], without however
pinning down the functor which should give the equivalence. An alternative
formulation was proposed by Zhu, working instead with sheaves on the stack of
$G$-isocrystals [Zhu21].

These notes should be regarded as a minor supplement to the works mentioned
above. More concretely, we are guided by the following motivating questions,
in the setting of Fargues-Scholze:

  1. 1.

Can we give an unconditional formulation of the categorical local Langlands
conjecture?

  2. 2.

What properties and compatibilities should the categorical conjecture enjoy?
What additional conjectures do these properties suggest?

  3. 3.

How does the categorical conjecture encode the classical local Langlands
conjecture?

  4. 4.

How do the non-basic strata in $\mathrm{Bun}_{G}$ fit into the picture?

These notes are _not_ intended as a detailed introduction to the structures
required to formulate these questions. More specifically, we will assume the
reader is somewhat familiar with the modern expectations regarding the local
Langlands correspondence, as outlined in (say) [Kal16] and [Kal22]. We will
also assume some familiarity with [FS21], and with the philosophy of the
“classical” geometric Langlands program.

There are exercises scattered throughout the text. They vary widely in
difficulty, but none of them are open problems.

### Notation and conventions

We will introduce various notations throughout the text. Here we briefly
mention several perhaps non-standard conventions. If $X$ is a disjoint union
of finite type (derived) Artin stacks, we write $\mathrm{Coh}(X)$ for the
category of bounded coherent complexes on $X$ with $X$ regarded as an ind-
algebraic stack. In particular, any object in $\mathrm{Coh}(X)$ automatically
has quasicompact support. On the other hand, we write $\mathrm{Perf}(X)$ for
all dualizable objects in $\mathrm{QCoh}(X)$. This leads to the slightly
unfortunate fact that $\mathrm{Perf}$ may not be contained in $\mathrm{Coh}$.

If $X$ is an object in a triangulated or stable $\infty$-category
$\mathcal{D}$, we say $X$ admits a filtration with graded pieces
$A_{1},\dots,A_{n}$ if there exists a sequence of maps $0=X_{0}\to X_{1}\to
X_{2}\to\cdots\to X_{n}=X$ with $A_{j}\simeq\mathrm{cone}(X_{j-1}\to X_{j})$
for all $1\leq j\leq n$. Similar remarks apply to functors between stable
$\infty$-categories.

##  1 The categorical conjecture

Fix a prime $p$, and a nonarchimedean local field $E$ with uniformizer
$\varpi$ and residue field $\mathbf{F}_{q}$ of char. $p$. Fix also a prime
$\ell\neq p$. Let $\Lambda$ be a $\mathbf{Z}_{\ell}$-algebra containing a
fixed choice of $\sqrt{q}$.

Let $G$ be a connected reductive group over $E$. We often assume that $G$ is
quasisplit, both for simplicity and because the categorical conjecture
requires this assumption. Let $Q$ be the minimal finite quotient of $W_{E}$
which acts nontrivially on $\hat{G},$ and put ${}^{L}G=\hat{G}\rtimes Q$,
which we regard as a linear algebraic group over
$\mathbf{Z}_{\ell}[\sqrt{q}]$. We write $\mathrm{Rep}_{\Lambda}(^{L}G)$ and
$\mathrm{Rep}_{\Lambda}(\hat{G})$ for the evident tensor categories of
algebraic representations on finite projective $\Lambda$-modules.

###  1.1 The automorphic side

The basic geometric object is the stack $\mathrm{Bun}_{G}$ of $G$-bundles on
the Fargues-Fontaine curve. We will not review this in detail here, since it
is now a classical object.111I am joking. The key feature of its geometry is
that it has a Harder-Narasimhan stratification by locally closed substacks
$\mathrm{Bun}_{G}^{b}$ indexed by elements of the Kottwitz set $b\in B(G)$
[Far20]. Each stratum is a classifying stack for a certain group v-sheaf
$\tilde{G}_{b}$, which is an extension of the constant group sheaf
$\underline{G_{b}(E)}$ by a connected unipotent group. By a deep theorem of
Viehmann [Vie23], the topology on $|\mathrm{Bun}_{G}|\cong B(G)$ is exactly
the Newton partial order topology on $B(G)$. We write $b\preceq b^{\prime}$ if
$b^{\prime}\in\overline{\\{b\\}}$, so the minimal elements are exactly the
basic $b$.

On the automorphic side, the key player is the category
$D(\mathrm{Bun}_{G},\Lambda)=D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)$ of
sheaves on $\mathrm{Bun}_{G}$. This is glued semi-orthogonally from analogous
categories of sheaves attached to each Harder-Narasimhan stratum, which turn
out to be much simpler. More precisely, for each stratum we have a canonical
t-exact tensor equivalence

$D(\mathrm{Bun}_{G}^{b},\Lambda)\cong D(G_{b}(E),\Lambda),$

where the right-hand side is simply the derived category of the abelian
category of smooth $\Lambda[G_{b}(E)]$-modules. This equivalence is induced by
the functors $s_{b\natural}$ and $s_{b}^{\ast}$ associated with the
tautological map $s_{b}:[\ast/\underline{G_{b}(E)}]\to\mathrm{Bun}_{G}^{b}$,
which turn out to be mutually inverse t-exact tensor equivalences. We will
_always_ identify $D(\mathrm{Bun}_{G}^{b},\Lambda)$ and $D(G_{b}(E),\Lambda)$
in this way.

For any $b\in B(G)$, there are functors
$i_{b}^{\ast}:D(\mathrm{Bun}_{G},\Lambda)\to D(G_{b}(E),\Lambda)$ and
$i_{b!}:D(G_{b}(E),\Lambda)\to D(\mathrm{Bun}_{G},\Lambda)$, which do “exactly
what you think they do.” The functor $i_{b!}$ has a right adjoint $i_{b}^{!}$,
while $i_{b}^{\ast}$ has a right adjoint $i_{b\ast}$ and also a left adjoint
$i_{b\sharp}$. This left adjoint is a special feature of the situation. The
three pushforwards are linked by natural transformations

$i_{b\sharp}\to i_{b!}\to i_{b\ast}$

which induce the identity after applying $i_{b}^{\ast}$. Note that
$i_{b\ast}A$ can only have nonzero stalks at points which are specializations
of $b$, while (more strangely) $i_{b\sharp}A$ can only have nonzero stalks at
points which are generizations of $A$.

Let us briefly recall the construction of these functors, especially the
pushforwards $i_{b\sharp}$ and $i_{b!}$ and the natural transformations
mentioned above. For any $b\in B(G)$, Fargues-Scholze construct an auxiliary
diagram of small v-stacks

---
$\textstyle{\mathcal{M}_{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{b}}$$\scriptstyle{\pi_{b}}$$\textstyle{\mathrm{Bun}_{G}}$$\textstyle{{}[\ast/\underline{G_{b}(E)}]}$

where $\pi_{b}$ is cohomologically smooth and surjects onto the open substack
$\mathrm{Bun}_{G}^{\preceq b}$ of points which are “more semistable” than $b$.
Moreover, the map $q_{b}$ has a canonical section given by a closed immersion
$[\ast/\underline{G_{b}(E)}]\to\mathcal{M}_{b}$. Puncturing $\mathcal{M}_{b}$
along this section we get an auxiliary diagram

---
$\textstyle{\mathcal{M}_{b}^{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{b}^{\circ}}$$\scriptstyle{\pi_{b}^{\circ}}$$\textstyle{\mathrm{Bun}_{G}}$$\textstyle{{}[\ast/\underline{G_{b}(E)}]}$

together with a natural open immersion
$j_{b}:\mathcal{M}_{b}^{\circ}\to\mathcal{M}_{b}$. The functor $i_{b\sharp}$
turns out to be given by the formula
$i_{b\sharp}=\pi_{b\natural}q_{b}^{\ast}$. The natural transformation
$j_{b\natural}j_{b}^{\ast}\to\mathrm{id}$ then induces a natural
transformation
$\pi_{b\natural}^{\circ}q_{b}^{\circ\ast}\to\pi_{b\natural}q_{b}^{\ast}$, and
we define $i_{b!}$ as the cofiber of this map.222This definition may seem
surprisingly complicated, but the issue is that in the $D_{\mathrm{lis}}$
formalism, we only have a priori access to $\natural$-pushforwards along
cohomologically smooth maps. Consequently, we are forced to construct $i_{b!}$
indirectly since $i_{b}$ itself is not cohomologically smooth. From this
construction the map $i_{b\sharp}\to i_{b!}$ is tautological. An easy
application of base change shows that $i_{b}^{\ast}i_{b!}\cong\mathrm{id}$, so
by adjunction we get the claimed transformation $i_{b!}\to i_{b\ast}$.
Finally, $i_{b}^{!}$ is defined as the right adjoint of $i_{b!}$.333We will
hardly use $i_{b}^{!}$ in these notes, but we mention it for completeness.

For our purposes, it will be very convenient to “renormalize” these functors.
To make the relevant definition, we note that $G_{b}$ is an inner form of a
Levi subgroup $M_{\\{b\\}}\subset G$, namely the centralizer of the Newton
cocharacter $\nu_{b}$. Let $P_{\\{b\\}}$ be the dynamic parabolic of
$\nu_{b}$, and let $\delta_{b}:M_{\\{b\\}}(E)\to q^{\mathbf{Z}}$ be the usual
modulus character. Since $G_{b}$ and $M_{\\{b\\}}$ have canonically isomorphic
cocenters, we can regard $\delta_{b}$ as a cocharacter $G_{b}(E)\to
q^{\mathbf{Z}}$. Since we have fixed a choice of $\sqrt{q}$ in our coefficient
ring, we can take the square root to get a character
$\delta_{b}^{1/2}:G_{b}(E)\to\Lambda^{\times}$.

######  Definition 1.1.1 (Renormalized functors).

For $?\in\\{\sharp,!,\ast\\}$, we define
$i_{b?}^{\mathrm{ren}}:D(G_{b}(E),\Lambda)\to D(\mathrm{Bun}_{G},\Lambda)$ by
the formula

$i_{b?}^{\mathrm{ren}}A=i_{b?}(A\otimes\delta_{b}^{1/2})[-\left\langle
2\rho_{G},\nu_{b}\right\rangle].$

Similarly, for $?\in\\{\ast,!\\}$ we define
$i_{b}^{?\mathrm{ren}}A=(\delta_{b}^{-1/2}\otimes i_{b}^{?}A)[\left\langle
2\rho_{G},\nu_{b}\right\rangle]$.

Note that when $b$ is basic, we have not changed the functors at all. The
importance of this renormalization will become clear later. For now let us
just remark that the renormalized functors interact cleanly with duality
(Proposition 1.1.4), have good (semi)perversity properties (Exercise 1.2.2),
are closely related to geometric Eisenstein series (Remark 1.4.8), and
preserve _L-_ parameters (Theorem 1.4.2).

######  Proposition 1.1.2.

One has the following a priori vanishing results:

_i._ $R\mathrm{Hom}(i_{b\sharp}A,i_{b^{\prime}\sharp}A^{\prime})=0$ unless
$b\preceq b^{\prime}$,

_ii._ $R\mathrm{Hom}(i_{b!}A,i_{b^{\prime}!}A^{\prime})=0$ unless
$b^{\prime}\preceq b$,

_iii._ $R\mathrm{Hom}(i_{b\sharp}A,i_{b^{\prime}!}A^{\prime})=0$ unless
$b=b^{\prime}$,

_iv. $R\mathrm{Hom}(i_{b\sharp}A,i_{b^{\prime}\ast}A^{\prime})=0$ _unless _
$b^{\prime}\preceq b$._
The obvious variants hold for renormalized pushforwards.

###### Proof.

Parts i., iii. and iv. are exercises using various adjunctions together with
the support properties of $i_{b\ast}$ and $i_{b\sharp}$ mentioned above, and
ii. can be deduced from i. by duality. ∎

####  1.1.1 Finiteness and duality

There are two natural finiteness conditions one can impose on objects of
$D(G_{b}(E),\Lambda)$. On one hand, we can consider the _compact_ _objects_ in
this category. It turns out that any compact object is built from basic
compact objects, i.e. representations of the form
$\mathrm{ind}_{K}^{G_{b}(E)}\Lambda$ where $K\subset G_{b}(E)$ is a pro-$p$
compact open subgroup and $\mathrm{ind}_{K}^{G_{b}(E)}$ denotes compact
induction, by finitely many shifts, cones and retracts. It is not hard to see
that $D(G_{b}(E),\Lambda)$ is compactly generated.

######  Remark 1.1.3.

When $\Lambda$ is a field of characteristic zero, a difficult theorem of
Bernstein asserts that $A\in D(G_{b}(E),\Lambda)$ is compact iff it has
bounded cohomological amplitude and each $H^{n}(A)$ is a finitely generated
representation. As a particular consequence of this theorem, we note that the
standard truncation functors on $D(G_{b}(E),\Lambda)$ preserve compact objects
when $\Lambda$ is a field of characteristic zero. We will use this fact later
in the construction of the hadal t-structure.

On the other hand, we can consider _ULA objects_ , namely those objects $A\in
D(G_{b}(E),\Lambda)$ for which $A^{K}$ is a perfect complex of
$\Lambda$-modules for all pro-$p$ open compacts $K$. Note that if $\Lambda$ is
a field and $A$ has bounded cohomological amplitude, $A$ is ULA exactly when
each $H^{n}(A)$ is an admissible representation in the usual sense. One should
think of the ULA condition as a very mild variant on admissibility, which is
somehow more conceptual.

These two finiteness conditions come with two associated dualities. On the
subcategory $D(G_{b}(E),\Lambda)^{\mathrm{ULA}}$ of ULA objects, we have the
operation of _smooth duality_ $\mathbf{D}_{\mathrm{sm}}$, sending $A$ to
$R\mathscr{H}\mathrm{om}(A,\Lambda)$, where
$R\mathscr{H}\mathrm{om}(-,\Lambda)$ is the internal hom towards the trivial
representation in $D(G_{b}(E),\Lambda)$. This operation makes sense for any
$A$, but on the subcategory of ULA objects it restricts to an involutive anti-
equivalence. When $\Lambda$ is a field and $A=\pi$ is an admissible
representation concentrated in degree zero, $\mathbf{D}_{\mathrm{sm}}\pi$ is
just the usual smooth dual $\pi^{\vee}$, i.e. the smooth vectors in the
abstract dual $\mathrm{Hom}_{\Lambda}(\pi,\Lambda)$.

On the subcategory $D(G_{b}(E),\Lambda)^{\omega}$ of compact objects, we also
have _cohomological duality $\mathbf{D}_{\mathrm{coh}}$_, which by definition
sends $A$ to the total external derived hom
$R\mathrm{Hom}(A,\mathcal{C}_{c}^{\infty}(G_{b}(E),\Lambda))$. It is a fun
exercise to check that $\mathbf{D}_{\mathrm{coh}}$ sends any basic compact
object $\mathrm{ind}_{K}^{G_{b}(E)}\Lambda$ to itself, but noncanonically so
(the isomorphism depending on a choice of $\Lambda$-valued Haar measure on
$G_{b}(E)$), and therefore defines an involutive anti-equivalence on compact
objects.

These finiteness conditions and their associated dualities have perfect
adaptations to sheaves on $\mathrm{Bun}_{G}$. Namely, smooth duality matches
with Verdier duality, which is given by exactly the same formula, namely
$\mathbf{D}_{\mathrm{Verd}}(A)=R\mathscr{H}\mathrm{om}(A,\Lambda)$. This
operation makes sense for all sheaves, and on the subcategory
$D(\mathrm{Bun}_{G},\Lambda)^{\mathrm{ULA}}$ of ULA sheaves it induces an
involutive antiequivalence. We omit the _definition_ of ULA sheaves on
$\mathrm{Bun}_{G}$, and simply note that they are (non-definitionally)
characterized by the property that their $\ast$-restriction to each stratum is
ULA in $D(G_{b}(E),\Lambda)$.

Cohomological duality matches with the more exotic operation of _Bernstein-
Zelevinsky duality_ , denoted $\mathbf{D}_{\mathrm{BZ}}$. This is only defined
on compact sheaves, where it again yields an involutive anti-equivalence. This
duality is characterized by the formula
$R\mathrm{Hom}(\mathbf{D}_{\mathrm{BZ}}(A),B)=\pi_{\natural}(A\otimes B)$
where $\pi:\mathrm{Bun}_{G}\to\ast$ is the structure map and $\pi_{\natural}$
is the left adjoint of $\pi^{\ast}$. Compact sheaves also have a useful
concrete characterization, by the property that their $\ast$-restriction to
each stratum is compact and is identically zero on all but finitely many
strata.

The finiteness conditions and dualities on $\mathrm{Bun}_{G}$ and on the
individual strata are linked as follows.

######  Proposition 1.1.4.

_o._ The functors $i_{b!}$, $i_{b\sharp}$, and $i_{b}^{\ast}$ preserve compact
objects. The functors $i_{b!}$, $i_{b\ast}$, and $i_{b}^{!}$ preserve ULA
objects. Likewise for the renormalized functors.

_i._ For any $A\in D(G_{b}(E),\Lambda)$ we have
$\mathbf{D}_{\mathrm{Verd}}i_{b!}^{\mathrm{ren}}A\simeq
i_{b\ast}^{\mathrm{ren}}\mathbf{D}_{\mathrm{sm}}A$, and if $A$ is ULA we also
have $\mathbf{D}_{\mathrm{Verd}}i_{b\ast}^{\mathrm{ren}}A\simeq
i_{b!}^{\mathrm{ren}}\mathbf{D}_{\mathrm{sm}}A$

_ii._ For any $A\in D(G_{b}(E),\Lambda)^{\omega}$ we have
$\mathbf{D}_{\mathrm{BZ}}i_{b!}^{\mathrm{ren}}A\simeq
i_{b\sharp}^{\mathrm{ren}}\mathbf{D}_{\mathrm{coh}}A$ and
$\mathbf{D}_{\mathrm{BZ}}i_{b\sharp}^{\mathrm{ren}}A\simeq
i_{b!}^{\mathrm{ren}}\mathbf{D}_{\mathrm{coh}}A$.

Note that the duality compatibilies here would be much uglier to state with
the naive pushforwards.

###### Sketch.

The key point, via some calesthenics with the appropriate diagram

---
$\textstyle{[\ast/\underline{G_{b}(E)}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{b}}$$\scriptstyle{i_{b}^{\prime}}$$\textstyle{\mathrm{Bun}_{G}^{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{b}}$$\scriptstyle{f_{b}}$$\textstyle{\mathrm{Bun}_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\ast}$

is the observation that $i_{b}^{!}\Lambda\simeq
f_{b}^{!}\Lambda\simeq\delta_{b}[-2\left\langle
2\rho_{G},\nu_{b}\right\rangle]$. Getting the shift correct here is not hard,
but pinning down the twist is much harder, and I only know how to do it by an
indirect argument using [FS21, Proposition IX.5.3 and Theorem IX.7.2]. In some
cases a direct argument is also possible, as observed by Hamann [Ham22]. I am
also informed that Hamann and Imai have worked out a direct proof which covers
all cases [HI23]. ∎

######  Remark 1.1.5.

It is instructive to recall that if $\Lambda$ is a field of characteristic
zero and $A=\pi$ is an irreducible admissible $G(E)$-representation
concentrated in degree zero, then $\pi$ is both compact _and_ ULA, so both
dualities make sense.444Again, compactness is not obvious, and follows from
the theorem of Bernstein mentioned earlier. The smooth dual of $\pi$ is again
an irreducible admissible representation, and the operation of smooth duality
is expected to interact very cleanly with the local Langlands parametrization
[Kal13]. The effect of cohomological duality is much less obvious: it follows
from deep work of Aubert/Bernstein/Schneider-Stuhler/Zelevinsky that
$\mathbf{D}_{\mathrm{coh}}(\pi)\simeq\mathrm{Zel}(\pi)[-d_{\pi}]$ for some
nonnegative integer $d_{\pi}$ and some irreducible admissible representation
$\mathrm{Zel}(\pi)$. The operation $\pi\mapsto\mathrm{Zel}(\pi)$ is the
Aubert-Zelevinsky involution.555Some authors would say that
$\mathrm{Zel}(\pi)^{\vee}$ is the Aubert-Zelevinsky involution. The integer
$d_{\pi}$ is just the dimension of the component of the Bernstein variety
containing $\pi$, but the representation $\mathrm{Zel}(\pi)$ is mysterious in
general, and it is hard to say much about it beyond the simple observation
that it has the same supercuspidal support as $\pi^{\vee}$.666It is at least
true that $\mathrm{Zel}(\pi)=\pi^{\vee}$ when $\pi$ is supercuspidal.

As a sample of what can happen, we recall that $\mathrm{Zel}(-)$ interchanges
the trivial representation $\mathbf{1}$ with the Steinberg representation
$\mathrm{St}$. Although $\mathrm{St}$ has the same semisimple $L$-parameter as
the trivial representation, the associated monodromy operators are very
different (in fact maximally different). In particular, cohomological duality
interacts in a somewhat mysterious way with the local Langlands
parametrization. Nevertheless, it will transpire that the categorical local
Langlands equivalence should intertwine Bernstein-Zelevinsky duality on
$\mathrm{Bun}_{G}$ with (twisted) Grothendieck-Serre duality on the stack of
_L_ -parameters.

###  1.2 t-structures

First we briefly recall the perverse t-structure, which makes sense for any
coefficient ring $\Lambda$. The existence of this t-structure has been well-
understood by experts for years (see e.g. [FS21, Remark I.10.3]), although it
doesn’t seem to be recorded in the literature.

######  Proposition 1.2.1.

There is a perverse t-structure ${}^{p}D^{\leq 0},\,^{p}D^{\geq 0}$ on
$D(\mathrm{Bun}_{G},\Lambda)$ uniquely characterized by the condition that
$A\in D(\mathrm{Bun}_{G},\Lambda)$ lies in ${}^{p}D^{\leq 0}$ (resp.
${}^{p}D^{\geq 0}$) if $i_{b}^{\ast}A$ sits in cohomological degrees
$\leq\left\langle 2\rho_{G},\nu_{b}\right\rangle$ for all $b$ (resp. if
$i_{b}^{!}A$ sits in cohomological degrees $\geq\left\langle
2\rho_{G},\nu_{b}\right\rangle$ for all $b$). When $\Lambda$ is
cohomologically regular in the sense that $\mathrm{Perf}(\Lambda)$ is
preserved by standard truncation, the perverse truncation functors preserve
the ULA property. When $\Lambda$ is a field, Verdier duality exchanges
${}^{p}D_{\mathrm{}}^{\leq 0}\cap D^{\mathrm{ULA}}$ and $\,{}^{p}D^{\geq
0}\cap D^{\mathrm{ULA}}$.

######  Exercise 1.2.2.

i. Check that $i_{b!}^{\mathrm{ren}}$ sends $D^{\leq 0}(G_{b}(E),\Lambda)$
into ${}^{p}D^{\leq 0}(\mathrm{Bun}_{G},\Lambda)$, and that
$i_{b\ast}^{\mathrm{ren}}$ sends $D^{\geq 0}(G_{b}(E),\Lambda)$ into
${}^{p}D^{\geq 0}(\mathrm{Bun}_{G},\Lambda)$.

ii. Prove that if $\Lambda$ is a field, there is a natural bijection between
irreducible objects $A\in\mathrm{Perv}(\mathrm{Bun}_{G},\Lambda)$ and pairs
$(b,\pi)$ where $b\in B(G)$ and $\pi\in\Pi(G_{b})$ is an irreducible smooth
representation, defined by sending a pair $(b,\pi)$ to the intermediate
extension sheaf

$i_{b!\ast}^{\mathrm{ren}}\pi\overset{\mathrm{def}}{=}\mathrm{im}\left({}^{p}H^{0}(i_{b!}^{\mathrm{ren}}\pi)\to\,^{p}H^{0}(i_{b\ast}^{\mathrm{ren}}\pi)\right).$

Prove that $\mathbf{D}_{\mathrm{Verd}}(i_{b!\ast}^{\mathrm{ren}}\pi)\simeq
i_{b!\ast}^{\mathrm{ren}}\pi^{\vee}$.

Now we come to the first really new construction in these notes.

Warning. In what follows, we assume that $\Lambda$ is a field _of
characteristic zero_. However, the following construction works verbatim for
any coefficient ring $\Lambda$ with the property that the standard truncation
functors on $D(G_{b}(E),\Lambda)$ preserve the subcategory of compact objects
for all $b\in B(G)$.777For any fixed $G$, this condition holds with
$\Lambda=\overline{\mathbf{F}_{\ell}}$ or $\overline{\mathbf{Z}_{\ell}}$ for
all but finitely many $\ell$. It would be interesting to explicate this finite
set of bad primes when $G=\mathrm{GL}_{n}$.

Define ${}^{h}D^{\leq 0}(\mathrm{Bun}_{G},\Lambda)^{\omega}$ to be the full
subcategory of $D(\mathrm{Bun}_{G},\Lambda)^{\omega}$ generated under finite
extensions by objects of the form $i_{b\sharp}^{\mathrm{ren}}A$ with $A\in
D^{\leq 0}(G_{b}(E),\Lambda)^{\omega}$. Likewise, define ${}^{h}D^{\geq
0}(\mathrm{Bun}_{G},\Lambda)^{\omega}$ to be the full subcategory of
$D(\mathrm{Bun}_{G},\Lambda)^{\omega}$ generated under finite extensions by
objects of the form $i_{b!}^{\mathrm{ren}}A$ with $A\in D^{\geq
0}(G_{b}(E),\Lambda)^{\omega}$.

######  Theorem 1.2.3.

The pair

$\left({}^{h}D^{\leq 0}(\mathrm{Bun}_{G},\Lambda)^{\omega},{}^{h}D^{\geq
0}(\mathrm{Bun}_{G},\Lambda)^{\omega}\right)$

defines a t-structure on $D(\mathrm{Bun}_{G},\Lambda)^{\omega}$, called the
_hadal_ 888Pronouned “HEY-dull”. See https://en.wikipedia.org/wiki/Hadal_zone.
Unlike perverse sheaves, which propagate downwards from the maximal points in
their support, hadal sheaves begin their life deep in the Newton strata of
$\mathrm{Bun}_{G}$, and then bubble up to the surface (i.e. the basic
$b$’s)._t-structure._ We write ${}^{h}\tau^{\leq n}$ and ${}^{h}\tau^{\geq n}$
for the truncation functors associated with this t-structure, and we write
$\mathrm{Had}(\mathrm{Bun}_{G},\Lambda)$ for the abelian category of _hadal
sheaves_ defined as its heart.

Of course we also write

${}^{h}H^{n}={}^{h}\tau^{\leq n}\circ{}^{h}\tau^{\geq n}\cong{}^{h}\tau^{\geq
n}\circ{}^{h}\tau^{\leq
n}:D(\mathrm{Bun}_{G},\Lambda)^{\omega}\to\mathrm{Had}(\mathrm{Bun}_{G},\Lambda)$

for the $n$th hadal cohomology functor.

###### Proof.

If $U\subset\mathrm{Bun}_{G}$ is any open substack, set
$D(U):=D(U,\Lambda)^{\omega}$ for brevity. Define ${}^{h}D^{\leq 0}(U)$ and
${}^{h}D^{\geq 0}(U)$ inside $D(U)$ analogously with the definition for
$U=\mathrm{Bun}_{G}$, but only allowing $b\in|U|$ in the specification of the
generators. It is clear that if $j:U\to V$ is any inclusion of open substacks
of $\mathrm{Bun}_{G}$, $j_{!}$ carries ${}^{h}D^{\leq 0}(U)$ fully faithfully
into ${}^{h}D^{\leq 0}(V)$, and likewise for ${}^{h}D^{\geq 0}$. It is also
clear from the definitions that ${}^{h}D^{\leq 0}(U)[1]\subset{}^{h}D^{\leq
0}(U)$ and ${}^{h}D^{\geq 0}(U)[-1]\subset{}^{h}D^{\geq 0}(U)$. Finally, we
observe that ${}^{h}D^{\geq 1}(U)$ is exactly the right orthogonal of
${}^{h}D^{\leq 0}(U)$ inside $D(U)$. This is an easy but very enlightening
exercise with the definitions, which we leave to the reader.

It is now clearly enough to prove that if $U\subset\mathrm{Bun}_{G}$ is a
_quasicompact_ open substack, the pair

$\left({}^{h}D^{\leq 0}(U),{}^{h}D^{\geq 0}(U)\right)$

defines a t-structure on $D(U)$. We will prove this by induction on the
maximal length of any chain of specializations inside $|U|$. When there are no
nontrivial such chains, $U$ is a finite disjoint union of open strata
$[\ast/\underline{G_{b}(E)}]$ with $b$ basic. In this case the result is
clear, since for basic $b$

$\left({}^{h}D^{\leq 0}([\ast/\underline{G_{b}(E)}]),{}^{h}D^{\geq
0}([\ast/\underline{G_{b}(E)}])\right)$

is just the standard t-structure on $D([\ast/\underline{G_{b}(E)}])\cong
D(G_{b}(E),\Lambda)^{\omega}$.

We now proceed by induction. Fix a quasicompact open substack $U$, and let
$b\in|U|$ be the closed point in a chain of maximal length. Let $j:V\to U$ be
the inclusion of the open substack with $|V|=|U|\smallsetminus b$, and let
$i_{b}:\mathrm{Bun}_{G}^{b}\to U$ be the inclusion of the closed substack
associated with $b$ as usual. By the induction hypothesis, we already have
access to the hadal t-structure and its truncation functors
${}^{h}\tau_{V}^{\leq n}$ and ${}^{h}\tau_{V}^{\geq n}$ on $D(V)$. By the
observations in the first paragraph of the proof, our only remaining task is
to show that any given $A\in D(U)$ can be fit into a distinguished triangle

$E\to A\to C\overset{[1]}{\to}$

with $E\in\,^{h}D^{\leq 0}(U)$ and $C\in\,^{h}D^{\geq 1}(U)$. For this, we
define objects $B,C\in D(U)$ inductively by requiring that they sit in
distinguished triangles

$i_{b\sharp}^{\mathrm{ren}}\tau^{\leq 0}i_{b}^{\ast\mathrm{ren}}A\to A\to
B\overset{[1]}{\to}$

and

$j_{!}\,^{h}\tau_{V}^{\leq 0}j^{\ast}B\to B\to C\overset{[1]}{\to}.$

By the octahedral axiom, we then get an object $E\in D(U)$ together with a
diagram

|  |
---|---|---
$\textstyle{i_{b\sharp}^{\mathrm{ren}}\tau^{\leq
0}i_{b}^{\ast\mathrm{ren}}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[1]}$$\textstyle{j_{!}\,^{h}\tau_{V}^{\leq
0}j^{\ast}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[1]}$

where the row and column are both distinguished triangles. It is clear that
$E$ sits in ${}^{h}D^{\leq 0}(U)$, since it is an extension of two objects in
this category, so we just need to check that $C$ sits in ${}^{h}D^{\geq
1}(U)$, which we’ve already noted is the right orthogonal of ${}^{h}D^{\leq
0}(U)$. It is therefore enough to check that
$\mathrm{Hom}(i_{b^{\prime}\sharp}^{\mathrm{ren}}F,C)=0$ for any
$b^{\prime}\in|U|$ and any $F\in D^{\leq
0}(G_{b^{\prime}}(E),\Lambda)^{\omega}$. To check this we divide into two
disjoint cases:

Case 1: $b^{\prime}=b$. In this case

$\displaystyle\mathrm{Hom}(i_{b\sharp}^{\mathrm{ren}}F,C)$
$\displaystyle=\mathrm{Hom}(i_{b\sharp}^{\mathrm{ren}}F,B)$
$\displaystyle\simeq\mathrm{Hom}(F,i_{b}^{\ast\mathrm{ren}}B)$
$\displaystyle=0$

where the first isomorphism follows from the triangle defining $C$ and the
fact that $\mathrm{Hom}(i_{b\sharp}^{\mathrm{ren}}F,j_{!}-)=0$, and the final
vanishing follows from the fact that
$i_{b}^{\ast\mathrm{ren}}B\simeq\tau^{\geq 1}i_{b}^{\ast\mathrm{ren}}A$ by
consideration of the triangle defining $B$.

Case 2: $b^{\prime}\in|V|$. In this case
$i_{b^{\prime}\sharp}^{\mathrm{ren}}F=j_{!}i_{b^{\prime}\sharp}^{\mathrm{ren}}F\in
j_{!}\,^{h}D^{\leq 0}(V)$, with the evident abuse of notation, so

$\mathrm{Hom}_{U}(i_{b^{\prime}\sharp}^{\mathrm{ren}}F,C)=\mathrm{Hom}_{V}(i_{b^{\prime}\sharp}^{\mathrm{ren}}F,j^{\ast}C).$

But $j^{\ast}C\simeq\,^{h}\tau_{V}^{\geq 1}j^{\ast}B$ by consideration of the
triangle defining $C$, and
$i_{b^{\prime}\sharp}^{\mathrm{ren}}F\in\,^{h}D^{\leq 0}(V)$, so we get the
desired vanishing. ∎

Our next goal is Theorem 1.2.7, which gives an explicit classification of
irreducible hadal sheaves. This requires several preparatory lemmas.

######  Lemma 1.2.4.

Fix $b\in B(G)$, and let $\pi$ be any finitely generated smooth
$G_{b}(E)$-representation.

_i._ The hadal sheaf ${}^{h}H^{0}(i_{b\sharp}^{\mathrm{ren}}\pi)$ does not
have any nonzero hadal quotient supported on $\mathrm{Bun}_{G}^{\prec b}$.

_ii._ The hadal sheaf ${}^{h}H^{0}(i_{b!}^{\mathrm{ren}}\pi)$ does not have
any nonzero hadal subobject supported on $\mathrm{Bun}_{G}^{\prec b}$.

###### Proof.

We prove the first claim; the second is analogous. Let
$j:\mathrm{Bun}_{G}^{\prec b}\to\mathrm{Bun}_{G}$ be the evident open
immersion. Let $F$ be a hadal sheaf supported on $\mathrm{Bun}_{G}^{\prec b}$,
so $F\cong j_{!}j^{\ast}F$. Since $i_{b\sharp}^{\mathrm{ren}}$ is right
t-exact for the hadal t-structure, we compute that

$\displaystyle\mathrm{Hom}({}^{h}H^{0}(i_{b\sharp}^{\mathrm{ren}}\pi),j_{!}j^{\ast}F)$
$\displaystyle\cong\mathrm{Hom}(i_{b\sharp}^{\mathrm{ren}}\pi,j_{!}j^{\ast}F)$
$\displaystyle\cong\mathrm{Hom}(\pi,i_{b}^{\ast\mathrm{ren}}j_{!}j^{\ast}F)$
$\displaystyle=0,$

since $i_{b}^{\ast}j_{!}=0$. ∎

######  Lemma 1.2.5.

Let $A\in\mathrm{Had}(\mathrm{Bun}_{G},\Lambda)$ be a hadal sheaf, and let $b$
be a maximally special point in the support of $A$, i.e. we assume that
$i_{b}^{\ast}A\not\simeq 0$ and that $i_{b^{\prime}}^{\ast}A=0$ for all
$b\prec b^{\prime}$ in the Newton partial order. Then
$i_{b}^{\ast\mathrm{ren}}A\in D(G_{b}(E),\Lambda)$ is concentrated in degree
zero.

###### Proof.

Since $i_{b}^{\ast\mathrm{ren}}$ is left t-exact with respect to the hadal
t-structure on $\mathrm{Bun}_{G}$ and the standard t-structure on
$D(G_{b}(E),\Lambda)$, we already know that $i_{b}^{\ast\mathrm{ren}}A$ is
concentrated in degrees $\geq 0$. To see that it is also concentrated in
degrees $\leq 0$, pick a quasicompact open substack $U\subset\mathrm{Bun}_{G}$
containing $\mathrm{supp}A$, and such that $b\in U$ is a closed point in $U$.
Write $V=U\smallsetminus\mathrm{Bun}_{G}^{b}$ as in the proof of Theorem
1.2.3, whose notation we will refer to in what follows. As in the proof of
Theorem 1.2.3, we may regard $A$ as a hadal sheaf in $D(U)$. Since
$A\simeq\,^{h}\tau_{U}^{\leq 0}A$ by assumption, the constructions in that
proof give a distinguished triangle

$i_{b\sharp}^{\mathrm{ren}}\tau^{\leq 0}i_{b}^{\ast\mathrm{ren}}A\to A\to
j_{!}\,^{h}\tau_{V}^{\leq 0}j^{\ast}B\overset{[1]}{\to},$

upon noting that the object $E$ we constructed there is exactly the connective
truncation of $A$. Then $i_{b}^{\ast\mathrm{ren}}j_{!}=0$ and
$i_{b}^{\ast\mathrm{ren}}i_{b\sharp}^{\mathrm{ren}}\cong\mathrm{id}$, so
applying $i_{b}^{\ast\mathrm{ren}}$ to this triangle gives

$i_{b}^{\ast\mathrm{ren}}A\simeq\tau^{\leq 0}i_{b}^{\ast\mathrm{ren}}A,$

as desired. ∎

######  Lemma 1.2.6.

Fix any $b\in B(G)$. Then the functor on finitely generated
$G_{b}(E)$-representations given by $M\mapsto
i_{b}^{\ast\mathrm{ren}}\,{}^{h}H^{0}(i_{b\sharp}^{\mathrm{ren}}M)$ is
naturally isomorphic to the identity functor.

In particular, if $M\to N$ is a nonzero map of finitely generated
$G_{b}(E)$-representations, the induced map
$\,{}^{h}H^{0}(i_{b\sharp}^{\mathrm{ren}}M)\to\,^{h}H^{0}(i_{b\sharp}^{\mathrm{ren}}N)$
is necessarily nonzero.

###### Proof.

Write $U=\mathrm{Bun}_{G}^{\preceq b}$ and $V=\mathrm{Bun}_{G}^{\prec b}$ as
in the proof of Theorem 1.2.3, whose notation we will partly reuse. Since
$A:=i_{b\sharp}^{\mathrm{ren}}M$ is connective for the hadal t-structure,
${}^{h}H^{0}(A)\simeq\,^{h}\tau_{U}^{\geq 0}A$. Rerunning the proof of Theorem
1.2.3 with the appropriate shifts, we inductively define distinguished
triangles

$i_{b\sharp}^{\mathrm{ren}}\tau^{\leq-1}i_{b}^{\ast\mathrm{ren}}A\to A\to
B\overset{[1]}{\to}$

and

$j_{!}\,^{h}\tau_{V}^{\leq-1}j^{\ast}B\to
B\to{}^{h}H^{0}(A)\overset{[1]}{\to}.$

Applying $i_{b}^{\ast\mathrm{ren}}$ to the second triangle we get
$i_{b}^{\ast\mathrm{ren}}{}^{h}H^{0}(A)\simeq i_{b}^{\ast\mathrm{ren}}B$, so
then using this and applying $i_{b}^{\ast\mathrm{ren}}$ to the first triangle
we get a distinguished triangle

$i_{b}^{\mathrm{ren}\ast}i_{b\sharp}^{\mathrm{ren}}\tau^{\leq-1}i_{b}^{\ast\mathrm{ren}}A\to
i_{b}^{\ast\mathrm{ren}}A\to
i_{b}^{\ast\mathrm{ren}}{}^{h}H^{0}(A)\overset{[1]}{\to}.$

But $i_{b}^{\ast\mathrm{ren}}A\cong M$, so the first term vanishes
identically, giving the result. ∎

######  Theorem 1.2.7.

For every pair $(b,\pi)$ with $b\in B(G)$ and $\pi$ an irreducible smooth
$G_{b}(E)$-representation, there is a unique _irreducible_ hadal sheaf
$\mathscr{G}_{b,\pi}$ characterized by the requirements that _
$\mathrm{supp}\,\mathscr{G}_{b,\pi}\subseteq\mathrm{Bun}_{G}^{\preceq b}$_and
_$i_{b}^{\ast\mathrm{ren}}\mathscr{G}_{b,\pi}\simeq\pi$_ , and given
explicitly by the formula

$\mathscr{G}_{b,\pi}\simeq\mathrm{im}\left(\,{}^{h}H^{0}(i_{b\sharp}^{\mathrm{ren}}\pi)\to\,^{h}H^{0}(i_{b!}^{\mathrm{ren}}\pi)\right).$

Moreover, every irreducible hadal sheaf arises from a uniquely associated pair
$(b,\pi)$ in this way.

We will sometimes write $i_{b\sharp!}^{\mathrm{ren}}\pi$ for the sheaf
$\mathscr{G}_{b,\pi}$, in analogy with the notation
$i_{b!\ast}^{\mathrm{ren}}$ for intermediate extension of perverse sheaves.

###### Proof.

Let $A$ be an irreducible hadal sheaf. Pick $b$ a maximally special point in
the support of $A$, so $i_{b}^{\ast\mathrm{ren}}A$ is concentrated in degree
zero by Lemma 1.2.5. Then
$i_{b\sharp}^{\mathrm{ren}}i_{b}^{\ast\mathrm{ren}}A$ is connective for the
hadal t-structure, so via the adjunction
$i_{b\sharp}^{\mathrm{ren}}i_{b}^{\ast\mathrm{ren}}\to\mathrm{id}$ we get a
distinguished triangle

$i_{b\sharp}^{\mathrm{ren}}i_{b}^{\ast\mathrm{ren}}A\to A\to
K\overset{[1]}{\to}$

of connective objects, such that $b\notin\mathrm{supp}K$. Since the hadal
truncation functors are compatible with $!$-extension along open substacks,
taking hadal cohomology gives a long exact sequence

$0\to\,^{h}H^{-1}(K)\to\,^{h}H^{0}(i_{b\sharp}^{\mathrm{ren}}i_{b}^{\ast\mathrm{ren}}A)\overset{\alpha}{\to}A\to\,^{h}H^{0}(K)\to
0$

of hadal sheaves whose outer terms have support disjoint from $b$. Then
$\alpha$ is nonzero (e.g. by noting that it induces an isomorphism between
nonzero objects after applying $i_{b}^{\ast\mathrm{ren}}$), so the
irreducibility hypothesis on $A$ implies that $\alpha$ is surjective. Since
the source of $\alpha$ has support contained in $\mathrm{Bun}_{G}^{\preceq
b}$, we then deduce additionally that
$\mathrm{supp}A\subseteq\mathrm{Bun}_{G}^{\preceq b}$.

Next, we show that the point $b$ is uniquely determined. Let $b^{\prime}$ be
any other maximally special point in the support of $A$. By a dual version of
the argument in the first paragraph of the proof, we get an injective map

$A\overset{\alpha^{\prime}}{\to}\,^{h}H^{0}(i_{b^{\prime}!}^{\mathrm{ren}}i_{b^{\prime}}^{\ast\mathrm{ren}}A).$

Composing this with the surjective map $\alpha$ constructed in the first
paragraph, we get a nonzero map

$\,{}^{h}H^{0}(i_{b\sharp}^{\mathrm{ren}}i_{b}^{\ast\mathrm{ren}}A)\overset{\alpha^{\prime}\circ\alpha}{\longrightarrow}\,^{h}H^{0}(i_{b^{\prime}!}^{\mathrm{ren}}i_{b^{\prime}}^{\ast\mathrm{ren}}A).$

But $i_{b\sharp}^{\mathrm{ren}}i_{b}^{\ast\mathrm{ren}}A$ is connective and
$i_{b^{\prime}!}^{\mathrm{ren}}i_{b^{\prime}}^{\ast\mathrm{ren}}A$ is
coconnective, by two applications of Lemma 1.2.5, so this is the same as the
datum of a nonzero map

$i_{b\sharp}^{\mathrm{ren}}i_{b}^{\ast\mathrm{ren}}A\to
i_{b^{\prime}!}^{\mathrm{ren}}i_{b^{\prime}}^{\ast\mathrm{ren}}A.$

But if such a nonzero map exists, then necessarily $b=b^{\prime}$ by
Proposition 1.1.2.iii.

Next we show that $i_{b}^{\ast\mathrm{ren}}A$, which is concentrated in degree
zero by Lemma 1.2.5, is actually an irreducible smooth representation. Let
$B\subseteq i_{b}^{\ast\mathrm{ren}}A$ be any subrepresentation. Consider the
maps

$\,{}^{h}H^{0}(i_{b\sharp}^{\mathrm{ren}}B)\overset{\beta}{\to}\,^{h}H^{0}(i_{b\sharp}^{\mathrm{ren}}i_{b}^{\ast\mathrm{ren}}A)\overset{\alpha}{\to}A.$

By the surjectivity of $\alpha$ and some general nonsense, we get an exact
sequence

$0\to\ker\beta\to\ker\alpha\circ\beta\to\ker\alpha\to\mathrm{coker}\,\beta\to\mathrm{coker}\,\alpha\circ\beta\to
0.$

From the first paragraph of the proof, we already know that $\ker\alpha$ has
support contained in $\mathrm{Bun}_{G}^{\prec b}$, so then also any quotient
of $\ker\alpha$ has the same support property. Now, since $A$ is irreducible,
$\mathrm{coker}\,\alpha\circ\beta$ is either $\simeq 0$ or $\simeq A$. If it
is $\simeq 0$, then

$\mathrm{coker}\beta\simeq\,^{h}H^{0}(i_{b\sharp}^{\mathrm{ren}}(i_{b}^{\ast\mathrm{ren}}A/B))$

is also quotient of $\ker\alpha$ and thus is supported in
$\mathrm{Bun}_{G}^{\prec b}$, so it must vanish identically, which then forces
$B=i_{b}^{\ast\mathrm{ren}}A$ by Lemma 1.2.6. If it is $\simeq A$, then
$\mathrm{im}\beta\subseteq\ker\alpha$, and we get an exact sequence

$0\to\ker\beta\to\,^{h}H^{0}(i_{b\sharp}^{\mathrm{ren}}B)\overset{\tau}{\to}\ker\alpha\to\mathrm{coker}\,\beta\to
A\to 0.$

But we already know that $\ker\alpha$ is supported in $\mathrm{Bun}_{G}^{\prec
b}$, while $\,{}^{h}H^{0}(i_{b\sharp}^{\mathrm{ren}}B)$ cannot have any
quotient with this support property by Lemma 1.2.4. This forces $\tau=0$ and
then $\ker\beta\overset{\sim}{\to}\,^{h}H^{0}(i_{b\sharp}^{\mathrm{ren}}B)$,
so then $\beta$ is the zero map. But then the natural inclusion map $B\to
i_{b}^{\ast\mathrm{ren}}A$ giving rise to $\beta$ is also the zero map by
Lemma 1.2.6, and thus $B\simeq 0$. Therefore, either $B=0$ or
$B=i_{b}^{\ast\mathrm{ren}}A$, so $i_{b}^{\ast\mathrm{ren}}A$ is irreducible.

Summarizing our efforts so far, we have produced from the irreducible hadal
sheaf $A$ a canonical pair $(b,\pi)$ as in the statement of the theorem, with
$b$ the unique maximally special point in the support of $A$, and with
$\pi=i_{b}^{\ast\mathrm{ren}}A$ irreducible. To reconstruct $A$ from this
datum, observe that in the course of our arguments, we obtained maps

$\,{}^{h}H^{0}(i_{b\sharp}^{\mathrm{ren}}\pi)\twoheadrightarrow
A\hookrightarrow\,^{h}H^{0}(i_{b!}^{\mathrm{ren}}\pi)$

whose composite is the canonical map induced by the natural transformation
$i_{b\sharp}^{\mathrm{ren}}\to i_{b!}^{\mathrm{ren}}$. This implies that the
association

$(b,\pi)\mapsto\mathscr{G}_{b,\pi}\overset{\mathrm{def}}{=}\mathrm{im}\left(\,{}^{h}H^{0}(i_{b\sharp}^{\mathrm{ren}}\pi)\to\,^{h}H^{0}(i_{b!}^{\mathrm{ren}}\pi)\right)$

defines an inverse to our recipe for extracting the pair $(b,\pi)$ from $A$.

Finally, we need to see that for _every_ pair $(b,\pi)$, the sheaf
$\mathscr{G}_{b,\pi}$ defined in the previous paragraph is irreducible. Fix
any such pair. Pick any irreducible hadal sheaf $A$ together with a nonzero
map $f:A\to\mathscr{G}_{b,\pi}$. By our arguments so far, we already know that
$A\simeq\mathscr{G}_{b^{\prime},\pi^{\prime}}$ for some pair
$(b^{\prime},\pi^{\prime})$. Composing the maps

$\,{}^{h}H^{0}(i_{b^{\prime}\sharp}^{\mathrm{ren}}\pi^{\prime})\twoheadrightarrow\mathscr{G}_{b^{\prime},\pi^{\prime}}\simeq
A\overset{f}{\to}\mathscr{G}_{b,\pi}\hookrightarrow\,^{h}H^{0}(i_{b!}^{\mathrm{ren}}\pi)$

gives a nonzero map
$\,{}^{h}H^{0}(i_{b^{\prime}\sharp}^{\mathrm{ren}}\pi^{\prime})\to\,^{h}H^{0}(i_{b!}^{\mathrm{ren}}\pi)$.
Arguing as in the second paragraph of the proof, the datum of such a nonzero
map is equivalent to the datum of a nonzero map
$i_{b^{\prime}\sharp}^{\mathrm{ren}}\pi^{\prime}\to i_{b!}^{\mathrm{ren}}\pi$.
But if such a nonzero map exists, then necessarily $b=b^{\prime}$ by
Proposition 1.1.2.iii. The various adjunctions easily imply a general
isomorphism
$\mathrm{Hom}(i_{b\sharp}^{\mathrm{ren}}A,i_{b!}^{\mathrm{ren}}B)\cong\mathrm{Hom}(A,B)$,
so now we also get a nonzero map $\pi^{\prime}\to\pi$, and thus an isomorphism
$\pi\simeq\pi^{\prime}$. Therefore,
$\mathscr{G}_{b,\pi}\simeq\mathscr{G}_{b^{\prime},\pi^{\prime}}$ is
irreducible. ∎

######  Remark 1.2.8.

Just as the perverse t-structure interacts well with Verdier duality, the
hadal t-structure should interact cleanly with Bernstein-Zelevinsky duality.
However, the precise statement is subtle, and we refer to Remark 2.3.5 for a
more detailed explanation of what we expect in this direction.

######  Exercise 1.2.9.

Check that the hadal t-structure on $D(\mathrm{Bun}_{G},\Lambda)^{\omega}$ is
bounded and nondegenerate.

###  1.3 The spectral side

Here we give a very brief recollection on the stack of _L-_ parameters and its
coarse moduli, mostly just to set notation; we regard this as the “easy” side
of the categorical conjecture. For a much more detailed treatment, see
[DHKM20], [Zhu21], or [FS21, Chapter VIII].

Let $\Lambda$ be any $\mathbf{Z}_{\ell}[\sqrt{q}]$-algebra. We write
$\mathrm{Par}_{G,\Lambda}=Z^{1}(W_{E},\hat{G})_{\Lambda}/\hat{G}$ for the
stack of $\ell$-adically continuous _L_ -parameters regarded as an Artin stack
locally of finite type over $\mathrm{Spec}\,\Lambda$, with its tautological
map $\tau_{G}:\mathrm{Par}_{G,\Lambda}\to B\hat{G}_{\Lambda}$. Note that we
can define an analogous stack for any linear algebraic group, but for non-
reductive groups the correct object turns out to be a _derived_ Artin stack
(see [Zhu21, Section 2.3]). We will only need this extra generality for
parabolic subgroups, in the definition of spectral Eisenstein series.

We will primarily be interested in the case when
$\Lambda=\overline{\mathbf{Q}_{\ell}}$, in which case we will drop $\Lambda$
from the notation. When $\Lambda=\overline{\mathbf{Q}_{\ell}}$ we write
$X_{G}^{\mathrm{spec}}$ for the coarse quotient of $\mathrm{Par}_{G}$, with
its canonical map $q:\mathrm{Par}_{G}\to X_{G}^{\mathrm{spec}}$. The
$\overline{\mathbf{Q}_{\ell}}$-points of $\mathrm{Par}_{G}$ parametrize
isomorphism classes of $\ell$-adically continuous _L-_ parameters
$\phi:W_{E}\to\,^{L}G(\overline{\mathbf{Q}_{\ell}})$, which we will simply
call _L_ -parameters. If $\phi$ is any _L_ -parameter, we write
$S_{\phi}=\mathrm{Cent}_{\hat{G}}(\phi)$ for its centralizer group, and we
write $i_{\phi}:BS_{\phi}\to\mathrm{Par}_{G}$ for the associated locally
closed immersion. Note that by [Zhu21, Lemma 3.1.8], we can also regard
$\mathrm{Par}_{G}$ as a moduli stack of Weil-Deligne parameters.

The closed points of $X_{G}^{\mathrm{spec}}$ are in canonical bijection with
_semisimple_ _L_ -parameters $W_{E}\to\,^{L}G(\overline{\mathbf{Q}_{\ell}})$.
Here we say an _L-_ parameter is semisimple if it is Frobenius-semisimple and
has open kernel. Note that $X_{G}^{\mathrm{spec}}$ is a disjoint union of
affine varieties over $\mathrm{Spec}\,\overline{\mathbf{Q}_{\ell}}$, and
$\mathcal{O}(X_{G}^{\mathrm{spec}})\cong\mathcal{O}(\mathrm{Par}_{G})$.
Moreover, each component of $X_{G}^{\mathrm{spec}}$ is a quotient of a torus
by a finite group, and in particular is Cohen-Macaulay. Note that any _L_
-parameter has a unique semisimplification, corresponding to the image of the
point $\phi\in\mathrm{Par}_{G}$ along $q$. Conversely, if $\phi$ is a
semisimple parameter, the fiber of $q$ over the associated closed point
$x_{\phi}\in X_{G}^{\mathrm{spec}}$ is a moduli space of $L$-parameters with
constant semisimplification $\phi$. Each such fiber contains a unique _closed_
$\overline{\mathbf{Q}_{\ell}}$-point, corresponding to the actual parameter
$\phi$. In particular, $q$ induces a bijection from the _closed_
$\overline{\mathbf{Q}_{\ell}}$-points of $\mathrm{Par}_{G}$ onto the closed
points of $X_{G}^{\mathrm{spec}}$.

It is instructive to understand the fibers of $q$ more explicitly.

######  Proposition 1.3.1.

Fix a semisimple L-parameter $\phi$. The reduced fiber of $q$ over the
associated closed point $x_{\phi}\in X_{G}^{\mathrm{spec}}$ admits the
explicit presentation

$q^{-1}(x_{\phi})^{\mathrm{red}}\simeq\left\\{(u,N)\in\mathcal{U}_{S_{\phi}}\times\mathfrak{g}^{\mathrm{ad}\phi(I_{E})}\mid\mathrm{ad}\phi(\mathrm{Fr})\cdot
N=q^{-1}N,\mathrm{ad}u\cdot N=N\right\\}/S_{\phi}$

where $S_{\phi}$ acts by simultaneous conjugation.

Here $\mathcal{U}_{S_{\phi}}$ denotes the variety of unipotent elements in
$S_{\phi}$. Note that the closed substack cut out by $u=1$ is exactly the
Vogan variety

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

parametrizing Frobenius-semisimple _L_ -parameters with semisimplification
$\phi$. We also note that the associated closed immersion $V_{\phi}\to
q^{-1}(x_{\phi})^{\mathrm{red}}$ has a natural retraction, given by forgetting
$u$. In some cases, e.g. when $\phi(\mathrm{Fr})$ is regular semisimple, the
Vogan variety is the entire fiber. At the other extreme, if $\phi$ is the
trivial _L_ -parameter (so $S_{\phi}=\hat{G}$), then only $N=0$ can occur, but
_any_ $u$ can occur, and the fiber is the entire quotient
$\mathcal{U}_{\hat{G}}/\hat{G}$. In general, the geometry of the fiber
involves variation of both $u$ and $N$. Note that in the “classical” local
Langlands correspondence, only Frobenius-semisimple _L_ -parameters are
relevant.

######  Exercise 1.3.2 (Hellmann).

Take $G=\mathrm{GL}_{4}$, and let $\phi$ be the semsimple parameter which is
trivial on inertia and with $\phi(\mathrm{Fr})=\mathrm{diag}(1,q,q,q^{2})$.
Explicate the finite topological space $|q^{-1}(x_{\phi})|$ as a set, and draw
all of the nontrivial specializations within it.

###  1.4 Categorical conjecture

We can now start to put the two sides together. For simplicitly, let $L$ be an
algebraic extension of $\mathbf{Q}_{\ell}(\sqrt{q})$, and let
$\Lambda\in\left\\{L,\mathcal{O}_{L}\right\\}$. In all that follows, we assume
either that $\Lambda=L$ or that $\ell$ is a very good prime for $G$ in the
sense of [FS21].

The essential carriers of information in Fargues-Scholze are the Hecke
operators, which we briefly recall (see [FS21, Section IX.2]).

######  Theorem 1.4.1.

For any $V\in\mathrm{Rep}_{\Lambda}(^{L}G)$, there is a naturally associated
functor

$T_{V}:D(\mathrm{Bun}_{G},\Lambda)\to D(\mathrm{Bun}_{G},\Lambda)^{BW_{E}}$

where $D(\mathrm{Bun}_{G},\Lambda)^{BW_{E}}$ denotes the appropriate category
of $W_{E}$-equivariant objects in $D(\mathrm{Bun}_{G},\Lambda)$. More
generally, for any finite set $I$ and any
$V\in\mathrm{Rep}_{\Lambda}((^{L}G)^{I})$, there is a naturally associated
functor

$T_{V}:D(\mathrm{Bun}_{G},\Lambda)\to
D(\mathrm{Bun}_{G},\Lambda)^{BW_{E}^{I}}.$

Composing with the forgetful functor to $D(\mathrm{Bun}_{G},\Lambda)$, the
induced endofunctor

$T_{V}:D(\mathrm{Bun}_{G},\Lambda)\to D(\mathrm{Bun}_{G},\Lambda)$

depends only on the restriction of $V$ to the diagonally embedded copy of
$\hat{G}$ in ${}^{L}G^{I}$. As an endofunctor of
$D(\mathrm{Bun}_{G},\Lambda)$, $T_{V}$ preserves compact and ULA objects, and
has left and right adjoint given by $T_{V^{\vee}}$.

The first key construction in Fargues-Scholze linking the the spectral and
automorphic worlds is a natural map

$\mathcal{O}(\mathrm{Par}_{G,\Lambda})\to\mathfrak{Z}(D(\mathrm{Bun}_{G},\Lambda))$

which they construct using the Hecke operators together with V. Lafforgue’s
excursion operator formalism [FS21, Theorem IX.5.2]. Note that for any $b\in
B(G)$, the functor $i_{b!}^{\mathrm{ren}}$ defines a fully faithful embedding
$D(G_{b}(E),\Lambda)\to D(\mathrm{Bun}_{G},\Lambda)$, which induces a map in
the other direction on Bernstein centers. Post-composing with the above map,
we obtain a canonical map
$\mathcal{O}(\mathrm{Par}_{G,\Lambda})\overset{\Psi_{G}^{b}}{\to}\mathfrak{Z}(G_{b}(E),\Lambda)$.999We
write $\mathfrak{Z}(G_{b}(E),\Lambda)$ for the center of the category of
smooth $\Lambda[G_{b}(E)]$-modules. When
$\Lambda=\overline{\mathbf{Q}_{\ell}}$, this is the usual Bernstein center,
and we drop $\Lambda$ from the notation. In that case we also write
$X_{G_{b}}$ for the Bernstein variety, so
$\mathcal{O}(X_{G_{b}})\cong\mathfrak{Z}(G_{b}(E))$. When $b=1$, we simply
write $\Psi_{G}$ for this map. The following is a renormalized version of
[FS21, Theorem IX.7.2].

######  Theorem 1.4.2.

For any $G$ and any $b\in B(G)$, the diagram

---
$\textstyle{\mathcal{O}(\mathrm{Par}_{G,\Lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{G}^{b}}$$\textstyle{\mathfrak{Z}(G_{b}(E),\Lambda)}$$\textstyle{\mathcal{O}(\mathrm{Par}_{G_{b},\Lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{G_{b}}}$

commutes. Here the left vertical arrow is induced by the canonical finite map
$\mathrm{Par}_{G_{b},\Lambda}\to\mathrm{Par}_{G,\Lambda}$ associated with the
canonical L-embedding ${}^{L}G_{b}\to\,^{L}G$.

Note that for non-basic $b$, our definition of $\Psi_{G}^{b}$ differs from
that of Fargues-Scholze, and is in fact simpler to use, as reflected in the
fact that we have the canonical _L-_ embedding appearing in the previous
theorem rather than the twisted embedding used in Fargues-Scholze.

######  Exercise 1.4.3.

Check that the map $\Psi_{G}^{b}$ is unaltered upon replacing
$i_{b!}^{\mathrm{ren}}$ by $i_{b\sharp}^{\mathrm{ren}}$ in its definition.

When $\Lambda=\overline{\mathbf{Q}_{\ell}}$, it can be helpful to think more
geometrically: the datum of the map $\Psi_{G}$ is equivalent to the datum of a
map of (ind-)varieties $\Psi_{G}^{\mathrm{geom}}:X_{G}\to
X_{G}^{\mathrm{spec}}$. From this perspective, the Fargues-Scholze
construction of _L_ -parameters is extremely transparent: any irreducible
smooth representation $\pi$ determines a closed point in $X_{G}$, and thus a
closed point in $X_{G}^{\mathrm{spec}}$ via the map
$\Psi_{G}^{\mathrm{geom}}$. But closed points in $X_{G}^{\mathrm{spec}}$
correspond exactly to semisimple _L-_ parameters.

However, Fargues-Scholze go much further than this, and prove the following
substantial upgrade of the above construction.

######  Theorem 1.4.4.

There is a canonical $\Lambda$-linear $\otimes$-action of
$\mathrm{Perf}(\mathrm{Par}_{G,\Lambda})$ on $D(\mathrm{Bun}_{G},\Lambda)$
compatible with the action of Hecke operators and preserving the subcategory
$D(\mathrm{Bun}_{G},\Lambda)^{\omega}$ of compact objects.

We write $C\ast(-)\circlearrowright D(\mathrm{Bun}_{G},\Lambda)$ for the
spectral action of $C\in\mathrm{Perf}(\mathrm{Par}_{G,\Lambda})$.
Compatibility with Hecke operators means that the diagram

$\textstyle{\mathrm{End}\left(D(\mathrm{Bun}_{G},\Lambda)\right)}$$\textstyle{\mathrm{Rep}_{\Lambda}(^{L}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V\mapsto
T_{V}}$$\textstyle{\mathrm{Rep}_{\Lambda}(\hat{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V\mapsto\tau_{G}^{\ast}V\ast(-)}$

commutes.

Fargues-Scholze then formulate a deep conjectural refinement of these
constructions. To state this, we need to assume that $G$ is quasisplit. We
also choose a Whittaker datum, i.e. a Borel subgroup $B\subset G$ and a
generic character $\psi:U(E)\to\Lambda^{\times}$, where $U\subset B$ is the
unipotent radical. We will typically use $\psi$ as shorthand for the choice of
Whittaker datum. We may then define the space

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

of compactly supported Whittaker functions with coefficients in $\Lambda$. In
other words, $W_{\psi}\subset\mathcal{C}(G(E),\Lambda)$ is the space of
functions such that $f(ug)=\psi(u)f(g)$ for all $u\in U(E)$ and $g\in G(E)$,
$f$ is right-invariant by some open compact subgroup of $G(E)$, and the
support of $f$ has compact image in $U(E)\backslash G(E)$. Note that
$W_{\psi}$ is a “large” $G(E)$-representation, but nonetheless has excellent
properties (which we will recall in Appendix A).

######  Conjecture 1.4.5.

Let $\Lambda$ be as above, and containing all $p$-power roots of unity. Then
there is a natural $\Lambda$-linear equivalence of categories

$\mathbf{L}_{\mathfrak{\psi}}^{G}:\mathrm{Coh}_{\mathrm{Nilp}}(\mathrm{Par}_{G,\Lambda})\overset{\sim}{\to}D(\mathrm{Bun}_{G},\Lambda)^{\omega}$

which is compatible with the spectral action, and which (after ind-completion)
sends the structure sheaf $\mathcal{O}_{\mathrm{Par}_{G,\Lambda}}$ to
$i_{1!}W_{\psi}$.

Compatibility with the spectral action means that we should have
$\mathbf{L}_{\psi}^{G}(A\otimes B)\simeq A\ast\mathbf{L}_{\psi}^{G}(B)$ for
all $A\in\mathrm{Perf}(\mathrm{Par}_{G,\Lambda})$ and
$B\in\mathrm{Coh_{Nilp}}(\mathrm{Par}_{G,\Lambda})$. Applying this
compatibility with $A\in\mathrm{Perf^{qc}}(\mathrm{Par}_{G,\Lambda})$ and
$B=\mathcal{O}$, in combination with the expectation that
$\mathbf{L}_{\psi}^{G}(\mathcal{O})=i_{1!}W_{\psi}$, we deduce that
necessarily we should have $\mathbf{L}_{\psi}^{G}(A)=A\ast i_{1!}W_{\psi}$ for
$A\in\mathrm{Perf^{qc}}(\mathrm{Par}_{G,\Lambda})$.101010Fargues likes to
advocate the perspective that $A\mapsto A\ast i_{1!}W_{\psi}$ is a kind of
non-abelian Fourier transform, and $\mathbf{L}_{\psi}^{G}$ should be some kind
of “continuous” extension of it from $\mathrm{Perf}$ to $\mathrm{Coh}$. In
particular, we see that $A\mapsto A\ast i_{1!}W_{\psi}$ should map
$\mathrm{Perf^{qc}}(\mathrm{Par}_{G,\Lambda})$ towards _compact_ objects in
$D(\mathrm{Bun}_{G},\Lambda)$. This is not obvious! When $\Lambda$ is a field,
this is closely related to showing that the map $\pi\mapsto\varphi_{\pi}$ has
finite fibers. We also see that $A\mapsto A\ast i_{1!}W_{\psi}$ should be
_fully faithful_ as a functor from $\mathrm{Perf}$ towards
$D(\mathrm{Bun}_{G},\Lambda)$, which again is far from obvious.

What further conditions do we expect $\mathbf{L}_{\psi}^{G}$ to satisfy?

#### Compatibility with the central grading

Using the inclusion $Z(\hat{G})^{\Gamma}\subset\hat{G}$, we get a
decomposition

$\mathrm{Coh_{Nilp}}(\mathrm{Par}_{G,\Lambda})\cong\oplus_{\chi\in
X^{\ast}(Z(\hat{G})^{\Gamma})}\mathrm{Coh_{Nilp}}(\mathrm{Par}_{G,\Lambda})^{Z(\hat{G})^{\Gamma}=\chi}.$

On the other hand, by a classic theorem of Kottwitz we have a canonical
bijection

$\displaystyle\pi_{0}(\mathrm{Bun}_{G})=\pi_{1}(G)_{\Gamma}$
$\displaystyle\cong X^{\ast}(Z(\hat{G})^{\Gamma})$ $\displaystyle\alpha$
$\displaystyle\mapsto\chi_{\alpha}$

which induces a corresponding decomposition

$D(\mathrm{Bun}_{G},\Lambda)^{\omega}\cong\oplus_{\alpha\in\pi_{1}(G)_{\Gamma}}D(\mathrm{Bun}_{G,\alpha},\Lambda)^{\omega}.$

Writing
$\mathrm{Bun}_{G}=\coprod_{\alpha\in\pi_{1}(G)_{\Gamma}}\mathrm{Bun}_{G,\alpha}$,
we expect that $\mathbf{L}_{\mathfrak{\psi}}^{G}$ should restrict to
compatible equivalences

$\mathrm{Coh_{Nilp}}(\mathrm{Par}_{G,\Lambda})^{Z(\hat{G})^{\Gamma}=\chi_{\alpha}}\overset{\sim}{\to}D(\mathrm{Bun}_{G,\alpha},\Lambda)^{\omega}$

for all $\alpha\in\pi_{1}(G)_{\Gamma}$.

#### Compatibility with duality

We let $\mathbf{D}_{\mathrm{GS}}=R\mathscr{H}\mathrm{om}(-,\omega)$ denote
Grothendieck-Serre duality functor on
$\mathrm{Coh}(\mathrm{Par}_{G,\Lambda})$. (We note for later use that
$\omega=\mathcal{O}_{\mathrm{Par}_{G,\Lambda}}$ canonically.) Let
$c:\mathrm{Par}_{G,\Lambda}\overset{\sim}{\to}\mathrm{Par}_{G,\Lambda}$ be the
involution defined by composition with the Chevalley involution at the level
of _L-_ parameters. Note that $\mathbf{D}_{\mathrm{GS}}$ and $c^{\ast}$
commute, so _twisted Grothendieck-Serre duality_

$\mathbf{D}_{\mathrm{twGS}}=c^{\ast}\mathbf{D}_{\mathrm{GS}}$

still defines an involutive anti-equivalence on
$\mathrm{Coh}(\mathrm{Par}_{G,\Lambda})$. The compatibility of categorical
local Langlands with duality can now be formulated as follows.

######  Conjecture 1.4.6.

There is a natural equivalence of functors

$\mathbf{L}_{\psi^{-1}}^{G}\circ\mathbf{D}_{\mathrm{twGS}}\simeq\mathbf{D}_{\mathrm{BZ}}\circ\mathbf{L}_{\psi}^{G}$

from $\mathrm{Coh}_{\mathrm{Nilp}}(\mathrm{Par}_{G,\Lambda})$ towards
$D(\mathrm{Bun}_{G},\Lambda)^{\omega}$.

Note that this is again a meta-conjecture, since we have not actually given a
candidate for the functor $\mathbf{L}_{\psi}^{G}$.

#### Compatibility with Eisenstein series

To explain this, we need to first formulate our expectations for Eisenstein
series. A much more detailed discussion will appear in [HHS].

Expectation. For any parabolic $P=MU\subset G$, there is a canonically
associated functor

$\mathrm{Eis}_{P}=\mathrm{Eis}_{P}^{G}:D(\mathrm{Bun}_{M},\Lambda)\to
D_{\mathrm{}}(\mathrm{Bun}_{G},\Lambda)$

with the following properties.

1\. There is a natural equivalence of functors $\mathrm{Eis}_{P}\circ
i_{1!}^{M}\simeq i_{1!}\circ i_{P}^{G}$, where $i_{1!}$ and
$i_{1!}^{M}:D(M(E),\Lambda)\to D(\mathrm{Bun}_{G},\Lambda)$ are the
appropriate extension by zero functors, and $i_{P}^{G}$ denotes the functor of
normalized parabolic induction. More generally, if $b\in B(M)$ is any element
whose image in $B(G)$ is basic, we expect that $\mathrm{Eis}_{P}\circ
i_{b!}^{M}\simeq i_{b!}\circ i_{P_{b}}^{G_{b}}$.

2\. $\mathrm{Eis}_{P}$ is compatible with composition: for any inclusion of
parabolics $P_{1}=M_{1}U_{1}\subset P_{2}=M_{2}U_{2}$, $P_{1}\cap M_{2}$ is a
parabolic in $M_{2}$ with Levi $M_{1}$, and there should be a natural
equivalence
$\mathrm{Eis}_{P_{1}}^{G}\simeq\mathrm{Eis}_{P_{2}}^{G}\circ\mathrm{Eis}_{P_{1}\cap
M_{2}}^{M_{2}}$. This equivalence should be compatible with triple composition
in the evident sense.

3\. $\mathrm{Eis}_{P}$ is compatible with any extension of scalars
$\Lambda\to\Lambda^{\prime}$.

4\. $\mathrm{Eis}_{P}$ commutes with all direct sums, and preserves compact
objects.

5\. $\mathrm{Eis}_{P}$ preserves ULA objects with quasicompact support.

6\. When $\Lambda$ is killed by a power of $\ell$ (so
$D_{\mathrm{lis}}=D_{\mathrm{\acute{e}t}}$), $\mathrm{Eis}_{P}$ is the functor
$p_{!}(\mathrm{IC}_{\mathrm{Bun}_{P},\mathbf{Z}_{\ell}[\sqrt{q}]}\otimes_{\mathbf{Z}_{\ell}[\sqrt{q}]}q^{\ast}(-))$,
where

---
$\textstyle{\mathrm{Bun}_{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{p}$$\textstyle{\mathrm{Bun}_{G}}$$\textstyle{\mathrm{Bun}_{M}}$

is the usual diagram, and
$\mathrm{IC}_{\mathrm{Bun}_{P},\mathbf{Z}_{\ell}[\sqrt{q}]}\in
D_{\mathrm{\acute{e}t}}(\mathrm{Bun}_{P},\mathbf{Z}_{\ell}[\sqrt{q}])$ is a
certain explicit invertible object (a square root of the dualizing complex on
$\mathrm{Bun}_{P}$, which can be described explicitly [HI23]).

Of course, when $\Lambda$ is a torsion ring, the formula in 6. can and should
be taken as the definition of $\mathrm{Eis}_{P}$. With this definition,
properties 1.-3. are relatively easy, but 4. and 5\. seem to lie significantly
deeper. The formula in 6. should in fact be applicable for any coefficient
ring, once the sheaf-theoretic machinery is sufficiently developed.

We now expect the following compatibility with categorical local Langlands.

######  Conjecture 1.4.7.

For all standard parabolics $P=MU\subset G$, there is an equivalence
$\mathbf{L}_{\psi}^{G}\circ\mathrm{Eis}_{P}^{\mathrm{spec}}\simeq\mathrm{Eis}_{P}\circ\mathbf{L}_{\psi_{M}}^{M}$,
where $\mathrm{Eis}_{P}^{\mathrm{spec}}=p_{\ast}^{\mathrm{spec}}\circ
q^{\mathrm{spec}\ast}$ is the spectral Eisenstein functor associated with the
diagram

---
$\textstyle{\mathrm{Par}_{P,\Lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{\mathrm{spec}}}$$\scriptstyle{p^{\mathrm{spec}}}$$\textstyle{\mathrm{Par}_{G,\Lambda}}$$\textstyle{\mathrm{Par}_{M,\Lambda}}$

of (derived) Artin stacks. This equivalence is compatible with composition in
the evident sense.

Note: Zhu defines the spectral Eisenstein functor by the formula
$p_{\ast}^{\mathrm{spec}}\circ q^{\mathrm{spec}!}$. However, there is an
isomorphism $q^{\mathrm{spec}!}\simeq q^{\mathrm{spec}\ast}$ because $q$ is
quasismooth, hence Gorenstein by [AG15, Corollary 2.2.7], so
$q^{\mathrm{spec}!}\mathcal{F}\simeq q^{\mathrm{spec\ast}}\mathcal{F}\otimes
q^{\mathrm{spec}!}\mathcal{O}$ [Gai13, Remark 7.2.7], and one can show that
$q^{\mathrm{spec}!}\mathcal{O}\simeq\mathcal{O}$ (see [Zhu21], Remark 2.3.8
and the comments after the proof of Proposition 2.3.9).

######  Remark 1.4.8.

The functors $i_{b!}^{\mathrm{ren}}$ and $i_{b\sharp}^{\mathrm{ren}}$ are
closely related to Eisenstein series. More precisely, let $b\in B(G)$ be any
element, and let $M\subset G$ be the centralizer of its Newton point
$\nu_{b}$; we may assume $b\in M(\breve{E})$, so $M_{b}=G_{b}$. Let $P$ and
$\overline{P}$ be the attracting and repelling dynamic parabolics associated
with $\nu_{b}$. We then expect (for any coefficient ring) that there are
natural isomorphisms

$i_{b!}^{\mathrm{ren}}\simeq\mathrm{Eis}_{P}^{G}\circ i_{b!}^{M}$

and

$i_{b\sharp}^{\mathrm{ren}}\simeq\mathrm{Eis}_{\overline{P}}^{G}\circ
i_{b!}^{M}$

as functors $D(M_{b}(E),\Lambda)\to D_{\mathrm{}}(\mathrm{Bun}_{G},\Lambda)$.
When $\Lambda$ is a torsion ring, this is an easy exercise (modulo the
identification of
$\mathrm{IC}_{\mathrm{Bun}_{P},\mathbf{Z}_{\ell}[\sqrt{q}]}$). See [FS21,
Example V.3.4] for a hint.

###  1.5 More conjectures

In this section we illustrate how the categorical conjecture leads naturally
to various additional conjectures.

Let us begin with the following question: How do Eisenstein series interact
with Hecke operators? More precisely, we could ask: for fixed $P=MU\subset G$
and $V\in\mathrm{Rep}(^{L}G)$, is there an intellegent way of rewriting the
composite functor $T_{V}\mathrm{Eis}_{P}$? This seems rather difficult at
first glance, but we can ask the same question on the spectral side. Here
things become simpler, since the spectral analogue of $T_{V}$ is tensoring
with the tautological vector bundle $\tau_{G}^{\ast}V$. We are now asking
whether the functor $\tau_{G}^{\ast}V\otimes\mathrm{Eis}_{P}^{\mathrm{spec}}$
can be rewritten in an intelligent way. This turns out to have a very
satisfying answer.

######  Proposition 1.5.1.

Choose a finite filtration $0=V_{0}\subset V_{1}\subset\cdots\subset
V_{m}=V|\hat{P}_{\Lambda}$ such that the $\hat{U}_{\Lambda}$-action on each
graded piece $W_{i}=V_{i}/V_{i-1}$ is trivial, i.e. such that each $W_{i}$ is
naturally inflated from $\mathrm{Rep}(\hat{M}_{\Lambda})$. Then
$\tau_{G}^{\ast}V\otimes\mathrm{Eis}_{P}^{\mathrm{spec}}(-)$ admits a
corresponding finite filtration with graded pieces
$\mathrm{Eis}_{P}^{\mathrm{spec}}(\tau_{M}^{\ast}W_{i}\otimes-)$.

###### Proof.

To begin, observe that we have a commutative diagram

$\textstyle{\mathrm{Par}_{M,\Lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{M}}$$\textstyle{\mathrm{Par}_{P,\Lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{P}}$$\scriptstyle{q^{\mathrm{spec}}}$$\scriptstyle{p^{\mathrm{spec}}}$$\textstyle{\mathrm{Par}_{G,\Lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{G}}$$\textstyle{B\hat{M}_{\Lambda}}$$\textstyle{B\hat{P}_{\Lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\hat{G}_{\Lambda}}$

of derived Artin stacks. We can now use the projection formula to write

$\displaystyle\tau_{G}^{\ast}V\otimes\mathrm{Eis}_{P}^{\mathrm{spec}}(-)$
$\displaystyle=\tau_{G}^{\ast}V\otimes
p_{\ast}^{\mathrm{spec}}q^{\mathrm{spec}\ast}(-)$ $\displaystyle\simeq
p_{\ast}^{\mathrm{spec}}(p^{\mathrm{spec}\ast}\tau_{G}^{\ast}V\otimes
q^{\mathrm{spec}\ast}(-))$ $\displaystyle\simeq
p_{\ast}^{\mathrm{spec}}(\tau_{P}^{\ast}(V|\hat{P}_{\Lambda})\otimes
q^{\mathrm{spec}\ast}(-)).$

Then by assumption $\tau_{P}^{\ast}(V|\hat{P}_{\Lambda})$ has a finite
filtration with graded pieces $q^{\mathrm{spec}\ast}\tau_{M}^{\ast}W_{i}$, so
the functor
$p_{\ast}^{\mathrm{spec}}(\tau_{P}^{\ast}(V|\hat{P}_{\Lambda})\otimes
q^{\mathrm{spec}\ast}(-))$ acquires a finite filtration with graded pieces

$\displaystyle
p_{\ast}^{\mathrm{spec}}(q^{\mathrm{spec}\ast}\tau_{M}^{\ast}W_{i}\otimes
q^{\mathrm{spec}\ast}(-))$ $\displaystyle\simeq
p_{\ast}^{\mathrm{spec}}q^{\mathrm{spec}\ast}(\tau_{M}^{\ast}W_{i}\otimes-)$
$\displaystyle=\mathrm{Eis}_{P}^{\mathrm{spec}}(\tau_{M}^{\ast}W_{i}\otimes-),$

using that $q^{\mathrm{spec}\ast}$ is symmetric monoidal. ∎

But now we can turn this into a conjecture on the automorphic side.

######  Conjecture 1.5.2.

Choose a finite filtration $0=V_{0}\subset V_{1}\subset\cdots\subset
V_{m}=V|\hat{P}_{\Lambda}$ such that the $\hat{U}_{\Lambda}$-action on each
graded piece $W_{i}=V_{i}/V_{i-1}$ is trivial, i.e. such that each $W_{i}$ is
naturally inflated from $\mathrm{Rep}(\hat{M}_{\Lambda})$. Then
$T_{V}\mathrm{Eis}_{P}^{\mathrm{}}(-)$ admits a corresponding finite
filtration with graded pieces $\mathrm{Eis}_{P}(T_{W_{i}}-)$.

We emphasize that while this conjecture takes place purely on the automorphic
side, it is forced on us by the categorical conjecture and the previous
proposition. This conjecture has been proved in many cases by Hamann [Ham22],
and a similar argument should work in general when $\Lambda$ is a torsion
ring, and for arbitrary coefficients once the sheaf-theoretic machinery
improves.

We can also start combining our predictions in more artful ways. For instance,
recall that we conjectured a natural equivalence
$\mathbf{L}_{\psi}^{G}\circ\mathrm{Eis}_{P}^{\mathrm{spec}}\simeq\mathrm{Eis}_{P}\circ\mathbf{L}_{\psi_{M}}^{M}$.
How does this interact with duality? Applying $\mathbf{D}_{\mathrm{BZ}}$ to
both sides, we compute

$\displaystyle\mathbf{D}_{\mathrm{BZ}}\circ\mathrm{Eis}_{P}\circ\mathbf{L}_{\psi_{M}}^{M}$
$\displaystyle\simeq\mathbf{D}_{\mathrm{BZ}}\circ\mathbf{L}_{\psi}^{G}\circ\mathrm{Eis}_{P}^{\mathrm{spec}}$
$\displaystyle\simeq\mathbf{L}_{\psi^{-1}}^{G}\circ\mathbf{D}_{\mathrm{twGS}}\circ\mathrm{Eis}_{P}^{\mathrm{spec}}$
$\displaystyle\overset{!}{\simeq}\mathbf{L}_{\psi^{-1}}^{G}\circ\mathrm{Eis}_{\overline{P}}^{\mathrm{spec}}\circ\mathbf{D}_{\mathrm{twGS}}^{M}$
$\displaystyle\simeq\mathrm{Eis}_{\overline{P}}\circ\mathbf{L}_{\psi_{M}^{-1}}^{M}\circ\mathbf{D}_{\mathrm{twGS}}^{M}$
$\displaystyle\simeq\mathrm{Eis}_{\overline{P}}\circ\mathbf{D}_{\mathrm{BZ}}^{M}\circ\mathbf{L}_{\psi_{M}}^{M}.$

Here we used several times the expected compatibility of the categorical
equivalence with duality, along with the equivalence
$\mathbf{L}_{\psi}^{G}\circ\mathrm{Eis}_{P}^{\mathrm{spec}}\simeq\mathrm{Eis}_{P}\circ\mathbf{L}_{\psi_{M}}^{M}$
and its variant
$\mathbf{L}_{\psi^{-1}}^{G}\circ\mathrm{Eis}_{\overline{P}}^{\mathrm{spec}}\simeq\mathrm{Eis}_{\overline{P}}\circ\mathbf{L}_{\psi_{M}^{-1}}^{M}$.
The only point which requires further analysis is the isomorphism labelled
“!”. We leave this as an enlightening exercise to the reader, with the key
hint being that the Chevalley involution exchanges $\hat{P}$ and
$\hat{\overline{P}^{\mathrm{}}}$. Anyway, recall now that
$\mathbf{L}_{\psi_{M}}^{M}$ is supposed to be an equivalence of categories, in
which case we may “cancel it out” from the first and last expressions. We have
thus arrived at the following conjecture, which again lives purely on the
automorphic side.

######  Conjecture 1.5.3.

For any given parabolic $P$ with opposite $\overline{P}$, there is a natural
equivalence of functors
$\mathbf{D}_{\mathrm{BZ}}\circ\mathrm{Eis}_{P}\simeq\mathrm{Eis}_{\overline{P}}\circ\mathbf{D}_{\mathrm{BZ}}^{M}$
from $D(\mathrm{Bun}_{M},\Lambda)^{\omega}$ to
$D(\mathrm{Bun}_{G},\Lambda)^{\omega}$.

This is a geometric analogue of the well-known fact that
$\mathbf{D}_{\mathrm{coh}}\circ i_{P}^{G}\simeq
i_{\overline{P}}^{G}\circ\mathbf{D}_{\mathrm{coh}}^{M}$ with $\mathbf{C}$\- or
$\overline{\mathbf{Q}_{\ell}}$-coefficients, which was first observed by
Bernstein and is in fact _equivalent_ to his famous second adjointness
theorem. With general coefficients, this isomorphism follows from recent work
of Dat-Helm-Kurinczuk-Moss [DHKM22].

######  Exercise 1.5.4.

1) Prove that Conjecture 1.5.3 implies second adjointness.

2) (Difficult.) Fix a torsion coefficient ring $\Lambda$ and a parabolic $P$.
Prove that the following two statements are _equivalent_ :

i. The functor $\mathrm{Eis}_{P}$ preserves compact objects, and Conjecture
1.5.3 is true.

ii. There is a natural equivalence of “constant term” functors
$q_{\ast}p^{!}\simeq\overline{q}_{!}\overline{p}^{\ast}$. (Here $\overline{p}$
and $\overline{q}$ refer to the obvious maps in the defining diagram for
$\mathrm{Eis}_{\overline{P}}$.)

Recall the map
$\mathcal{O}(X_{G}^{\mathrm{spec}})\cong\mathcal{O}(\mathrm{Par}_{G})\to\mathfrak{Z}(D(\mathrm{Bun}_{G},\overline{\mathbf{Q}_{\ell}}))$
discussed at the beginning of section 1.4. By the very nature of the
categorical center, this induces a canonical and functorial map
$\mathcal{O}(X_{G}^{\mathrm{spec}})\to\mathrm{End}(\mathcal{F})$ for any
$\mathcal{F}\in D(\mathrm{Bun}_{G},\overline{\mathbf{Q}_{\ell}})$.

######  Exercise 1.5.5.

Show that the categorical conjecture and Conjecture 1.4.7 together imply the
following: for any parabolic $P=MU\subset G$ and any $\mathcal{F}\in
D(\mathrm{Bun}_{M},\overline{\mathbf{Q}_{\ell}})$, the diagram

$\textstyle{\mathrm{End}(\mathcal{F})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{End}(\mathrm{Eis}_{P}(\mathcal{F}))}$$\textstyle{\mathcal{O}(X_{M}^{\mathrm{spec}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(X_{G}^{\mathrm{spec}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

commutes.

###  1.6 Finiteness conditions and spectral decomposition of sheaves

Now we set $\Lambda=\overline{\mathbf{Q}_{\ell}}$ and drop it from the
notation. Before continuing our discussion, it is very useful to analyze how
the various finiteness conditions on objects in $D(\mathrm{Bun}_{G})$
interact, and how they interact with “spectral decomposition” of sheaves.

We begin with the following observations. For any semisimple parameter $\phi$,
we can formally define the full subcategory
$D(\mathrm{Bun}_{G})_{\phi}^{\mathrm{ULA}}\subset
D(\mathrm{Bun}_{G})^{\mathrm{ULA}}$ of $\phi$_-local_ ULA sheaves spanned by
objects $A$ such that for all $b$ and $n$, every irreducible subquotient of
$H^{n}(i_{b}^{\ast\mathrm{ren}}A)$ has Fargues-Scholze parameter $\phi$. It is
easy to see that $D(\mathrm{Bun}_{G})_{\phi}^{\mathrm{ULA}}$ is a thick
triangulated subcategory stable under Hecke operators, and one can also prove
that it is stable under the perverse truncation functors. In fact, one can
prove that $D(\mathrm{Bun}_{G})_{\phi}^{\mathrm{ULA}}$ is canonically a direct
factor of $D(\mathrm{Bun}_{G})^{\mathrm{ULA}}$, in the sense that any ULA
sheaf $A$ has a canonical and functorial decomposition $A\cong A_{\phi}\oplus
A^{\phi}$ where $A_{\phi}\in D(\mathrm{Bun}_{G})_{\phi}^{\mathrm{ULA}}$ and

$\mathrm{Hom}(B,A^{\phi})=\mathrm{Hom}(A^{\phi},B)=0$

for all $B\in D(\mathrm{Bun}_{G})_{\phi}^{\mathrm{ULA}}$ (see [Han23b] for a
detailed statement and proof). One can also prove that Verdier duality induces
an involutive anti-equivalence

$\mathbf{D}_{\mathrm{Verd}}:D(\mathrm{Bun}_{G})_{\phi}^{\mathrm{ULA}}\overset{\sim}{\to}D(\mathrm{Bun}_{G})_{\phi^{\vee}}^{\mathrm{ULA}}.$

At the level of perverse sheaves, we get an evident category
$\mathrm{Perv}(\mathrm{Bun}_{G})_{\phi}^{\mathrm{ULA}}$ of $\phi$-local
perverse ULA sheaves, which is a direct factor of
$\mathrm{Perv}(\mathrm{Bun}_{G})^{\mathrm{ULA}}$, and Verdier duality induces
an exact anti-equivalence of abelian categories

$\mathbf{D}_{\mathrm{Verd}}:\mathrm{Perv}(\mathrm{Bun}_{G})_{\phi}^{\mathrm{ULA}}\overset{\sim}{\to}\mathrm{Perv}(\mathrm{Bun}_{G})_{\phi^{\vee}}^{\mathrm{ULA}}.$

######  Exercise 1.6.1.

Check that $\mathrm{Perv}(\mathrm{Bun}_{G})_{\phi}^{\mathrm{ULA}}$ is a Serre
subcategory of $\mathrm{Perv}(\mathrm{Bun}_{G})^{\mathrm{ULA}}$.

It will also be very useful to consider the following more restrictive
finiteness condition.

######  Definition 1.6.2.

A sheaf $A\in D(\mathrm{Bun}_{G})$ is _finite_ if it is both compact and ULA.
Equivalently, $A$ is finite if it has quasicompact support and
$\oplus_{n}H^{n}(i_{b}^{\ast}A)$ is a finite length $G_{b}(E)$-representation
for every $b$.

Finite sheaves clearly form a thick triangulated subcategory of
$D(\mathrm{Bun}_{G})$, which we denote $D(\mathrm{Bun}_{G})_{\mathrm{fin}}$.
The name is meant to suggest that such objects have finite length in some
sense. Indeed, one easily checks that $D(\mathrm{Bun}_{G})_{\mathrm{fin}}$ is
the thick triangulated subcategory of $D(\mathrm{Bun}_{G})$ consisting of
sheaves which can be obtained from objects of the form
$i_{b!}^{\mathrm{ren}}\pi$ (with $\pi$ any irreducible
$G_{b}(E)$-representation) via finitely many shifts, cones and retracts. This
category is stable under Hecke operators, since Hecke operators preserve
compactness and ULAness separately. We also note that for any two finite
sheaves $A,B$, $R\mathrm{Hom}(A,B)$ is a perfect complex of
$\overline{\mathbf{Q}_{\ell}}$-vector spaces, and in particular
$\mathrm{End}(A)$ is an Artinian $\overline{\mathbf{Q}_{\ell}}$-algebra. This
is a special case of the more general fact that $R\mathrm{Hom}(A,B)$ is
perfect whenever $A$ is compact and $B$ is ULA, which follows from [FS21,
Prop. VII.7.4 and Prop. VII.7.9].

Warning. Finite sheaves are _not_ stable under Verdier duality, except when
$G$ is a torus. Indeed, for non-toral groups it is easy to see that
$i_{1!}\overline{\mathbf{Q}_{\ell}}$ is a finite sheaf whose Verdier dual
$i_{1\ast}\overline{\mathbf{Q}_{\ell}}$ does not have quasicompact support.
However, we have the following important result.

######  Theorem 1.6.3.

If $\pi\in\Pi(G_{b})$ is irreducible, then $i_{b\sharp}^{\mathrm{ren}}\pi$ is
finite. In particular, finite sheaves are stable under Bernstein-Zelevinsky
duality.

###### Proof.

By Proposition 1.1.4 and an easy induction on support, it is enough to prove
the first statement. This immediately reduces to proving that
$i_{b^{\prime}}^{\ast\mathrm{ren}}i_{b\sharp}^{\mathrm{ren}}\pi$ is compact
and admissible for any $b^{\prime}\preceq b$. Compactness is clear, since both
$i_{b\sharp}^{\mathrm{ren}}$ and $i_{b^{\prime}}^{\ast\mathrm{ren}}$ preserve
compactness. For admissibility, we apply the following criterion: a compact
object $B\in D(G_{b^{\prime}}(E),\overline{\mathbf{Q}_{\ell}})$ is admissible
iff its support in the Bernstein variety $X_{G_{b^{\prime}}}$ is zero-
dimensional. The “only if” direction is clear; for the “if” direction, note
that any compact $B$ is admissible “over the Bernstein center” [Ber84], so
true admissibility follows if the action of the Bernstein center factors over
an Artinian quotient.

To apply this criterion, we note that the composite map

$c:X_{G_{b^{\prime}}}\overset{\Psi_{G_{b^{\prime}}}^{\mathrm{geom}}}{\longrightarrow}X_{G_{b^{\prime}}}^{\mathrm{spec}}\to
X_{G}^{\mathrm{spec}}$

has discrete fibers, and in fact it is a finite morphism after restricting to
any connected component of the source. This is clear for the second map (see
[DHKM22] for a more general statement which also works integrally), and for
the first map it is Lemma 1.6.4 below. Writing $x_{\pi}\in
X_{G}^{\mathrm{spec}}$ for the closed point corresponding to the semisimple
_L-_ parameter of $\pi$, we conclude by observing that the support of
$i_{b^{\prime}}^{\ast\mathrm{ren}}i_{b\sharp}^{\mathrm{ren}}\pi$ is contained
(set-theoretically) in $c^{-1}(x_{\pi})$, which follows from Theorem 1.4.2.
The admissibility criterion then applies, giving the desired result. ∎

######  Lemma 1.6.4.

For any $G$, the Fargues-Scholze map $\Psi_{G}^{\mathrm{geom}}:X_{G}\to
X_{G}^{\mathrm{spec}}$ has discrete fibers, and in fact is a finite morphism
after restricting to any connected component of $X_{G}$.

###### Proof.

This can be checked one component at a time in $X_{G}$. Then the usual
explicit description of individual Bernstein components (in terms of a fixed
pair $[M,\sigma]$ with $\sigma$ supercuspidal) plus compatibility of
everything with parabolic induction and twisting reduces us to the special
case of a cuspidal component $D$. In this case, one simply uses the fact that
the map from $D$ to the spectral Bernstein variety is compatible with
unramified twists, and unramified twisting has finite stabilizers in both
settings and is transitive on $D$. (Alternatively - but it’s basically the
same argument in different clothing - in the case of a cuspidal component one
can reduce the claim for $G$ to the analogous claims for $G^{\mathrm{der}}$
and $Z_{G}^{\circ}$ separately, using compatibility with products and central
isogenies. But for $G^{\mathrm{der}}$ it is trivial, since cuspidal components
of semisimple groups are zero-dimensional, and for $Z_{G}^{\circ}$ it is clear
since one has $X_{T}=X_{T}^{\mathrm{spec}}$ for $T$ any torus.) ∎

######  Remark 1.6.5.

I do not know how to prove this theorem with
$\overline{\mathbf{F}_{\ell}}$-coefficients. The problem is that I have no
idea how to prove Lemma 1.6.4 with $\overline{\mathbf{F}_{\ell}}$-coefficients
in general, although the same argument as written above works for $\ell$
outside an explicit finite set of bad primes.

######  Exercise 1.6.6.

Prove that for any compact objects $A,B\in D(\mathrm{Bun}_{G})^{\omega}$, the
Fargues-Scholze map
$\mathcal{O}(\mathrm{Par}_{G})\to\mathfrak{Z}(D(\mathrm{Bun}_{G}))$ naturally
makes $\mathrm{Hom}(A,B)$ into a _finitely generated_
$\mathcal{O}(\mathrm{Par}_{G})$-module. Hint: Reduce to the case where $A,B$
are $!$\- or $\sharp$-pushforwards from individual strata, then use Lemma
1.6.4 and Theorem 1.4.2 together with Bernstein’s basic finiteness theorems.

Now suppose $A$ is a finite sheaf. Then $A$ is ULA, so for any semisimple
parameter we have a decomposition $A\cong A_{\phi}\oplus A^{\phi}$ as before,
where $A_{\phi}$ and $A^{\phi}$ are both finite. Since $A$ is finite, it is
easy to see that $A_{\phi}=0$ for all but finitely many $\phi$, so we get a
canonical direct sum decomposition

$A\cong\bigoplus_{\phi}A_{\phi}$

in $D(\mathrm{Bun}_{G})_{\mathrm{fin}}$ where only finitely many summands are
nonzero. By the functoriality of this decomposition, we even get a canonical
direct sum decomposition of categories

$D(\mathrm{Bun}_{G})_{\mathrm{fin}}\cong\bigoplus_{\phi}D(\mathrm{Bun}_{G})_{\mathrm{fin},\phi}$

where of course we set
$D(\mathrm{Bun}_{G})_{\mathrm{fin},\phi}\overset{\mathrm{def}}{=}D(\mathrm{Bun}_{G})_{\mathrm{fin}}\cap
D(\mathrm{Bun}_{G})_{\phi}^{\mathrm{ULA}}$.

######  Exercise 1.6.7.

Check that Bernstein-Zelevinsky duality induces an involutive anti-equivalence

$\mathbf{D}_{\mathrm{BZ}}:D(\mathrm{Bun}_{G})_{\mathrm{fin},\phi}\overset{\sim}{\to}D(\mathrm{Bun}_{G})_{\mathrm{fin},\phi^{\vee}}.$

Finally, recall that have constructed the hadal t-structure (Theorem 1.2.3),
which is a certain t-structure on $D(\mathrm{Bun}_{G})^{\omega}$ with heart
denoted $\mathrm{Had}(\mathrm{Bun}_{G})$. We then write

$\mathrm{Had}(\mathrm{Bun}_{G})_{\mathrm{fin}}\overset{\mathrm{def}}{=}\mathrm{Had}(\mathrm{Bun}_{G})\cap
D(\mathrm{Bun}_{G})_{\mathrm{fin}}$

for the subcategory of finite hadal sheaves. Using Theorem 1.6.3, one can
check that the hadal truncation functors preserve
$D(\mathrm{Bun}_{G})_{\mathrm{fin}}$, so the pair

$\left({}^{h}D^{\leq 0}(\mathrm{Bun}_{G},\Lambda)^{\omega}\cap
D(\mathrm{Bun}_{G})_{\mathrm{fin}},{}^{h}D^{\geq
0}(\mathrm{Bun}_{G},\Lambda)^{\omega}\cap
D(\mathrm{Bun}_{G})_{\mathrm{fin}}\right)$

defines a t-structure on $D(\mathrm{Bun}_{G})_{\mathrm{fin}}$ with heart
$\mathrm{Had}(\mathrm{Bun}_{G})_{\mathrm{fin}}$. Finally, by the functoriality
of the direct sum decompositions above, we get a canonical decomposition

$\mathrm{Had}(\mathrm{Bun}_{G})_{\mathrm{fin}}\cong\bigoplus_{\phi}\mathrm{Had}(\mathrm{Bun}_{G})_{\mathrm{fin},\phi}.$

######  Exercise 1.6.8.

i. Check that $\mathrm{Had}(\mathrm{Bun}_{G})_{\mathrm{fin}}$ is a Serre
subcategory of $\mathrm{Had}(\mathrm{Bun}_{G})$.

ii. Check that every object in $\mathrm{Had}(\mathrm{Bun}_{G})_{\mathrm{fin}}$
is of finite length.

Warning. When $G$ is not a torus, the interplay between these finiteness
conditions can be very subtle. In particular, we highlight the following
phenomena:

1) The perverse truncation functors do not preserve the property of being
finite.

2) There are examples of irreducible perverse sheaves which are not finite
sheaves (the constant sheaf $\overline{\mathbf{Q}_{\ell}}$ on one component of
$\mathrm{Bun}_{G}$ has this property).

3) There are examples of finite sheaves which are perverse, but whose Jordan-
Holder series in the perverse category is infinite.

###  1.7 The categorical conjecture over $\overline{\mathbf{Q}_{\ell}}$,
unconditionally

We continue to fix $\Lambda=\overline{\mathbf{Q}_{\ell}}$ as our coefficient
ring. This leads to several simplifications in the formulation of the
categorical conjecture:

1) the nilpotent singular support condition on
$\mathrm{Coh}(\mathrm{Par}_{G})$ is automatic, and

2) each connected component of $\mathrm{Par}_{G}$ has the property that
$\mathrm{IndPerf}=\mathrm{QCoh}$.

Using 2), we can formally upgrade the spectral action to a monoidal action of
$\mathrm{QCoh}(\mathrm{Par}_{G})$ on $D(\mathrm{Bun}_{G})$. Acting on the
Whittaker sheaf in particular yields a functor

$\displaystyle a_{\psi}:\mathrm{QCoh}(\mathrm{Par}_{G})$ $\displaystyle\to
D(\mathrm{Bun}_{G})$ $\displaystyle\mathcal{F}$
$\displaystyle\mapsto\mathcal{F}\ast i_{1!}W_{\psi}$

where we choose to write “$a$” for _a_ ction. As noted in the discussion after
Conjecture 1.4.5, we expect that $a_{\psi}$ is fully faithful: under the
hoped-for equivalence
$D(\mathrm{Bun}_{G})\simeq\mathrm{IndCoh}(\mathrm{Par}_{G})$, $a_{\psi}$
should correspond to the natural fully faithful inclusion
$\mathrm{QCoh}\overset{\Xi}{\to}\mathrm{Ind}\mathrm{Coh}$. As noted earler, we
also expect that $a_{\psi}$ carries $\mathrm{Perf^{qc}}(\mathrm{Par}_{G})$
into $D(\mathrm{Bun}_{G})^{\omega}$. This is known unconditionally for many
groups: more precisely, for those $G$ which are _reasonable_ in the sense of
Definition A.1.1.

Warning. The category $\mathrm{IndCoh}$ contains two copies of $\mathrm{Coh}$:
the “native” copy coming from the tautological inclusion
$\mathcal{C}\subset\mathrm{Ind}(\mathcal{C})$, and a second “phantom” copy
$\Xi(\mathrm{Coh})$. These are not the same, and their overlap is actually
just $\mathrm{Perf}$. In particular, non-perfect objects in the “phantom” copy
are not compact when viewed as objects of $\mathrm{IndCoh}$. Translating into
the above picture, we see that for $\mathcal{F}$ a coherent complex on
$\mathrm{Par}_{G}$ which is not perfect, $a_{\psi}(\mathcal{F})$ should not be
compact. In particular, $a_{\psi}$ cannot be the functor realizing the
categorical local Langlands equivalence.

However, there is a closely related functor which _should_ realize this
equivalence.

######  Proposition 1.7.1.

There is a (unique) functor

$c_{\psi}:D(\mathrm{Bun}_{G})\to\mathrm{QCoh}(\mathrm{Par}_{G})$

such that for all $A\in D(\mathrm{Bun}_{G})$ and all
$\mathcal{F}\in\mathrm{QCoh}(\mathrm{Par}_{G})$, there is a natural
isomorphism

$R\mathrm{Hom}(i_{1!}W_{\psi},\mathcal{F}\ast A)\simeq
R\Gamma(\mathrm{Par}_{G},\mathcal{F}\otimes c_{\psi}(A)).$

The functor $c_{\psi}$ is motivated by classical geometric Langlands, where
the analogous beast is usually called the functor of _enhanced Whittaker
coefficient_ , and its construction in our setting is exactly the same as in
classical geometric Langlands (see [FR22, Section 10.2] and [Ras22]). As
justification for the name, note that

$\displaystyle R\Gamma(\mathrm{Par}_{G},c_{\psi}(A))$ $\displaystyle\simeq
R\mathrm{Hom}(i_{1!}W_{\psi},A)$
$\displaystyle=R\mathrm{Hom}(W_{\psi},i_{1}^{\ast}A)$

is exactly the space of Whittaker models of $i_{1}^{\ast}A$. Note also that
$c_{\psi}$ is $\mathrm{QCoh}$-linear, in the sense that
$c_{\psi}(\mathcal{G}\ast A)\simeq\mathcal{G}\otimes c_{\psi}(A)$ for all
$A\in D(\mathrm{Bun}_{G})$ and
$\mathcal{G}\in\mathrm{QCoh}(\mathrm{Par}_{G})$.

######  Proposition 1.7.2.

The functor $c_{\psi}$ is right adjoint to $a_{\psi}$.

###### Proof.

For $\mathcal{F}\in\mathrm{Perf}(\mathrm{Par}_{G})$ and $A\in
D(\mathrm{Bun}_{G})$ we compute

$\displaystyle\mathrm{Hom}(a_{\psi}(\mathcal{F}),A)$
$\displaystyle=\mathrm{Hom}(\mathcal{F}\ast i_{1!}W_{\psi},A)$
$\displaystyle\cong\mathrm{Hom}(i_{1!}W_{\psi},\mathbf{D}_{\mathrm{GS}}\mathcal{F}\ast
A)$
$\displaystyle\cong\mathrm{Hom}(\mathcal{O},\mathbf{D}_{\mathrm{GS}}\mathcal{F}\otimes
c_{\psi}(A))$ $\displaystyle\cong\mathrm{Hom}(\mathcal{F},c_{\psi}(A)).$

The only nontrivial point here is the second line, which follows from the fact
that for any given $\mathcal{F}\in\mathrm{Perf}(\mathrm{Par}_{G})$, the
endofunctor $A\mapsto\mathcal{F}\ast A$ on $D(\mathrm{Bun}_{G})$ is both left
and right adjoint to the endofunctor
$A\mapsto\mathbf{D}_{\mathrm{GS}}\mathcal{F}\ast A$. Since both sides of this
isomorphism convert colimits in $\mathcal{F}$ into limits, it formally extends
to an isomorphism valid for any
$\mathcal{F}\in\mathrm{IndPerf}=\mathrm{QCoh}$. ∎

Since $c_{\psi}$ is right adjoint to $a_{\psi}$, under the hoped-for
equivalence $D(\mathrm{Bun}_{G})\simeq\mathrm{IndCoh}(\mathrm{Par}_{G})$ it
should correspond to the natural functor
$\mathrm{IndCoh}\overset{\Psi}{\to}\mathrm{QCoh}$ right adjoint to $\Xi$. Now,
it is easy to see that $\Psi$ defines an equivalence from the “native” copy of
$\mathrm{Coh}$ in IndCoh onto the usual copy of $\mathrm{Coh}$ inside
$\mathrm{QCoh}$.111111This follows, for instance, from the observation that
the sequence of functors
$\mathrm{QCoh}\overset{\Xi}{\to}\mathrm{IndCoh}\overset{\Psi}{\to}\mathrm{QCoh}$
is obtained by ind-completing the tautological sequence of functors
$\mathrm{Perf}\overset{}{\to}\mathrm{Coh}\overset{}{\to}\mathrm{QCoh}$. We are
thus led to the following conjecture.

######  Conjecture 1.7.3.

The functor $c_{\psi}$ restricts to an equivalence of categories

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

In other words, $c_{\psi}$ should realize the categorical local Langlands
equivalence. Note that this is a precise and unconditional conjecture!
However, very little is obvious here. For instance, implicit in this
conjecture is the expectation that $c_{\psi}$ carries compact objects in
$D(\mathrm{Bun}_{G})$ to coherent complexes, which is already far from
obvious.

######  Exercise 1.7.4.

Prove that $c_{\psi}$ is compatible with the central grading.

In this setting, we can also formulate the duality compatibility
unconditionally.

######  Conjecture 1.7.5.

There is a natural equivalence of functors

$\mathbf{D}_{\mathrm{twGS}}\circ c_{\psi}\simeq
c_{\psi^{-1}}\circ\mathbf{D}_{\mathrm{BZ}}$

from $D(\mathrm{Bun}_{G})^{\omega}$ towards $\mathrm{Coh}(\mathrm{Par}_{G})$.

Here as before, $c_{\psi^{-1}}$ denotes the enhanced coefficient functor
associated with the dual Whittaker datum $(B,\psi^{-1})$. Composing with
$i_{1!}$ and using
$i_{1!}\mathbf{D}_{\mathrm{coh}}\simeq\mathbf{D}_{\mathrm{BZ}}i_{1!}$, we see
that Hellmann’s functor $R_{\psi}=c_{\psi}\circ i_{1!}$ should satisfy
$\mathbf{D}_{\mathrm{twGS}}\circ R_{\psi}\simeq
R_{\psi^{-1}}\circ\mathbf{D}_{\mathrm{coh}}$.

As some evidence for this, we have the following result. For the notion of a
reasonable group, see Definition A.1.1.

######  Proposition 1.7.6.

If $G$ is reasonable, there is a natural equivalence of functors

$\mathbf{D}_{\mathrm{BZ}}\circ a_{\psi}\simeq
a_{\psi^{-1}}\circ\mathbf{D}_{\mathrm{twGS}}$

from $\mathrm{Perf^{qc}}(\mathrm{Par}_{G})$ towards
$D(\mathrm{Bun}_{G})^{\omega}$.

###### Sketch.

This follows by combining Theorem A.0.1 and Proposition A.1.3.i with the
following result: For any $\mathcal{F}\in\mathrm{Perf}(\mathrm{Par}_{G})$ and
$A\in D(\mathrm{Bun}_{G})^{\omega}$, one has
$\mathbf{D}_{\mathrm{BZ}}(\mathcal{F}\ast
A)\simeq\mathbf{D}_{\mathrm{twGS}}(\mathcal{F})\ast\mathbf{D}_{\mathrm{BZ}}(A)$.
This can be deduced from [FS21, Theorem VIII.5.1 and Theorem IX.2.2]. ∎

######  Exercise 1.7.7.

(Difficult.) Assuming that $G$ is reasonable and $\ell$ is a very good prime
in the sense of [FS21], formulate and prove the appropriate variant of
Proposition 1.7.6 with $\Lambda=\mathbf{\overline{\mathbf{Z}_{\ell}}}$ or with
$\Lambda=\overline{\mathbf{F}_{\ell}}$. (Note that this can be done _without_
imposing some additional condition of being “$\ell$-integrally reasonable”.)

####  1.7.1 A variant with restricted variation

Recall that we defined the category of finite sheaves
$D(\mathrm{Bun}_{G})_{\mathrm{fin}}$.

######  Conjecture 1.7.8.

The functor $c_{\psi}$ restricts to an equivalence from
$D(\mathrm{Bun}_{G})_{\mathrm{fin}}$ towards the full subcategory
$\mathrm{Coh}(\mathrm{Par}_{G})_{\mathrm{fin}}\subset\mathrm{Coh}(\mathrm{Par}_{G})$
spanned by objects which are supported set-theoretically on finitely many
fibers of the map $q:\mathrm{Par}_{G}\to X_{G}^{\mathrm{spec}}$.

This is reminiscent of (and motivated by) the recent “restricted variation”
variant of classical geometric Langlands. The compatibility of $c_{\psi}$ with
the action of the spectral Bernstein center easily implies that for any $A\in
D(\mathrm{Bun}_{G})_{\mathrm{fin}}$, the quasicoherent complex $c_{\psi}(A)$
does in fact satisfy the desired support condition. One can be more precise:
for any finite sheaf $A$, the ring $\mathrm{End}(A)$ is Artinian, so the
natural map $\mathcal{O}(X_{G}^{\mathrm{spec}})\to\mathrm{End}(A)$ factors
over an Artinian quotient. Writing $Z_{A}\subset X_{G}^{\mathrm{spec}}$ for
the associated finite-length subscheme, the sheaf $c_{\psi}(A)$ then has
support contained in $q^{-1}(Z_{A})$. With a little more thought, it’s not
difficult to see that Conjecture 1.7.3 implies Conjecture 1.7.8. In fact, the
converse implication also holds, in the following form.

######  Proposition 1.7.9.

Suppose that the functor $c_{\psi}$ preserves compact objects, and that
Conjecture 1.7.8 is true. Then Conjecture 1.7.3 is true.

###### Proof.

This follows by a straightforward adaptation of the arguments in [AGK+23,
Section 21.4]. ∎

However, the coherence property of $c_{\psi}$ applied to any finite sheaf does
not seem easy. Here is some weak evidence in its favor.

######  Proposition 1.7.10.

Suppose $A\in D(\mathrm{Bun}_{G})_{\mathrm{fin}}$. Then for any
$\mathcal{F}\in\mathrm{Perf}(\mathrm{Par}_{G})$,
$R\Gamma(\mathrm{Par}_{G},\mathcal{F}\otimes
c_{\psi}(A))\in\mathrm{Perf}(\overline{\mathbf{Q}_{\ell}})$.

###### Sketch..

Reduce to the case where $\mathcal{F}$ is a vector bundle pulled back from
$V\in\mathrm{Rep}(B\hat{G})$, so then

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

by direct examination of the definitions. Then $A$ is finite and Hecke
operators preserve finite sheaves, so $i_{1}^{\ast}T_{V}A$ has only finitely
many nonzero cohomologies, each of finite length, and hence is supported on a
finite union of Bernstein components. If $e$ is the idempotent projector onto
this finite union of Bernstein components, then $eW_{\psi}$ is compact by a
result of Bushnell-Henniart cited in the discussion before Theorem A.0.1. Then

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

is perfect, as desired. ∎

Finally, we note that it can be useful to localize even further and study
Conjecture 1.7.8 “one semisimple parameter at a time”. More precisely, for any
fixed semisimple parameter $\phi$, define
$\mathrm{Coh}(\mathrm{Par}_{G})_{\phi}$ as the full subcategory of
$\mathrm{Coh}(\mathrm{Par}_{G})$ spanned by objects which are supported set-
theoretically on the fiber $q^{-1}(x_{\phi})$. Clearly we have a direct sum
decomposition of categories

$\mathrm{Coh}(\mathrm{Par}_{G})_{\mathrm{fin}}=\bigoplus_{\phi}\mathrm{Coh}(\mathrm{Par}_{G})_{\phi}.$

Then Conjecture 1.7.8 is obviously equivalent to the expectation that
$c_{\psi}$ induces an equivalence
$D(\mathrm{Bun}_{G})_{\mathrm{fin},\phi}\overset{\sim}{\to}\mathrm{Coh}(\mathrm{Par}_{G})_{\phi}$
for every $\phi$.

######  Exercise 1.7.11.

Show that if $G$ is reasonable (Definition A.1.1), the functor $a_{\psi}$
induces functors

$\mathrm{Perf}(\mathrm{Par}_{G})_{\mathrm{fin}}\to
D(\mathrm{Bun}_{G})_{\mathrm{fin}}$

and

$\mathrm{Perf}(\mathrm{Par}_{G})_{\phi}\to
D(\mathrm{Bun}_{G})_{\mathrm{fin},\phi}$

for every semisimple parameter $\phi$.

##  2 Compatibility with classical local Langlands

We continue to fix $\Lambda=\overline{\mathbf{Q}_{\ell}}$ as our coefficient
ring, and drop it from the notation. In what sense does the categorical local
Langlands conjecture encode classical local Langlands? The answer to this
question turns out to be surprisingly complicated. Let us begin with a naive
guess for what might be true, closely following [FS21, Remark I.10.3].

Best Hope. The categorical equivalence is t-exact with respect to the perverse
t-structure on $D(\mathrm{Bun}_{G})$ and some exotic perverse coherent
t-structure on $\mathrm{Coh}(\mathrm{Par}_{G})$, and thus induces a bijection
between irreducible objects in the hearts. On the $\mathrm{Bun}_{G}$ side, the
irreducibles are indexed by pairs $(b,\pi)$ where $b\in B(G)$ and
$\pi\in\mathrm{Irr}(G_{b}(E))$. On the $\mathrm{Par}_{G}$ side, the
irreducibles are indexed by pairs $(\phi,\rho)$ where $\phi$ is a Frobenius-
semisimple $L$-parameter and $\rho$ is an irreducible algebraic representation
of the centralizer $S_{\phi}$.

Unfortunately, this “best hope” seems slightly too simple, for several related
reasons:

  1. 1.

$\mathrm{Par}_{G}$ includes many points corresponding to $L$-parameters which
are _not_ Frobenius-semisimple, which play no role in the classical local
Langlands correspondence.

  2. 2.

The perverse t-structure on $D(\mathrm{Bun}_{G})$ is _not_ self-dual with
respect to $\mathbf{D}_{\mathrm{BZ}}$, which however is the natural duality
appearing in the categorical conjecture.

  3. 3.

The geometry of $\mathrm{Par}_{G}$ seems to preclude any simple direct
definition of the hoped-for t-structure, especially around _L_ -parameters
with nontrivial monodromy, since perverse coherence is not (naively) a
reasonable notion around these parameters.

One of our main points is that neverthless, the “best hope” should be true for
most $L$-parameters, in a sense we will make precise. The starting point for
this is a remarkable recent result of Bertoloni Meli-Oi [BMO23], which says
that the bijection on irreducibles predicted by the “best hope” is actually
true! We will then turn things around and use their results as a guide to
formulate some precise guesses for how the categorical LLC interacts with the
classical LLC.

Here is a brief and impressionistic outline of our current understanding. We
will spend most of this section developing this outline into a precise
collection of conjectures, and giving evidence for them.

  * •

After localizing around a _generous_ _L_ -parameter (Definition 2.1.5), the
perverse and hadal t-structures on (appropriately decorated versions of)
$D(\mathrm{Bun}_{G})$ should coincide, and they should match with the standard
t-structure on $\mathrm{Coh}(\mathrm{Par}_{G})$. Moreover, sheaves on
different strata of $\mathrm{Bun}_{G}$ will not interact with each other, and
Hecke operators should be t-exact.

  * •

After localizing around a _semisimple generic_ _L-_ parameter (Definition
2.3.1), the perverse t-structure on $D(\mathrm{Bun}_{G})$ should match the
standard t-structure on $\mathrm{Coh}(\mathrm{Par}_{G})$, and the hadal
t-structure on $D(\mathrm{Bun}_{G})$ should match some perverse coherent
t-structure on $\mathrm{Coh}(\mathrm{Par}_{G})$. Sheaves on different strata
of $\mathrm{Bun}_{G}$ will interact with each other in highly nontrivial ways,
but Hecke operators should still be t-exact for both t-structures.

Beyond the semisimple generic case, we have the following very natural general
question: The hadal t-structure is defined on the whole category
$D(\mathrm{Bun}_{G})^{\omega}$, so Conjecture 1.7.3 implies that it _must_
correspond to _some_ t-structure on $\mathrm{Coh}(\mathrm{Par}_{G})$. Can we
describe this t-structure intrinsically?

######  Remark 2.0.1.

It is of significant interest to extend these ideas to integral and torsion
coefficients. Some precise conjectures in these settings have been proposed by
Hamann and Hamann-Lee ([Ham23, Section 3] and [HL23, Section 6.2]). We note in
particular that our conditions of being “generous” and “semisimple generic”
match with the conditions of being “Langlands-Shahidi type” and “weakly
Langlands-Shahidi type”, respectively, as formulated in [HL23, Definition
6.2]. However, we emphasize that [HL23, Definition 6.2] also makes sense with
$\overline{\mathbf{F}_{\ell}}$-coefficients, which falls outside the scope of
our discussion. I would also like to point out that I landed on the notion of
generous parameters while trying to understand [Ham22]. Finally, we also refer
to [Ham23, HL23] for some very striking global applications of this
philosophy.

###  2.1 Generous _L_ -parameters

Let us begin by recalling the $B(G)_{\mathrm{bas}}$ form of the local
Langlands correspondence. Fix $G$ quasisplit and pinned as usual, and fix a
Whittaker datum $\psi$. For any $\phi$, we set
$S_{\phi}^{\natural}=S_{\phi}/(S_{\phi}\cap\hat{G}^{\mathrm{der}})^{\circ}$.
This is a disconnected reductive group whose identity component is a torus,
and there is a natural map of algebraic groups $Z(\hat{G})^{\Gamma}\to
S_{\phi}^{\natural}$ which turns out to be an isogeny.

The following form of the local Langlands conjecture was formulated by
Kottwitz in unpublished work. To the best of my knowledge it first appeared in
print in [Rap95] (see the discussion preceding Conjecture 5.1). We also refer
to [Kal16, Conjecture F] for a modern and slightly more precise formulation.

######  Conjecture 2.1.1 ($B(G)_{\mathrm{bas}}$ local Langlands
correspondence).

For each $b\in B(G)_{\mathrm{bas}}$, there is a natural finite-to-one map
$\Pi(G_{b})\to\Phi(G)$. Writing $\Pi_{\phi}(G_{b})$ for the fiber over $\phi$,
there is a natural bijection

$\iota_{\psi}:\coprod_{b\in
B(G)_{\mathrm{bas}}}\Pi_{\phi}(G_{b})\overset{\sim}{\to}\mathrm{Irr}(S_{\phi}^{\natural})$

depending only on the choice of Whittaker datum, such that the diagram

$\textstyle{\underset{b\in
B(G)_{\mathrm{bas}}}{\coprod}\Pi_{\phi}(G_{b})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\,\;\;\;\;\;\;\;\iota_{\psi}}$$\textstyle{\mathrm{Irr}(S_{\phi}^{\natural})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B(G)_{\mathrm{bas}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{G}}$$\textstyle{X^{\ast}(Z(\hat{G})^{\Gamma})}$

commutes. Here the left vertical arrow is the obvious projection, the right
vertical map is induced by restriction along $Z(\hat{G})^{\Gamma}\to
S_{\phi}^{\natural}$, and $\kappa_{G}$ is the Kottwitz isomorphism.

Of course, the word “natural” is doing a lot of heavy lifting here. However,
this conjecture is now known unconditionally for many groups.

######  Theorem 2.1.2 (Bertoloni Meli-Oi).

Suppose that the $B(G)_{\mathrm{bas}}$ LLC holds for $G$ and all its standard
Levi subgroups. Then for each $b\in B(G)$, there is a natural finite-to-one
map $\Pi(G_{b})\to\Phi(G)$. Writing $\Pi_{\phi}(G_{b})$ for the fiber over
$\phi$, there is a natural bijection

$\iota_{\psi}:\coprod_{b\in
B(G)_{\mathrm{}}}\Pi_{\phi}(G_{b})\overset{\sim}{\to}\mathrm{Irr}(S_{\phi})$

depending only on the choice of Whittaker datum, such that the diagram

$\textstyle{\underset{b\in
B(G)}{\coprod}\Pi_{\phi}(G_{b})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\;\;\;\;\;\;\iota_{\psi}}$$\textstyle{\mathrm{Irr}(S_{\phi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{G}}$$\textstyle{X^{\ast}(Z(\hat{G})^{\Gamma})}$

commutes. The left vertical map is the obvious projection and the right
vertical map is induced by restriction along $Z(\hat{G})^{\Gamma}\to
S_{\phi}$.

We like to think of this as follows: there is a natural bijection between
pairs $(\phi,\rho)$ where $\phi$ is a Frobenius-semisimple $L$-parameter and
$\rho\in\mathrm{Irr}(S_{\phi})$ is an irreducible algebraic representation,
and pairs $(b,\pi)$ where $b\in B(G)$ and $\pi$ is an irreducible smooth
representation of $G_{b}(E)$. We will sometimes call this the BM-O bijection.
We refer the reader to [BMO23] for the details behind this beautiful
construction.

Warning. For non-basic $b$, the set $\Pi_{\phi}(G_{b})$ appearing in Theorem
2.1.2 is usually not an _L_ -packet. Rather, it is the (finite) union of the
_L_ -packets attached to the parameters
$\phi^{\prime}:W_{E}\times\mathrm{SL}_{2}\to\,^{L}G_{b}$ such that
$i\circ\phi^{\prime}$ is conjugate to $\phi$, for $i:\,^{L}G_{b}\to\,^{L}G$
the evident map of $L$-groups. Said differently, the set $\Pi_{\phi}(G_{b})$
is the fiber of the composite map

$\Pi(G_{b})\to\Phi(G_{b})\to\Phi(G),$

and the second arrow is typically not injective.

######  Remark 2.1.3.

If $\phi$ is discrete, then $S_{\phi}\cong S_{\phi}^{\natural}$, and
$\Pi_{\phi}(G_{b})$ is empty for all non-basic $b$. In this case the $B(G)$
LLC contains the same information as the $B(G)_{\mathrm{bas}}$ LLC. In
general, the BM-O bijection extends the $B(G)_{\mathrm{bas}}$ correspondence
in the evident way, by restricting to
$\rho\in\mathrm{Irr}(S_{\phi}^{\natural})\subset\mathrm{Irr}(S_{\phi})$ on the
right-hand side and $b\in B(G)_{\mathrm{bas}}$ on the left-hand side.

######  Example 2.1.4.

Take $G=\mathrm{GL}_{2}$, and let $\phi=\chi_{1}\oplus\chi_{2}$ be a sum of
two random characters. By local class field theory we can also think of the
$\chi_{i}$’s as smooth characters of $E^{\times}$. Then
$S_{\phi}=\mathbf{G}_{m}^{2}$, so $\mathrm{Irr}(S_{\phi})=\mathbf{Z}^{2}$. How
to attach to an ordered pair $(m,n)\in\mathbf{Z}^{2}$ a pair $(b,\pi)$? It’s
easy to guess what to do for $b$: just take the isocrystal with slopes $m$ and
$n$. Now when $m=n$, $G_{b}=G$ and we can take
$\pi=i_{B}^{G}(\chi_{1}\boxtimes\chi_{2})$. When $m\neq n$, $G_{b}=T$ is the
usual maximal torus in $G$, and we take $\pi=\chi_{1}\boxtimes\chi_{2}$ or
$\pi=\chi_{2}\boxtimes\chi_{1}$ according to whether $m>n$ or $m<n$. Note that
$\Pi_{\phi}(G_{b})$ contains two elements when $b$ is not basic.

######  Definition 2.1.5.

A semisimple $L$-parameter
$\phi:W_{E}\to\,^{L}G(\overline{\mathbf{Q}_{\ell}})$ is _generous_ if
$\mathrm{Par}_{G}\times_{X_{G}^{\mathrm{spec}}}\\{x_{\phi}\\}\cong BS_{\phi}$.

The name is meant to suggest on one hand that such parameters are “generic”,
in two different ways - they form an open dense subset of
$X_{G}^{\mathrm{spec}}$, and they are generic in the technical sense of
Proposition 2.3.1.(2) \- and also that they have nice properties which make
the classical and categorical local Langlands correspondence simpler at these
parameters.

######  Example 2.1.6.

Suppose $G=\mathrm{GL}_{n}$. Then an _L-_ parameter is generous iff it is a
direct sum $\phi\simeq\phi_{1}\oplus\cdots\oplus\phi_{d}$ where the
$\phi_{i}$’s are pairwise-distinct supercuspidal parameters with
$\sum_{i}\mathrm{dim}\phi_{i}=n$, and with $\phi_{i}\not\simeq\phi_{j}(1)$ for
any $i\neq j$. Note that the associated $\mathrm{GL}_{n}(E)$-representation
$\pi$ is generic, and is irreducibly induced from a supercuspidal
representation of a Levi.

######  Proposition 2.1.7.

_1)_ A discrete $L$-parameter is generous iff it is supercuspidal.

_2)_ If $\phi$ is generous, then $S_{\phi}^{\circ}$ is a torus.

_3)_ If $\phi$ is generous, then $q$ is flat in a neighborhood of
$q^{-1}(x_{\phi})$, and there is a natural _regular_ closed immersion

$i_{\phi}:BS_{\phi}=\mathrm{Par}_{G}\times_{X_{G}^{\mathrm{spec}}}\\{x_{\phi}\\}\hookrightarrow\mathrm{Par}_{G},$

and $\mathrm{Par}_{G}$ is smooth in a neighborhood of $\mathrm{im}i_{\phi}$.

In particular, if $\phi$ is generous, the pushforward $i_{\phi\ast}$ sends
irreducible representations of $S_{\phi}$ towards perfect complexes on
$\mathrm{Par}_{G}$. We can now state the main conjecture relating classical
and categorical LLC at generous parameters.

######  Conjecture 2.1.8.

Suppose that $(\phi,\rho)$ and $(b,\pi)$ match under the BM-O bijection, where
$\phi$ is a generous L-parameter.

_i._ There is an isomorphism

$a_{\psi}(i_{\phi\ast}\rho)\overset{\mathrm{def}}{=}i_{\phi\ast}\rho\ast
i_{1!}W_{\psi}\simeq i_{b!}^{\mathrm{ren}}\pi$

in $D(\mathrm{Bun}_{G})$.

_ii._ The natural maps

$i_{b\sharp}^{\mathrm{ren}}\pi\overset{\sim}{\to}i_{b!}^{\mathrm{ren}}\pi\overset{\sim}{\to}i_{b\ast}^{\mathrm{ren}}\pi$

are isomorphisms.

This immediately suggests another conjecture, which in some cases can be
verified more easily.

######  Conjecture 2.1.9.

Suppose that $\phi$ is a generous L-parameter. Then

$\mathscr{F}_{\phi}=\underset{b\in
B(G),\pi\in\Pi_{\phi}(G_{b})}{\bigoplus}i_{b!}^{\mathrm{ren}}\pi^{\oplus\mathrm{dim}\iota_{\psi}(b,\pi)}$

is a Hecke eigensheaf with eigenvalue $\phi$.

This follows immediately from Conjecture 2.1.8, since that conjecture formally
implies an isomorphism

$\mathscr{F}_{\phi}\simeq i_{\phi\ast}\mathcal{O}(S_{\phi})\ast
i_{1!}W_{\psi}$

where $\mathcal{O}(S_{\phi})$ is the regular representation, and the right-
hand side is a Hecke eigensheaf of the stated type for completely formal
reasons. Note that the second part of Conjecture 2.1.8 implies additionally
that $\mathscr{F}_{\phi}$ is perverse.

######  Remark 2.1.10.

We emphasize that Conjecture 2.1.8 completely describes how to compute Hecke
operators on the atomic sheaves attached to generous parameters. More
precisely, suppose $\phi$ is generous and $(b,\pi)$ matches $(\phi,\rho)$
under the BM-O bijection. Let $V\in\mathrm{Rep}(^{L}G)$ be any representation,
with associated Hecke operator $T_{V}$. To compute
$T_{V}i_{b!}^{\mathrm{ren}}\pi$, simply decompose
$V|_{S_{\phi}}\otimes\rho\simeq\oplus_{j}\rho_{j}^{\oplus m_{j}}$ as a sum of
irreducible $S_{\phi}$-representations (with multiplicity). Let
$(b_{j},\pi_{j})$ be the pair matching $(\phi,\rho_{j})$ under the BM-O
bijection. Then Conjecture 2.1.8 predicts that

$T_{V}i_{b!}^{\mathrm{ren}}\pi\simeq\oplus_{j}i_{b_{j}!}^{\mathrm{ren}}\pi_{j}^{\oplus
m_{j}}.$

With slightly more effort, one can also describe the Galois action on
$T_{V}i_{b!}^{\mathrm{ren}}$ explicitly. The final outcome is the formula

$T_{V}i_{b!}^{\mathrm{ren}}\pi\simeq\bigoplus_{(b^{\prime},\pi^{\prime})}i_{b^{\prime}!}^{\mathrm{ren}}\pi^{\prime}\boxtimes\mathrm{Hom}_{S_{\phi}}(\iota_{\psi}(b,\pi)^{\vee}\otimes\iota_{\psi}(b^{\prime},\pi^{\prime}),V\circ\phi)$

in $D(\mathrm{Bun}_{G})^{BW_{E}}$, where the $W_{E}$-action comes via the
second factor in the evident sense. When $\phi$ is supercuspidal, this exactly
recovers the Kottwitz conjecture.

######  Remark 2.1.11.

If we know the first part of Conjecture 2.1.8 for $\phi$ and $\phi^{\vee}$,
then in fact the second part follows. More precisely, suppose that
$(\phi,\rho)$ and $(b,\pi)$ match under the BM-O bijection for $\psi$. Then we
expect that $(\phi^{\vee},\rho^{\vee})$ and $(b,\pi^{\vee})$ match under the
BM-O bijection for $\psi^{-1}$ [Kal13], so the first part of Conjecture 2.1.8
implies that

$i_{\phi^{\vee}\ast}\rho^{\vee}\ast i_{1!}W_{\psi^{-1}}\simeq
i_{b!}^{\mathrm{ren}}\pi^{\vee}.$

On the other hand, it is easy to see that
$i_{\phi^{\vee}\ast}\rho^{\vee}\simeq\mathbf{D}_{\mathrm{twGS}}(i_{\phi\ast}\rho)[\dim
S_{\phi}]$, so using the compatibility of the spectral action with duality
proved in Proposition 1.7.6, we compute that

$\displaystyle i_{\phi^{\vee}\ast}\rho^{\vee}\ast i_{1!}W_{\psi^{-1}}$
$\displaystyle\simeq\mathbf{D}_{\mathrm{BZ}}\left(i_{\phi\ast}\rho\ast
i_{1!}W_{\psi}\right)[\dim S_{\phi}]$
$\displaystyle\simeq\mathbf{D}_{\mathrm{BZ}}\left(i_{b!}^{\mathrm{ren}}\pi\right)[\dim
S_{\phi}]$ $\displaystyle\simeq
i_{b\sharp}^{\mathrm{ren}}\mathbf{D}_{\mathrm{coh}}(\pi)[\dim S_{\phi}]$

where we used Proposition 1.1.4 in the last line. Equating these two
calculations, we get an isomorphism $i_{b!}^{\mathrm{ren}}\pi^{\vee}\simeq
i_{b\sharp}^{\mathrm{ren}}\mathbf{D}_{\mathrm{coh}}(\pi)[\dim S_{\phi}]$. From
this we immediately see (by taking the stalk at $b$) that
$\pi^{\vee}\simeq\mathbf{D}_{\mathrm{coh}}(\pi)[\dim
S_{\phi}]\simeq\mathrm{Zel}(\pi)$, and then that
$i_{b!}^{\mathrm{ren}}\pi^{\vee}\simeq i_{b\sharp}^{\mathrm{ren}}\pi^{\vee}$.
Applying $\mathbf{D}_{\mathrm{BZ}}$ to this last isomorphism and using
Proposition 1.1.4 again, we get the same isomorphism for $\pi$. This implies
the first isomorphism in the second part of the conjecture.

To get the second isomorphism, we argue more generally as follows. Fix a
semisimple $L$-parameter $\phi$, and suppose that
$i_{b\sharp}^{\mathrm{ren}}\pi\overset{\sim}{\to}i_{b!}^{\mathrm{ren}}\pi$ is
an isomorphism for all $b$ and all $\pi\in\Pi(G_{b})$ with Fargues-Scholze
parameter $\phi$. It then follows that
$i_{b!}^{\mathrm{ren}}\pi\overset{\sim}{\to}i_{b\ast}^{\mathrm{ren}}\pi$ is an
isomorphism for all $b$ and all $\pi\in\Pi(G_{b})$ with Fargues-Scholze
parameter $\phi$. Indeed, it’s clearly enough to prove that
$i_{b^{\prime}}^{\ast\mathrm{ren}}i_{b\ast}^{\mathrm{ren}}\pi=0$ for all
$b^{\prime}\neq b$. Using that
$i_{b^{\prime}}^{\ast\mathrm{ren}}i_{b\ast}^{\mathrm{ren}}\pi$ is admissible
and left-bounded, with all of its irreducible constituents having Fargues-
Scholze parameter $\phi$, we easily reduce further to proving that
$R\mathrm{Hom}(\tau,i_{b^{\prime}}^{\ast\mathrm{ren}}i_{b\ast}^{\mathrm{ren}}\pi)=0$
for all $b^{\prime}\neq b$ and all $\tau\in\Pi(G_{b^{\prime}})$ with Fargues-
Scholze parameter $\phi$. But now we compute that

$\displaystyle
R\mathrm{Hom}_{G_{b^{\prime}}}(\tau,i_{b^{\prime}}^{\ast\mathrm{ren}}i_{b\ast}^{\mathrm{ren}}\pi)$
$\displaystyle\cong
R\mathrm{Hom}(i_{b^{\prime}\sharp}^{\mathrm{ren}}\tau,i_{b\ast}^{\mathrm{ren}}\pi)$
$\displaystyle\cong
R\mathrm{Hom}(i_{b^{\prime}!}^{\mathrm{ren}}\tau,i_{b\ast}^{\mathrm{ren}}\pi)$
$\displaystyle\cong
R\mathrm{Hom}_{G_{b}}(i_{b}^{\ast\mathrm{ren}}i_{b^{\prime}!}^{\mathrm{ren}}\tau,\pi)$
$\displaystyle=0$

where the first and third isomorphisms come from the obvious adjunctions, the
second isomorphism follows from our assumptions, and the last line follows
from the trivial vanishing
$i_{b}^{\ast\mathrm{ren}}i_{b^{\prime}!}^{\mathrm{ren}}\tau=0$ for
$b^{\prime}\neq b$.

######  Remark 2.1.12.

The logic in the last paragraph of the previous remark can also be reversed,
in the evident sense.

####  2.1.1 Example: Supercuspidal parameters

Here we exactly recover Fargues’s original eigensheaf conjecture. More
precisely, for supercuspidal $\phi$, Conjecture 2.1.9 was already formulated
by Fargues in 2014 [Far16], and Conjecture 2.1.8 is essentially stated in
[FS21]. Here some partial results are known. More precisely, Conjecture 2.1.8
is known for $\mathrm{GL}_{n}$ [ALB21, Han23a] and unramified
$\mathrm{U}_{2n+1}/\mathbf{Q}_{p}$ [BMHN22]. Additionally, Conjecture 2.1.9 is
known for $\mathrm{GSp}_{4}$ [Ham21] and $\mathrm{SO}_{2n+1}$ (H.,
unpublished) when $E/\mathbf{Q}_{p}$ is unramified with $p>2$.

####  2.1.2 Example: Generic toral parameters for $\mathrm{GL}_{n}$

We now take $G=\mathrm{GL}_{n}$. Let $\phi=\chi_{1}\oplus\cdots\oplus\chi_{n}$
be a direct sum of characters, which we also identify with characters
$\chi_{i}:E^{\times}\to\overline{\mathbf{Q}_{\ell}}^{\times}$ through the
usual reciprocity map. We assume $\phi$ is generous, so in particular
$S_{\phi}\cong\mathbf{G}_{m}^{n}$ and
$\mathrm{Irr}(S_{\phi})\cong\mathbf{Z}^{n}$. Our goal here is to sketch the
following result.

######  Theorem 2.1.13.

Notation as above, suppose also that $\phi$ is $\ell$-integral and generous
mod-$\ell$. Then Conjecture 2.1.8 is true for $\phi$.

The argument relies critically on deep work of Hamann [Ham22]. The extra
assumptions related to $\ell$-integrality are needed in Hamann’s work, and
will be irrelevant once the sheaf-theoretic machinery improves.

###### Proof.

Under the obvious bijection
$\mathbf{j}=(j_{1},\dots,j_{n})\in\mathbf{Z}^{n}\leftrightarrow\rho_{\mathbf{j}}\in\mathrm{Irr}(S_{\phi})$,
set $|\mathbf{j}|=\sum_{i}|j_{i}|$, and let
$(b_{\mathbf{j}},\pi_{\mathbf{j}}$) be the pair associated with
$(\phi,\rho_{\mathbf{j}})$ under the BM-O bijection. We will prove the
isomorphism $i_{\phi\ast}\rho_{\mathbf{j}}\ast i_{1!}W_{\psi}\simeq
i_{b_{\mathbf{j}}!}^{\mathrm{ren}}\pi_{\mathbf{j}}$ by induction on
$|\mathbf{j}|$.

Base case $|\mathbf{j}|=0$. This amounts to the assertion that

$\sigma:=W_{\psi}\otimes_{\mathfrak{Z}(G)}\mathfrak{Z}(G)/\mathfrak{m}_{\phi}\simeq
i_{B}^{G}(\chi_{1}\boxtimes\cdots\boxtimes\chi_{n}),$

which can be proved directly. Sketch: Using Theorem A.0.1 and the fact that
$X_{G}\cong X_{G}^{\mathrm{spec}}$ for $\mathrm{GL}_{n}$ [FS], together with
some standard structure theory a la Bernstein, it is easy to see that $\sigma$
is a finite-length admissible representation with $\sigma^{\mathrm{ss}}\simeq
i_{B}^{G}(\chi_{1}\boxtimes\cdots\boxtimes\chi_{n})^{\oplus m}$ for some
$m\geq 1$. To show that $m\leq 1$, it is enough to see that the Jacquet module
$j_{G}^{B}\sigma$ has length $\leq n!$. This can be done by noting that
$j_{G}^{B}W_{\psi}\simeq\mathcal{C}_{c}^{\infty}(T,\overline{\mathbf{Q}_{\ell}}$),
and then using the general fact that

$j_{G}^{P}(-\otimes_{\mathfrak{Z}(G)}\mathfrak{Z}(G)/I)\simeq
j_{G}^{P}(-)\otimes_{\mathfrak{Z}(M)}\mathfrak{Z}(M)/f(I)\mathfrak{Z}(M)$

as functors, where $f:\mathfrak{Z}(G)\to\mathfrak{Z}(M)$ is the usual map
induced by parabolic induction. In the case at hand, one concludes by noting
that
$\mathfrak{Z}(T)/f(\mathfrak{m}_{\phi})\mathfrak{Z}(T)\simeq\oplus_{\sigma\in
S_{n}}\mathfrak{Z}(T)/\mathfrak{m}_{\chi_{\sigma(1)}\boxtimes\cdots\boxtimes\chi_{\sigma(n)}}$.

Induction step. Suppose given $\mathbf{j}$. We can choose some
$\mathbf{j}^{\prime}$ with $|\mathbf{j}^{\prime}|=|\mathbf{j}|-1$ and
$V\in\\{\mathrm{std},\mathrm{std}^{\vee}\\}$ such that $\rho_{\mathbf{j}}\in
V|_{S_{\phi}}\otimes\rho_{\mathbf{j^{\prime}}}$. Note that
$V|_{S_{\phi}}\otimes\rho_{\mathbf{j^{\prime}}}$ is multiplicity free, so we
can decompose it as a sum
$V|_{S_{\phi}}\otimes\rho_{\mathbf{j^{\prime}}}\simeq\rho_{\mathbf{j}_{1}}\oplus\cdots\oplus\rho_{\mathbf{j}_{n}}$
where $\mathbf{j}^{\prime}$ and $\mathbf{j}_{i}$ have the same component
except in the $i$th spot. We will now compute
$T_{V}i_{b_{\mathbf{j}^{\prime}}!}^{\mathrm{ren}}\pi_{\mathbf{j}^{\prime}}\in
D(\mathrm{Bun}_{G})^{BW_{E}}$ in two different ways, and then use the Weil
group action to break apart the results of the calculation.

_First method._ By the induction hypothesis we have
$i_{b_{\mathbf{j^{\prime}}}!}^{\mathrm{ren}}\pi_{\mathbf{j}^{\prime}}\simeq
i_{\phi\ast}\rho_{\mathbf{j}^{\prime}}\ast i_{1!}W_{\psi}$, so we compute that

$\displaystyle
T_{V}i_{b_{\mathbf{j^{\prime}}}!}^{\mathrm{ren}}\pi_{\mathbf{j}^{\prime}}$
$\displaystyle\simeq T_{V}(i_{\phi\ast}\rho_{\mathbf{j}^{\prime}}\ast
i_{1!}W_{\psi})$ $\displaystyle\simeq(V\otimes
i_{\phi\ast}\rho_{\mathbf{j}^{\prime}})\ast i_{1!}W_{\psi}$
$\displaystyle\simeq
i_{\phi\ast}(V|_{S_{\phi}}\otimes\rho_{\mathbf{j}^{\prime}})\ast
i_{1!}W_{\psi}$ $\displaystyle\simeq\oplus_{1\leq k\leqslant
n}(i_{\phi\ast}\rho_{\mathbf{j}_{k}}\ast i_{1!}W_{\psi})\boxtimes\chi_{k}^{\pm
1}$

where the $\pm$ sign is chosen according to whether $V=\mathrm{std}$ or
$V=\mathrm{std}^{\vee}$. Note that the Weil group action in the fourth line
comes from the tautological Weil group action on $V|_{S_{\phi}}$.

_Second method._ Hamann’s results show that
$i_{b_{\mathbf{j^{\prime}}}!}^{\mathrm{ren}}\pi_{\mathbf{j}^{\prime}}\simeq\mathrm{Eis}_{B}(i_{b_{\mathbf{j}^{\prime}}!}^{T}\chi)$
for all $\mathbf{j}^{\prime}$ [Ham22, Theorem 9.1]. Now using the filtered
commutation of $\mathrm{Eis}$ with Hecke operators, together with the
genericity, we compute that

$\displaystyle T_{V}\mathrm{Eis}_{B}(i_{b_{\mathbf{j}^{\prime}}!}^{T}\chi)$
$\displaystyle\simeq\mathrm{Eis}_{B}(T_{V|\hat{T}}i_{b_{\mathbf{j}^{\prime}}!}^{T}\chi)$
$\displaystyle\simeq\mathrm{Eis}_{B}(\oplus_{1\leq k\leq
n}i_{b_{\mathbf{j}_{k}}!}^{T}\chi\boxtimes\chi_{k}^{\pm 1})$
$\displaystyle\simeq\oplus_{1\leq k\leq
n}\mathrm{Eis}_{B}(i_{b_{\mathbf{j}_{k}}!}^{T}\chi)\boxtimes\chi_{k}^{\pm 1}$
$\displaystyle\simeq\oplus_{1\leq k\leq
n}i_{b_{\mathbf{j}_{k}}!}^{\mathrm{ren}}\pi_{\mathbf{j}_{k}}\boxtimes\chi_{k}^{\pm
1}$

with the same sign convention, where we have used Hamann’s results again.

Equating the outcomes of these two calculations, and using the fact that
$\chi_{1},\dots,\chi_{k}$ are distinct as characters of $W_{E}$, we get the
desired result. ∎

Next, we explain how the finiteness conditions and t-structures should behave
around generous parameters.

######  Proposition 2.1.14.

Let $\phi$ be a generous L-parameter, and assume Conjecture 2.1.8 for $\phi$.
Assume also that the BM-O bijection for $\phi$ exhausts all pairs $(b,\pi)$
such that $i_{b!}^{\mathrm{ren}}\pi$ has Fargues-Scholze parameter $\phi$.

Then the functors $i_{b!}^{\mathrm{ren}}$ and $i_{b}^{\ast\mathrm{ren}}$
induce a canonical direct product decomposition

$D(\mathrm{Bun}_{G})_{\phi}^{\mathrm{ULA}}\cong\prod_{b}D(G_{b}(E),\overline{\mathbf{Q}_{\ell}})_{\phi}^{\mathrm{ULA}}$

identifying the perverse t-structure on the left-hand side with the product of
the standard t-structures on the right-hand side, and the Hecke action of
$\mathrm{Rep}(\hat{G})$ on $D(\mathrm{Bun}_{G})_{\phi}^{\mathrm{ULA}}$ is
perverse t-exact.

Moreover, the perverse and hadal t-structures on
$D(\mathrm{Bun}_{G})_{\mathrm{fin}}$ coincide, and the same functors as above
induce a canonical direct sum decomposition

$D(\mathrm{Bun}_{G})_{\mathrm{fin},\phi}\cong\bigoplus_{b}D(G_{b}(E),\overline{\mathbf{Q}_{\ell}})_{\mathrm{fin},\phi}$

compatible with the decomposition above and identifying the hadal (=perverse)
t-structure on the left-hand side with the sum of the standard t-structures on
the right-hand side. Finally, the Hecke action of $\mathrm{Rep}(\hat{G})$ on
$D(\mathrm{Bun}_{G})_{\mathrm{fin},\phi}$ is hadal t-exact.

###### Sketch.

Using the second half of Conjecture 2.1.8 together with the exhaustion
assumption, it is clear that we have isomorphisms of functors
$i_{b!}^{\mathrm{ren}}\simeq i_{b\ast}^{\mathrm{ren}}\simeq
i_{b\sharp}^{\mathrm{ren}}$ and $i_{b}^{\ast\mathrm{ren}}\simeq
i_{b}^{!\mathrm{ren}}$ on the $\phi$-localized ULA/finite categories, which
forces these functors to be t-exact with respect to the perverse/hadal
t-structure on $\mathrm{Bun}_{G}$ and the standard t-structures on the
representation categories. Moreover, all the gluing functors
$i_{b^{\prime}}^{\ast\mathrm{ren}}i_{b\ast}^{\mathrm{ren}}$ between
$\phi$-localized ULA sheaves on different strata vanish identically. This
implies the direct product decomposition and the identifications of
t-structures as stated. For the statements regarding the Hecke action, it is
enough to prove that any sheaf of the form $T_{V}i_{b!}^{\mathrm{ren}}\pi$ is
perverse/hadal. This follows from the discussion in Remark 2.1.10. ∎

######  Exercise 2.1.15.

Fix a generous parameter $\phi$. Assume that the hypotheses of Proposition
2.1.14 hold for $\phi$ and $\phi^{\vee}$, and that the BM-O bijections for
$\phi$ and $\phi^{\vee}$ satisfy the expected compatibility with smooth
duality as in Remark 2.1.11. How do the decompositions of categories in
Proposition 2.1.14 interact with the relevant dualities?

At this point, it is hard not to state the following (unconditional!)
conjecture.

######  Conjecture 2.1.16.

If $\phi$ is a generous parameter, the functor

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

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

Of course, as noted above, we expect that the perverse and hadal t-structures
should coincide on the left-hand side. However, if we said “perverse” instead
of “hadal” in the formulation of this conjecture, it would no longer be
entirely unconditional, because it is not clear a priori that perverse
truncations preserve $D(\mathrm{Bun}_{G})_{\mathrm{fin},\phi}$.

###  2.2 The trivial _L_ -parameter

In this section we assume for simplicity that $G$ is split. Let $\phi$ be the
_trivial_ _L_ -parameter. Clearly $S_{\phi}=\hat{G}$, so
$\mathrm{Irr}(S_{\phi})\cong X^{\ast}(\hat{T})^{+}=X_{\ast}(T)^{+}$ by usual
highest weight theory. Therefore, for any $\lambda\in X_{\ast}(T)^{+}$, the
BM-O bijection defines an associated pair $(b_{\lambda},\pi_{\lambda})$ which
can be described totally explicitly. In fact, $b_{\lambda}$ is just the
element $\lambda(\varpi)\in G(E)$, and $G_{b_{\lambda}}=C_{G}(\lambda)$ is the
standard Levi subgroup of $G$ centralizing $\lambda$. The representation
$\pi_{\lambda}$ turns out to be the (irreducible) normalized parabolic
induction $i_{B}^{G_{b_{\lambda}}}(\mathbf{1})$ of the trivial representation,
where $B\subset G_{b_{\lambda}}$ is any choice of Borel. Our goal here is to
formulate a conjecture describing the coherent sheaves on $\mathrm{Par}_{G}$
associated with the sheaves $i_{b_{\lambda}!}^{\mathrm{ren}}\pi_{\lambda}$ and
$i_{b_{\lambda}\sharp}^{\mathrm{ren}}\pi_{\lambda}$ on $\mathrm{Bun}_{G}$. It
turns out that a precise guess for these sheaves is forced on us by the
expected compatibility of the categorical equivalence with Eisenstein functors
and with duality. However, the situation is much more complicated than for
generous _L-_ parameters.

To begin, observe that the fiber of the map $\mathrm{Par}_{G}\to
X_{G}^{\mathrm{spec}}$ over the trivial _L_ -parameter is exactly the quotient
stack $\mathcal{N}/\hat{G}$, where $\mathcal{N}\subset\hat{\mathfrak{g}}$ is
the nilpotent cone with its usual $\hat{G}$-action. In particular, we have a
canonical closed immersion
$\nu:\mathcal{N}/\hat{G}\hookrightarrow\mathrm{Par}_{G}$. Now, recall the
$\hat{G}$-equivariant diagram of schemes

$\textstyle{\tilde{\mathcal{N}}=T^{\ast}(\hat{G}/\hat{B})\cong\hat{\mathfrak{u}}\times^{\hat{B}}\hat{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{\pi}$$\textstyle{\hat{G}/\hat{B}}$$\textstyle{\mathcal{N}}$

where $\pi:\tilde{\mathcal{N}}\to\mathcal{N}$ is the Springer resolution. For
any $\lambda\in X^{\ast}(\hat{T})$, we have the usual equivariant line bundle