3.4.17]). Conversely, assume that (ii) is satisfied. Then $M$ is discrete
because $M=\varinjlim_{H}R\Gamma_{\mathrm{sm}}(H,M)$, where on the right-hand
side $H$ ranges through all compact open subgroups of $G$ (see [Man22, Lemma
3.4.19]). We claim that $M$ lies in
$\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})$, i.e. that the
natural morphism
$M^{\varphi}\otimes_{\mathbf{F}_{p}}\mathcal{O}^{a}_{C}\mathbin{\xrightarrow{\sim}}M$
is an isomorphism. By the colimit property we just used the collection of
functors $R\Gamma_{\mathrm{sm}}(H,-)$, where $H$ ranges through a basis of
compact open subgroups of $G$, is conservative, hence it is enough to show
that
$R\Gamma_{\mathrm{sm}}(H,M^{\varphi}\otimes_{\mathbf{F}_{p}}\mathcal{O}^{a}_{C})=R\Gamma_{\mathrm{sm}}(H,M)$
for all $H$ in the basis. Noting that that $R\Gamma_{\mathrm{sm}}(H,-)$
commutes with $(-)^{\varphi}$ (which is formal) and with
$-\otimes_{\mathbf{F}_{p}}\mathcal{O}^{a}_{C}/\pi$ (as used above), the claim
boils down to showing that for $M_{H}:=R\Gamma_{\mathrm{sm}}(H,M)$ we have
$M_{H}^{\varphi}\otimes_{\mathbf{F}_{p}}\mathcal{O}^{a}_{C}/\pi\mathbin{\xrightarrow{\sim}}M_{H}$.
But by assumption $M_{H}$ is dualizable, hence the desired equivalence follows
from the Riemann-Hilbert correspondence (see [Man22, Theorem 3.9.23]). From
the Riemann-Hilbert correspondence we also deduce that
$R\Gamma_{\mathrm{sm}}(H,M^{\varphi})$ is dualizable, hence $M^{\varphi}$ is
admissible, as desired.

We have now finished the proof that (i) and (ii) are equivalent. To prove that
they are also equivalent to (iii) we make the following computation. Let
$X=\operatorname{Spa}C$, $\Lambda=\mathcal{O}_{C}/\pi$,
$\omega_{G}:=f^{!}\Lambda^{a}$ and for every compact open subgroup $H\subset
G$ let $f_{H}\colon X/H\to X$ be the natural map. Then from 3.12 and 3.14 we
get:

$\displaystyle f_{H*}M^{\vee}$
$\displaystyle=f_{H*}\operatorname{\underline{\operatorname{Hom}}}(M,\Lambda^{a})=f_{H*}\operatorname{\underline{\operatorname{Hom}}}(M\otimes\omega_{G},\omega_{G})=\operatorname{\underline{\operatorname{Hom}}}(f_{H*}(M\otimes\omega_{G}),\Lambda^{a})$
$\displaystyle=(f_{H*}(M\otimes\omega_{G}))^{\vee}.$

Note also that for small enough $H$ the action of $H$ on $\omega_{G}$ is
trivial, so that $-\otimes\omega_{G}$ is just a shift in the $\infty$-category
of $H$-representations. Consequently for small enough $H$ the functor
$R\Gamma_{\mathrm{sm}}(H,-)$ preserves duals up to shift. Since the functors
$R\Gamma_{\mathrm{sm}}(H,-)$ are also conservative (for $H$ varying in a basis
of compact open subgroups of $G$) it follows that (iii) is equivalent to the
claim that $M$ and $M^{\vee}$ are bounded and for a basis $H$ of compact open
subgroups of $G$ we have that both $M_{H}:=R\Gamma_{\mathrm{sm}}(H,M)$ and
$M_{H}^{\vee}$ are discrete and the natural morphism
$M_{H}\mathbin{\xrightarrow{\sim}}M_{H}^{\vee\vee}$ is an isomorphism. By 3.25
this implies (ii). Moreover, if (ii) is satisfied then all the conditions in
(iii) follow immediately except possibly that $M^{\vee}$ is also bounded. To
see that $M^{\vee}$ is indeed bounded, note first that
$M^{\vee}=\varinjlim_{H}R\Gamma_{\mathrm{sm}}(H,M^{\vee})$ (see [Man22, Lemma
3.4.19]) and by the above computation the right-hand side is (up to a constant
shift) isomorphic to $\varinjlim_{H}M_{H}^{\vee}$. On the other hand, the
discrete dualization functor $N\mapsto(N^{\vee})_{\omega}$ is $t$-exact on
discrete objects in
$\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}_{C}/\pi)$
(see [Man22, Lemma 3.7.15]). In particular, if $M_{H}$ is concentrated in
homological degrees $[a,b]$ then $M_{H}^{\vee}$ is concentrated in homological
degrees $[-b,-a]$. The fact that $M$ is bounded implies that we can choose $a$
and $b$ independently of $H$, which then implies that $M^{\vee}$ is also
bounded.

It remains to prove the last part of the claim, so assume that $M$ is
admissible and bounded. It follows immediately from (iii) that also $M^{\vee}$
(computed in
$\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(G,\mathcal{O}_{C}/\pi)$)
is admissible (and bounded), so in particular it is discrete and comes from an
$\mathbf{F}_{p}$-representation via the Riemann-Hilbert correspondence. It is
thus formal that this dual can be computed in any of the claimed
$\infty$-categories. ∎

######  Remark 3.27.

It follows from the proof that in 3.25 it is enough to require that there is
_some_ isomorphism $M\cong M^{\vee\vee}$ instead of insisting that the natural
map $M\to M^{\vee\vee}$ is an isomorphism. It follows that in condition (iii)
of 3.26 it is enough to require the existence of _some_ isomorphism $M\cong
M^{\vee\vee}$.

##  4\. Duality and applications

###  4.1. The main result

Fix a finite extension $F/\mathbf{Q}_{p}$, let $G:=\operatorname{GL}_{n}(F)$,
let $D$ be the central division algebra over $F$ of invariant $1/n$ and let
$C$ be the completed algebraic closure of $F$. In the following we will work
with small v-stacks over $\operatorname{Spa}C$, so in particular the final
object $*$ will denote $\operatorname{Spa}C$ and
$B\underline{H}=[\operatorname{Spa}C/\underline{H}]$ for any locally profinite
group $H$. Recall the fundamental diagram from the introduction:

${X}$${B\underline{G}}$${B\underline{D^{\times}}}$$\scriptstyle{f}$$\scriptstyle{g}$

Here $X$ is obtained from the infinite level Lubin-Tate tower by modding out
the commuting actions of $G$ and $D^{\times}$, so that in particular we have
$X\cong[\mathbf{P}^{n-1}_{C}/\underline{D^{\times}}]\cong[\Omega_{C}^{n-1}/\underline{G}]$,
where
$\Omega_{C}^{n-1}=\mathbf{P}^{n-1}_{C}\setminus\mathbf{P}^{n-1}_{C}(\mathbf{Q}_{p})$
denotes Drinfeld space. It follows that $f$ and $g$ are smooth and that $g$ is
proper. Recall also that for every locally profinite group $H$ which has
locally finite $p$-cohomological dimension (e.g. $H=G$ or $H=D^{\times}$) we
have

$\displaystyle\operatorname{\mathcal{D}}_{\mathrm{\acute{e}t}}(B\underline{H},\mathbf{F}_{p})$
$\displaystyle=\operatorname{\mathcal{D}}^{\mathrm{sm}}(H,\mathbf{F}_{p}),$
$\displaystyle\operatorname{\operatorname{\mathcal{D}}^{a}_{\scalebox{0.5}{$\square$}}}(\mathcal{O}^{+}_{B\underline{H}}/p)^{\varphi}$
$\displaystyle=\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(H,\mathcal{O}_{C}/p)^{\varphi},$

and the upper $\infty$-category embedds fully faithfully into the lower one
via the Riemann-Hilbert functor. As explained in the introduction, we can now
define the Jacquet-Langlands functor as follows.

######  Definition 4.1.

In the above setup, we define the _$p$ -adic Jacquet-Langlands functors_

$\displaystyle\mathcal{J}$
$\displaystyle\colon\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})\to\operatorname{\mathcal{D}}^{\mathrm{sm}}(D^{\times},\mathbf{F}_{p}),$
$\displaystyle A\mapsto
g_{{\mathrm{\acute{e}t}},*}f_{\mathrm{\acute{e}t}}^{*}A,$
$\displaystyle\mathcal{J}_{\scalebox{0.5}{$\square$}}$
$\displaystyle\colon\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(G,\mathcal{O}_{C}/p)^{\varphi}\to\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(D^{\times},\mathcal{O}_{C}/p)^{\varphi},$
$\displaystyle A\mapsto g_{*}f^{*}A,$

where we implicitly use the identifications of sheaves on the classifying
stack with smooth representations discussed above.

The functor $\mathcal{J}_{\scalebox{0.5}{$\square$}}$ is defined in the
setting of a full 6-functor formalism and should thus be seen as the “actual”
Jacquet-Langlands functor, whereas $\mathcal{J}$ is a more naively defined
shadow of $\mathcal{J}_{\scalebox{0.5}{$\square$}}$. In fact, it is completely
formal that for all
$A\in\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})$ we have

(1)
$\displaystyle\mathcal{J}(A)=(\mathcal{J}_{\scalebox{0.5}{$\square$}}(A\otimes\mathcal{O}^{+a}_{B\underline{G}}/p))^{\varphi}.$

In other words, $\mathcal{J}(A)$ remembers the $\varphi$-invariants of the
Jacquet-Langlands correspondent
$\mathcal{J}_{\scalebox{0.5}{$\square$}}(A\otimes\mathcal{O}^{+}_{B\underline{G}}/p)$
of $A$. It turns out that if $A$ is admissible then in fact these
$\varphi$-invariants capture the whole Jacquet-Langlands correspondent. In
fact, the Jacquet-Langlands correspondent of an admissible representation is
itself admissible and hence lies in the image of the Riemann-Hilbert functor,
as is shown in the following result.

######  Theorem 4.2.

In the above setup, let
$A\in\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})$ be
admissible. Then:

  1. (i)

$\mathcal{J}(A)\in\operatorname{\mathcal{D}}^{\mathrm{sm}}(D^{\times},\mathbf{F}_{p})$
is admissible.

  2. (ii)

The natural primitive comparison map

$\displaystyle\mathcal{J}(A)\otimes\mathcal{O}^{+a}_{B\underline{D^{\times}}}/p\mathbin{\xrightarrow{\sim}}\mathcal{J}_{\scalebox{0.5}{$\square$}}(A\otimes\mathcal{O}^{+a}_{B\underline{G}}/p)$

induced by Eq. 1 is an isomorphism.

  3. (iii)

There is a natural isomorphism

$\displaystyle(\mathcal{S}_{D^{\times}}\circ\mathcal{J})(A)=(\mathcal{J}\circ\mathcal{S}_{G})(A)[2n-2](n-1)$

in $\operatorname{\mathcal{D}}^{\mathrm{sm}}(D^{\times},\mathbf{F}_{p})$.

Recall that Scholze already proved (i) and (ii). We will prove (i)–(iii)
simultaneously, independently of Scholze’s results. In fact, we will see that
all (i)–(iii) follow formally from the following unconditional duality for
$\mathcal{J}_{\scalebox{0.5}{$\square$}}$:

######  Lemma 4.3.

In the above setup, there is a natural isomorphism

$\displaystyle(-)^{\vee}\circ\mathcal{J}_{\scalebox{0.5}{$\square$}}=(\mathcal{J}_{\scalebox{0.5}{$\square$}}\circ(-)^{\vee})[2n-2](n-1)$

of functors
$\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(G,\mathcal{O}_{C}/p)^{\varphi}\to\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(D^{\times},\mathcal{O}_{C}/p)^{\varphi}$.

###### Proof.

For every
$A\in\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(G,\mathcal{O}_{C}/p)^{\varphi}$
we get the following chain of natural equivalences (using that $g$ is proper
and smooth and that $f$ is smooth):

$\displaystyle\operatorname{\underline{\operatorname{Hom}}}(g_{*}f^{*}A,\mathcal{O}^{+a}_{B\underline{D^{\times}}}/p)$
$\displaystyle=\operatorname{\underline{\operatorname{Hom}}}(g_{!}f^{*}A,\mathcal{O}^{+a}_{B\underline{D^{\times}}}/p)$
$\displaystyle=g_{*}\operatorname{\underline{\operatorname{Hom}}}(f^{*}A,g^{!}(\mathcal{O}^{+a}_{B\underline{D^{\times}}}/p))$
$\displaystyle=g_{*}\operatorname{\underline{\operatorname{Hom}}}(f^{*}A,\mathcal{O}^{+a}_{X}/p)[2n-2](n-1)$
$\displaystyle=g_{*}f^{*}\operatorname{\underline{\operatorname{Hom}}}(A,\mathcal{O}^{+a}_{B\underline{G}}/p)[2n-2](n-1).$

In the last step we used that by the smoothness of $f$, $f^{*}$ commutes with
internal hom (see [Man22, Proposition 3.8.7]. ∎

###### Proof of Theorem 4.2.

Fix an admissible
$A\in\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{F}_{p})$ as in the
claim and let us denote
$A^{\prime}:=A\otimes\mathcal{O}_{B\underline{G}}^{+a}/p$ the corresponding
object in
$\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(G,\mathcal{O}_{C}/p)^{\varphi}$
given by the Riemann-Hilbert embedding. Then $A^{\prime}$ is admissible (by
definition) and hence by 3.26 both $A^{\prime}$ and $A^{\prime\vee}$ are
bounded and discrete and the natural map
$A^{\prime}\mathbin{\xrightarrow{\sim}}A^{\prime\vee\vee}$ is an isomorphism.
Clearly $\mathcal{J}_{\scalebox{0.5}{$\square$}}$ preserved bounded and
discrete objects (this is true for $f^{*}$ like for any pullback and it
follows for $g_{*}$ from the fact that $g$ is proper and $p$-bounded). Thus it
follows from 4.3 that both
$\mathcal{J}_{\scalebox{0.5}{$\square$}}(A^{\prime})$ and
$\mathcal{J}_{\scalebox{0.5}{$\square$}}(A^{\prime})^{\vee}$ are bounded and
discrete. Moreover, 4.3 also implies that there is an isomorphism
$\mathcal{J}_{\scalebox{0.5}{$\square$}}(A^{\prime})\cong\mathcal{J}_{\scalebox{0.5}{$\square$}}(A^{\prime})^{\vee\vee}$,
so altogether it follows from 3.26 (and 3.27) that
$\mathcal{J}_{\scalebox{0.5}{$\square$}}(A^{\prime})$ is admissible. This
immediately implies (i). It also implies (ii) because we now know that
$\mathcal{J}_{\scalebox{0.5}{$\square$}}(A^{\prime})$ lies in the image of the
Riemann-Hilbert functor. Now (iii) follows from 4.3 and the last part of 3.26.
∎

######  Remark 4.4.

In general, both sides of the duality isomorphism are somewhat intractible.
However, in certain favorable situations it reduces to a more concrete
statement. We record one useful instance of this in the following proposition.
Recall from Proposition 3.7 that for any admissible smooth
$\pi\in\mathrm{Rep}_{\mathbf{F}_{p}}^{\mathrm{sm}}(G)$, the derived dual
$\mathcal{S}_{G}^{\dim_{G}\pi}(\pi)$ is always nonzero. Following [Koh17], we
say $\pi$ is Cohen-Macaulay if $\mathcal{S}_{G}^{i}(\pi)=0$ for all
$i\neq\dim_{G}\pi$.

######  Proposition 4.5.

Let $\pi\in\mathrm{Rep}_{\mathbf{F}_{p}}^{\mathrm{sm}}(G)$ be an admissible
representation which is Cohen-Macaulay, and such that there is an integer $c$
with $\mathcal{J}^{i}(\pi)=0$ for all $i\neq c$. Then there are isomorphisms

$\mathcal{S}_{D^{\times}}^{i}(\mathcal{J}^{c}(\pi))\simeq\mathcal{J}^{i+2n-2-c-\dim_{G}\pi}(\check{\pi})$

for all $i$, where $\check{\pi}=\mathcal{S}_{G}^{\dim_{G}\pi}(\pi)$ is the
unique nonzero derived dual of $\pi$.

###### Proof.

Evaluate both sides of the duality isomorphism
$\mathcal{S}_{D^{\times}}\circ\mathcal{J})\cong(\mathcal{J}\circ\mathcal{S}_{G})[2n-2](n-1)$
on $\pi$. Since $\mathcal{J}(\pi)=\mathcal{J}^{c}(\pi)[-c]$ by assumption, the
left-hand side evaluates to
$\mathcal{S}_{D^{\times}}(\mathcal{J}^{c}(\pi)[-c])\cong\mathcal{S}_{D^{\times}}(\mathcal{J}^{c}(\pi))[c]$.
Next, since $\mathcal{S}_{G}(\pi)=\check{\pi}[-\dim_{G}\pi]$ by assumption,
the right-hand side simplifies to
$\mathcal{J}(\check{\pi}[-\dim_{G}\pi])[2n-2](n-1)\cong\mathcal{J}(\check{\pi})[2n-2-\dim_{G}\pi](n-1)$.
Equating both sides and rearranging the shifts, we get
$\mathcal{S}_{D^{\times}}(\mathcal{J}^{c}(\pi))=\mathcal{J}(\check{\pi})[2n-2-\dim_{G}\pi-c](n-1)$.
Passing to cohomology on both sides (and ignoring the Tate twists), we get the
result. ∎

######  Remark 4.6.

We explain how to show a variant of our duality isomorphism in the setting of
$\mathbf{Z}/p^{n}$ or $\mathbf{Z}_{p}$-coefficients. We explain the first
case; the second follows from a limit argument. First some preliminary
observations. Let $H$ be any $p$-adic Lie group. One easily checks that
Proposition 3.9 extends to the setting of $\mathbf{Z}/p^{n}$-coefficients,
yielding a symmetric monoidal equivalence
$\operatorname{\mathcal{D}}^{\mathrm{sm}}(H,\mathbf{Z}/p^{n})\cong\operatorname{\mathcal{D}}_{{\mathrm{\acute{e}t}}}(B\underline{H},\mathbf{Z}/p^{n})$.
On these categories we again have a duality functor $\mathcal{S}_{H}(-)$,
given by internal hom towards the monoidal unit. We say
$A\in\operatorname{\mathcal{D}}^{\mathrm{sm}}(H,\mathbf{Z}/p^{n})$ is
admissible if $R\Gamma_{{\mathrm{sm}}}(H,A)$ is a perfect complex of
$\mathbf{Z}/p^{n}$-modules for a basis of open subgroups $H$. Admissible
complexes are automatically bounded, and are preserved by duality and
$-\otimes_{\mathbf{Z}/p^{n}}^{\mathbf{L}}\mathbf{F}_{p}$, and the latter
functor is conservative on admissible complexes. Conversely, if $A$ is given
and
$A\otimes_{\mathbf{Z}/p^{n}}^{\mathbf{L}}\mathbf{F}_{p}\in\operatorname{\mathcal{D}}^{\mathrm{sm}}(H,\mathbf{F}_{p})$
is admissible, then $A$ is necessarily admissible.

With this in mind, we _define_
$\mathcal{J}:\operatorname{\mathcal{D}}^{\mathrm{sm}}(G,\mathbf{Z}/p^{n})\to\operatorname{\mathcal{D}}^{\mathrm{sm}}(D^{\times},\mathbf{Z}/p^{n})$
as
$\mathcal{J}=g_{{\mathrm{\acute{e}t}}\ast}f^{\ast}_{{\mathrm{\acute{e}t}}}$.
This is compatible with the $\mathbf{F}_{p}$ version of $\mathcal{J}$ via
$-\otimes_{\mathbf{Z}/p^{n}}^{\mathbf{L}}\mathbf{F}_{p}$. Using this
observation and the final statement in the previous paragraph together with
the finiteness properties of $\mathcal{J}$ with $\mathbf{F}_{p}$-coefficients,
it is easy to see that $\mathcal{J}$ preserves admissibility in this setting.
Now, the key observation is that there is a natural map
$\mathcal{J}(A)\otimes\mathcal{J}(\mathcal{S}_{G}(A))\to\mathbf{Z}/p^{n}[2-2n](1-n)$,
defined as the composition

$\displaystyle\mathcal{J}(A)\otimes\mathcal{J}(\mathcal{S}_{G}(A))$
$\displaystyle\to\mathcal{J}(A\otimes\mathcal{S}_{G}(A))$
$\displaystyle\to\mathcal{J}(\mathbf{Z}/p^{n})$ $\displaystyle\to\tau^{\geq
2n-2}\mathcal{J}(\mathbf{Z}/p^{n})$
$\displaystyle\cong\mathbf{Z}/p^{n}[2-2n](1-n).$

Here the first arrow is obtained from the lax monoidal structure of
$\mathcal{J}$, the second arrow is induced by the evaluation map
$A\otimes\mathcal{S}_{G}(A)\to\mathbf{Z}/p^{n}$, and the final isomorphism is
induced by the usual computation of the étale cohomology of
$\mathbf{P}^{n-1}$. By adjunction (and rearranging the shifts and twists),
this map induces a map
$\mathcal{J}(\mathcal{S}_{G}(A))[2n-2](n-1)\to\mathcal{S}_{D^{\times}}(\mathcal{J}(A))$.
When $A$ is admissible, the source and target of this map are also admissible,
so to see it is an isomorphism, it suffices to show this after applying
$-\otimes_{\mathbf{Z}/p^{n}}^{\mathbf{L}}\mathbf{F}_{p}$. But all operations
here are compatible with reduction mod $p$, so we now conclude by invoking the
duality isomorphism with $\mathbf{F}_{p}$-coefficients.

###  4.2. Applications to Gelfand-Kirillov dimension

In this section we prove Theorems 1.5 and 1.6, along with some related
results. We will freely use the basic properties of dimension, in particular
the results of Propositions 3.7-3.8 and the surrounding discussion.

###### Proof of Theorem 1.5.

Let $d=\dim_{G}(\pi)$, so
$\mathcal{S}_{G}(\pi)\in\operatorname{\mathcal{D}}^{\mathrm{sm}}(G)^{[0,d]}$.
Since $\mathcal{J}(-)$ carries
$\operatorname{\mathcal{D}}^{\mathrm{sm}}(G)^{[a,b]}$ into
$\operatorname{\mathcal{D}}^{\mathrm{sm}}(D^{\times})^{[a,b+2n-2]}$, we deduce
that

$(\mathcal{J}\circ\mathcal{S}_{G})(\pi)[2n-2]\in\operatorname{\mathcal{D}}^{\mathrm{sm}}(D^{\times})^{\leq
d}.$

By the duality isomorphism
$(\mathcal{S}_{D^{\times}}\circ\mathcal{J})(\pi)\cong(\mathcal{J}\circ\mathcal{S}_{G})(\pi)[2n-2]$,
we get that
$M=(\mathcal{S}_{D^{\times}}\circ\mathcal{J})(\pi)\in\operatorname{\mathcal{D}}^{\mathrm{sm}}(D^{\times})^{\leq
d}$. Now look at the fourth quadrant spectral sequence

$E_{2}^{i,j}=\mathcal{S}_{D^{\times}}^{i}(\mathcal{J}^{-j}(\pi))\Rightarrow
H^{i+j}(M).$

Let $c=\max_{i}\dim_{D^{\times}}\mathcal{J}^{i}(\pi)$, and let $i_{0}$ be the
least $i$ realizing this maximum. Then $E_{2}^{c,-i_{0}}$ is $c$-dimensional
as an $D^{\times}$-representation, and as we turn the pages of the spectral
sequence it only interacts with representations of dimension $<c$ via the
incoming differentials. Therefore a nonzero subquotient survives to the
$E_{\infty}$-page, giving that $H^{c-i_{0}}(M)\neq 0$. But
$M\in\operatorname{\mathcal{D}}^{\mathrm{sm}}(D^{\times})^{\leq d}$, so this
implies that $c-i_{0}\leq d$, and therefore $c\leq d+i_{0}\leq d+N_{\pi}$, as
desired. ∎

We conjecture the following strengthening of Theorem 1.5.

######  Conjecture 4.7.

For any $\pi\in\mathrm{Rep}_{\mathbf{F}_{p}}^{\mathrm{adm}}(G)$, we have
$\dim_{D^{\times}}\mathcal{J}^{i}(\pi)\leq\dim_{G}\pi$ for all $i\geq 0$.

In many cases, this can be proved. In particular, we have the following
result, which can be proved by an easy adaptation of the proof of Theorem 1.5.

######  Proposition 4.8.

Suppose $\pi\in\mathrm{Rep}_{\mathbf{F}_{p}}^{\mathrm{adm}}(G)$ is such that
for all $i>n-1$ and all $j\geq 0$, $\mathcal{J}^{i}(\pi)=0$ and
$\mathcal{J}^{i}(\mathcal{S}_{G}^{j}(\pi))=0$. Then Conjecture 4.7 is true for
$\pi$.

The hypotheses of this proposition are satisfied for any representation of the
form $\pi=\mathrm{Ind}_{P}^{G}\sigma$, where $P$ is any parabolic subgroup
contained in the standard maximal parabolic $P_{n-1,1}$ and $\sigma$ is any
admissible smooth representation of the Levi quotient $P\twoheadrightarrow M$.
To see this, note that the main theorem of [JLH21] shows that
$\mathcal{J}^{i}\circ\mathrm{Ind}_{P}^{G}=0$ for all $i>n-1$. It’s then enough
to observe that for all $j$ we have
$\mathcal{S}_{G}^{j}(\mathrm{Ind}_{P}^{G}\sigma)=\mathrm{Ind}_{P}^{G}\delta_{P}\mathcal{S}_{M}^{j-d}(\sigma)$
for a certain integer $d$ and smooth character $\delta_{P}$ (use [Koh17,
Theorem 4.7 and Corollary 5.3]), so the main theorem of [JLH21] again applies.

We also observe that Conjecture 4.7 implies that $\mathcal{J}^{2n-2}$ is very
small, and can be explicitly computed.

######  Proposition 4.9.

Suppose that Conjecture 4.7 is true. Then for any
$\pi\in\mathrm{Rep}_{\mathbf{F}_{p}}^{\mathrm{adm}}(G)$, we have
$\dim_{\mathbf{F}_{p}}\mathcal{J}^{2n-2}(\pi)<\infty$, and in fact there is an
isomorphism

$\mathcal{J}^{2n-2}(\pi)=\mathcal{S}_{D^{\times}}^{0}(\mathcal{J}^{0}(\mathcal{S}_{G}^{0}(\pi))).$

The final isomorphism here implies (assuming Conjecture 4.7) that
$\mathcal{J}^{2n-2}(\pi)$ is nonzero _exactly_ when $\pi$ admits a $G$-stable
quotient $\pi\twoheadrightarrow\tau$ such that
$\dim_{\mathbf{F}_{p}}\tau<\infty$ and $\tau^{\mathrm{SL}_{n}(F)}\neq 0$. We
also note that the following proof shows that
$\mathcal{J}^{2n-2}(\pi)=\mathcal{S}_{D^{\times}}^{0}(\mathcal{J}^{0}(\mathcal{S}_{G}^{0}(\pi)))$
whenever $\dim_{D^{\times}}\mathcal{J}^{2n-2}(\pi)=0$, independently of
Conjecture 4.7.

###### Proof.

Set $M=(\mathcal{J}\circ\mathcal{S}_{G})(\pi)$. By the duality theorem, we get
an $E_{2}$ spectral sequence

$E_{2}^{i,j}=\mathcal{S}^{i}_{D^{\times}}(\mathcal{J}^{-j}(\pi))\Rightarrow
H^{i+j+2n-2}(M).$

Suppose that $\dim_{\mathbf{F}_{p}}\mathcal{J}^{2n-2}(\pi)=\infty$, i.e. that
$c:=\dim_{D^{\times}}\mathcal{J}^{2n-2}(\pi)>0$. Then $E_{2}^{c,-(2n-2)}$ is
nonzero and $c$-dimensional, and a c-dimensional quotient of it survives to
the $E_{\infty}$-page, showing that $\dim_{D^{\times}}H^{c}(M)\geq c$. On the
other hand, we have a second spectral sequence
$\mathcal{J}^{i}(\mathcal{S}_{G}^{j}(\pi))\Rightarrow H^{i+j}(M)$, showing
that $H^{c}(M)$ has a finite filtration whose gradeds are subquotients of
$\mathcal{J}^{i}(\mathcal{S}_{G}^{c-i}(\pi))$ for varying $i$. Since
$\dim_{G}\mathcal{S}_{G}^{c-i}(\pi)\leq c-i$, Conjecture 4.7 then implies that
$\dim_{D^{\times}}\mathcal{J}^{i}(\mathcal{S}_{G}^{c-i}(\pi))\leq c-i<c$ for
$i>0$. But when $i=0$,
$\dim_{D^{\times}}\mathcal{J}^{0}(\mathcal{S}_{G}^{c}(\pi))=0$ by the explicit
calculation of $\mathcal{J}^{0}$. Therefore $H^{c}(M)$ has a finite filtration
whose gradeds have dimension $<c$, so $\dim_{D^{\times}}H^{c}(M)<c$. This is a
contradiction.

For the final statement, note that the two spectral sequences show
(unconditionally) that

$\mathcal{S}_{D^{\times}}^{0}(\mathcal{J}^{2n-2}(\pi))=\mathcal{J}^{0}(\mathcal{S}_{G}^{0}(\pi))$

for all $\pi\in\mathrm{Rep}_{\mathbf{F}_{p}}^{\mathrm{adm}}(G)$. If
$\mathcal{J}^{2n-2}(\pi)$ is zero-dimensional, then
$\mathcal{S}_{D^{\times}}^{0}(\mathcal{J}^{2n-2}(\pi))=\mathcal{S}_{D^{\times}}(\mathcal{J}^{2n-2}(\pi))$
is the full derived dual of $\mathcal{J}^{2n-2}(\pi)$, so the result follows
from biduality. ∎

###### Proof of Theorem 1.6.

Let $\check{\pi}$ be the unique nonzero derived dual of $\pi$. The duality
isomorphism then simplifies to an isomorphism

$\mathcal{S}_{D^{\times}}(\mathcal{J}(\pi))\cong\mathcal{J}(\check{\pi})[2n-2-\dim_{G}\pi].$

Since $\mathcal{J}(\check{\pi})$ is concentrated in degrees $\geq 0$, the
right-hand side, and then also the left-hand side, is concentrated in degrees
$\geq\dim_{G}\pi-(2n-2)$.

If $\mathcal{J}(\pi)$ is zero, there is nothing to prove. Otherwise, let
$c=\max_{i}\dim_{D^{\times}}\mathcal{J}^{i}(\pi)$, and let $i_{0}$ be any $i$
realizing this maximum. Arguing as in the proof of Theorem 1.5, we see that
the left-hand side has nonzero cohomology in degree $c-i_{0}$. Therefore
$c\geq\dim_{G}\pi-(2n-2)+i_{0}\geq\dim_{G}\pi-(2n-2)$, giving the desired
result. ∎

###  4.3. The case of $\mathrm{GL}_{2}$

Now we specialize to the case $n=2$, with $F/\mathbf{Q}_{p}$ still arbitrary.
Set $d=[F:\mathbf{Q}_{p}]$. Let $B\subset G=\mathrm{GL}_{2}(F)$ be the upper-
triangular Borel subgroup, with $T=F^{\times}\times F^{\times}\subset B$ the
diagonal maximal torus. Let $\omega:F^{\times}\to\mathbf{F}_{p}^{\times}$ be
the character defined as the composition

$F^{\times}\overset{\mathrm{Nm}_{F/\mathbf{Q}_{p}}}{\longrightarrow}\mathbf{Q}_{p}^{\times}\overset{x\mapsto
x|x|}{\longrightarrow}\mathbf{Z}_{p}^{\times}\overset{\mathrm{red}}{\to}\mathbf{F}_{p}^{\times}.$

If $\chi_{1}\otimes\chi_{2}:T\to\mathbf{F}_{p}^{\times}$ is any smooth
character, we can consider the smooth induction
$\mathrm{Ind}_{B}^{G}(\chi_{1}\otimes\chi_{2})$. This is irreducible iff
$\chi_{1}\neq\chi_{2}$. Moreover,
$\mathcal{S}_{G}^{i}(\mathrm{Ind}_{B}^{G}(\chi_{1}\otimes\chi_{2}))=0$ for all
$i\neq d$, and

$\mathcal{S}_{G}^{d}(\mathrm{Ind}_{B}^{G}(\chi_{1}\otimes\chi_{2}))\cong\mathrm{Ind}_{B}^{G}(\omega\chi_{1}^{-1}\otimes\omega^{-1}\chi_{2}^{-1}).$

In particular, if we set
$\pi(\chi_{1},\chi_{2})=\mathrm{Ind}_{B}^{G}(\omega\chi_{1}\otimes\chi_{2})$,
then

$\mathcal{S}_{G}^{d}(\pi(\chi_{1},\chi_{2}))\otimes(\chi_{1}\chi_{2}\omega\circ\det)\cong\pi(\chi_{2},\chi_{1}),$

and moreover $\pi(\chi_{1},\chi_{2})$ and its dual
$\mathcal{S}_{G}^{d}(\pi(\chi_{1},\chi_{2}))$ are both irreducible iff
$\chi_{1}\chi_{2}^{-1}\neq\omega^{\pm 1}$.

Assume then that $\chi_{1},\chi_{2}$ is a pair of characters such that
$\chi_{1}/\chi_{2}\neq\omega^{\pm 1}$. As noted in the introduction, we then
have $\mathcal{J}^{i}(\pi(\chi_{1},\chi_{2}))=0$ for all $i\neq 1$, and
likewise with $\chi_{1}$ and $\chi_{2}$ swapped. With this vanishing in mind,
we set $\tau(\chi_{1},\chi_{2})=\mathcal{J}^{1}(\pi(\chi_{1},\chi_{2}))$.

######  Theorem 4.10.

Notation and assumptions as above, exactly one of the following is true.

i. $\tau(\chi_{1},\chi_{2})=\tau(\chi_{2},\chi_{1})=0.$

ii. Both $\tau(\chi_{1},\chi_{2})$ and $\tau(\chi_{2},\chi_{1})$ are nonzero,
and are exchanged by the functor
$\mathcal{S}_{D^{\times}}^{d}(-)\otimes(\chi_{1}\chi_{2}\omega\circ\mathrm{Nm})$.
Moreover, $\mathcal{S}_{D^{\times}}^{i}(-)$ applied to either representation
vanishes for all $i\neq d$.

###### Proof.

Evaluate the duality isomorphism
$(\mathcal{S}_{D^{\times}}\circ\mathcal{J})\cong(\mathcal{J}\circ\mathcal{S}_{G})[2]$
on the representation $\pi(\chi_{1},\chi_{2})$. Since
$\mathcal{J}(\pi(\chi_{1},\chi_{2}))=\tau(\chi_{1},\chi_{2})[-1]$, the left-
hand side simplifies to $\mathcal{S}_{D^{\times}}(\tau(\chi_{1},\chi_{2}))[1]$
by the results recalled above. On the other hand,
$\mathcal{S}_{G}(\pi(\chi_{1},\chi_{2}))\cong\pi(\chi_{2},\chi_{1})\otimes(\chi_{1}\chi_{2}\omega\circ\det)^{-1}[-d]$,
so the right-hand side simplifies to
$\mathcal{J}(\pi(\chi_{2},\chi_{1}))\otimes(\chi_{1}\chi_{2}\omega\circ\mathrm{Nm})^{-1}[2-d]\cong\tau(\chi_{2},\chi_{1})\otimes(\chi_{1}\chi_{2}\omega\circ\mathrm{Nm})^{-1}[1-d]$.
Thus, canceling the shifts, we get an isomorphism

$\mathcal{S}_{D^{\times}}(\tau(\chi_{1},\chi_{2}))\cong\tau(\chi_{2},\chi_{1})\otimes(\chi_{1}\chi_{2}\omega\circ\mathrm{Nm})^{-1}[-d],$

which immediately gives the desired result. ∎

####  4.3.1. A non-generic case

In this section we sketch some consequences of our duality theorem in the
setting of _reducible_ principal series representations of
$\mathrm{GL}_{2}(\mathbf{Q}_{p})$. Maintain the previous notation, but with
$F=\mathbf{Q}_{p}$. The principal series representation $\pi(\omega^{-1},1)$
is reducible of length two, and sits in a short exact sequence

$0\to\mathbf{1}\to\pi(\omega^{-1},1)\to\mathrm{St}\to 0\;\;\;\;\;(\dagger)$

where $\mathrm{St}$ is the mod $p$ Steinberg representation of
$\mathrm{GL}_{2}(\mathbf{Q}_{p})$. To ease notation, set
$\kappa=\mathcal{S}^{1}_{G}(\mathrm{St})$. Note that
$\mathcal{S}_{G}(\mathbf{1})=\mathbf{1}[0]$ and
$\mathcal{S}_{G}(\pi(\omega^{-1},1))=\pi(1,\omega^{-1})[-1]$, where
$\pi(1,\omega^{-1})$ is an irreducible principal series. The associated long
exact sequence for $\mathcal{S}^{i}_{G}(-)$ is now easy to analyze: the
embedding
$\mathcal{S}^{0}_{G}(\mathrm{St})\hookrightarrow\mathcal{S}^{0}_{G}(\pi(\omega^{-1},1))=0$
implies that the long exact sequence degenerates to a short exact sequence

$0\to\mathbf{1}\to\kappa\to\pi(1,\omega^{-1})\to 0\;\;\;\;(\ddagger)$

and that $\mathcal{S}^{i}_{G}(\mathrm{St})$ for all $i\neq 1$. Biduality then
implies that the previous sequence is non-split, since otherwise the
nonvanishing of $\mathcal{S}^{0}_{G}(-)$ applied to the first term would
contradict the vanishing of $\mathcal{S}^{0}_{G}(\kappa)$. Note that $\kappa$
is the unique non-split extension of $\pi(1,\omega^{-1})$ by $\mathbf{1}$.

Using the main result in [Lud17] together with the explicit computation of
$\mathcal{J}^{0}$, the following vanishing results are clear:

  * •

$\mathcal{J}(\mathbf{1})=R\Gamma(\mathbf{P}^{1}_{C},\mathbf{F}_{p})=\mathbf{1}\oplus\mathbf{1}[-2]$

  * •

$\mathcal{J}^{i}(\pi(1,\omega^{-1}))=0$ for all $i\neq 1$.

  * •

$\mathcal{J}^{i}(\pi(\omega^{-1},1))=0$ for all $i\neq 0,1$, and
$\mathcal{J}^{0}(\pi(\omega^{-1},1))=\mathbf{1}$.

  * •

$\mathcal{J}^{i}(\mathrm{St})=0$ for all $i\neq 1$.

The long exact sequence in $\mathcal{J}^{i}$’s associated with the sequence
$(\dagger)$ then degenerates to a short exact sequence

$0\to\mathcal{J}^{1}(\pi(\omega^{-1},1))\to\mathcal{J}^{1}(\mathrm{St})\to\mathbf{1}\to
0\;\;\;\;\;(\dagger^{\prime}).$

Next, observe that $\mathcal{J}^{2}$ is right-exact, so applying it to the
exact sequence $(\ddagger)$ gives a surjection
$\mathbf{1}=\mathcal{J}^{2}(\mathbf{1})\twoheadrightarrow\mathcal{J}^{2}(\kappa)$.
This shows that $\dim_{D^{\times}}\mathcal{J}^{2}(\kappa)=0$. Using
Proposition 4.9 and the remark immediately afterwards, we then deduce from the
vanishing $\mathcal{S}^{0}_{G}(\kappa)=0$ that in fact
$\mathcal{J}^{2}(\kappa)=0$. Note that Hu-Wang proved the vanishing
$\mathcal{J}^{2}(\kappa)=0$ by global methods [HW22, Corollary 8.19], while
our argument is purely local. With this vanishing in hand, we see that the
long exact sequence in $\mathcal{J}^{i}$’s associated with the sequence
$(\ddagger)$ degenerates to a short exact sequence

$0\to\mathcal{J}^{1}(\kappa)\to\mathcal{J}^{1}(\pi(1,\omega^{-1}))\to\mathbf{1}\to
0\;\;\;\;\;(\ddagger^{\prime}).$

Next, we observe that the representations $\mathrm{St}$ and
$\pi(1,\omega^{-1})$ both satisfy the conditions of Proposition 4.5, with
$c=1$ and $\dim_{G}(-)=1$ in both cases. Applying that proposition, we get
duality isomorphisms

$\mathcal{S}^{i}_{D^{\times}}(\mathcal{J}^{1}(\mathrm{St}))\cong\mathcal{J}^{i}(\kappa)$

and

$\mathcal{S}^{i}_{D^{\times}}(\mathcal{J}^{1}(\pi(1,\omega^{-1})))\cong\mathcal{J}^{i}(\pi(\omega^{-1},1))$

for all $i\geq 0$. When $i=0$ these yield no new information. The vanishing of
the right-hand sides for $i\geq 2$ implies that $\mathcal{J}^{1}(\mathrm{St})$
and $\mathcal{J}^{1}(\pi(1,\omega^{-1}))$ are of dimension $\leq 1$; the exact
sequences $(\dagger^{\prime})$ and $(\ddagger^{\prime})$ then imply that
$\mathcal{J}^{1}(\kappa)$ and $\mathcal{J}^{1}(\pi(\omega^{-1},1))$ are also
of dimension $\leq 1$. Finally, in the case $i=1$, the left-hand side of each
isomorphism is either identically zero, or Cohen-Macaulay of dimension one.
This yields a dichotomy: either

  1. (1)

$\mathcal{J}^{1}(\mathrm{St})=\mathcal{J}^{1}(\pi(1,\omega^{-1}))=\mathbf{1}$
and $\mathcal{J}^{1}(\pi(\omega^{-1},1))=\mathcal{J}^{1}(\kappa)=0$, or

  2. (2)

all four representations $\mathcal{J}^{1}(\mathrm{St})$,
$\mathcal{J}^{1}(\pi(1,\omega^{-1}))$, $\mathcal{J}^{1}(\pi(\omega^{-1},1))$,
and $\mathcal{J}^{1}(\kappa)$ are one-dimensional, and the last two are Cohen-
Macaulay.

Just as in the situation of Theorem 1.7, we don’t currently see how to rule
out (1) by a purely local argument. However, using [HW22, Proposition
8.18.(i)] and their exact sequence (8.13), it is easy to see that (1) does not
occur. We leave the details to the interested reader.

##  5\. A partial Künneth formula

In this section only, we change notation slightly, to emphasize the dependence
of the Jacquet-Langlands functor on the integer $n$. More precisely, we write
$\mathcal{J}_{n}$ for the Jacquet-Langlands functor from
$\operatorname{\mathcal{D}}(\mathrm{GL}_{n}(F))$ towards
$\operatorname{\mathcal{D}}(D_{1/n}^{\times})$.

For any integer $0<d<n$, let $P_{n-d,d}\subset\mathrm{GL}_{n}(F)$ be the usual
upper-triangular parabolic with Levi
$\mathrm{GL}_{n-d}(F)\times\mathrm{GL}_{d}(F)$. Let
$\sigma_{1}\boxtimes\sigma_{2}$ be a smooth representation of the Levi, and
let
$\mathrm{Ind}_{P_{n-d,d}}^{\mathrm{GL}_{n}(F)}(\sigma_{1}\boxtimes\sigma_{2})$
be the usual parabolic induction to a smooth representation of
$\mathrm{GL}_{n}(F)$. The main result in this section is an alternative
formula for
$\mathcal{J}_{n}(\mathrm{Ind}_{P_{n-d,d}}^{\mathrm{GL}_{n}(F)}(\sigma_{1}\boxtimes\sigma_{2}))$,
in terms of the cohomology of an auxiliary space with coefficients in a
certain sheaf built from $\sigma_{1}$ and $\mathcal{J}_{d}(\sigma_{2})$. This
builds on the analysis begun by the first author in [JLH21, Appendix A]. When
$d=1$, this formula is implicit in [JLH21], as explained below.

To state the main result, we need to introduce the key auxiliary space. In
what follows we will freely use some standard notation related to relative
Fargues-Fontaine curves. In particular, for any perfectoid space
$S/\operatorname{Spa}C$, we have the adic relative Fargues-Fontaine curve
$\mathcal{X}_{S^{\flat}}$, which comes with a canonical closed immersion
$\iota:S\hookrightarrow\mathcal{X}_{S^{\flat}}$ realizing $S$ as a closed
Cartier divisor. We write $\mathcal{O}(1/n)$ for the usual stable vector
bundle of rank $n$ and degree $1$, which is defined functorially over any
$\mathcal{X}_{S^{\flat}}$.

######  Definition 5.1.

Let $W_{n,d}$ denote the moduli space over $\operatorname{Spd}C$ parametrizing
quotients of $\mathcal{O}(1/n)$ which are isomorphic to $\mathcal{O}(1/d)$
after pullback to any geometric point.

Note that $W_{n,d}$ carries a natural $\underline{D_{1/n}^{\times}}$-action by
bundle automorphisms.444Recall that the automorphism sheaf
$S\mapsto\mathrm{Aut}(\mathcal{O}(1/n)^{\oplus m}/\mathcal{X}_{S^{\flat}})$ is
exactly the sheaf $\underline{\mathrm{GL}_{m}(D_{1/n}^{\times})}$. It is easy
to see that $W_{n,d}$ is a spatial diamond, and we will see below that it is
proper over $\operatorname{Spd}C$. It is also easy to see that $W_{n,d}$ can
be defined equivalently as the moduli space of _sub_ bundles of
$\mathcal{O}(1/n)$ which are isomorphic to $\mathcal{O}^{n-d}$ after pullback
to any geometric point. Trivializing the subbundle in this alternative
description gives a canonical $\mathrm{GL}_{n-d}(F)$-torsor over $W_{n,d}$,
corresponding to a canonical map $h_{1}:W_{n,d}\to
B\underline{\mathrm{GL}_{n-d}(F)}$. On the other hand, trivializing the
quotient bundle in the original moduli problem gives a canonical
$D_{1/d}^{\times}$-torsor over $W_{n,d}$, corresponding to a canonical map
$h_{2}:W_{n,d}\to B\underline{D_{1/d}^{\times}}$.

With these preparations in hand, the main result in this section can be stated
as follows.

######  Theorem 5.2.

For any $\sigma_{1}\in\operatorname{\mathcal{D}}(\mathrm{GL}_{n-d}(F))$ and
$\sigma_{2}\in\operatorname{\mathcal{D}}(\mathrm{GL}_{d}(F))$, there is a
natural isomorphism

$\mathcal{J}_{n}(\mathrm{Ind}_{P_{n-d,d}}^{\mathrm{GL}_{n}(F)}(\sigma_{1}\boxtimes\sigma_{2}))\cong
R\Gamma(W_{n,d},h_{1,{\mathrm{\acute{e}t}}}^{\ast}\sigma_{1}\otimes
h_{2,{\mathrm{\acute{e}t}}}^{\ast}\mathcal{J}_{d}(\sigma_{2})).$

When $d=1$, this specializes to the isomorphism
$\mathcal{J}_{n}(\mathrm{Ind}_{P_{n-1,1}}^{\mathrm{GL}_{n}(F)}(\sigma_{1}\boxtimes\sigma_{2}))\cong
R\Gamma(\mathcal{M}_{P(F)},\mathcal{F}_{\sigma_{1}\boxtimes\sigma_{2}})$
implicit in the proof of [JLH21, Theorem 5.3.1] (where we freely use some
notation from loc. cit.). Indeed, $\mathcal{M}_{P(F)}=X_{n,1}=W_{n,1}$ by the
discussion below, and a direct calculation shows that
$\mathcal{F}_{\sigma_{1}\boxtimes\sigma_{2}}\cong
h_{1}^{\ast}\sigma_{1}\otimes h_{2}^{\ast}\sigma_{2}$, noting that
$\mathcal{J}_{1}(\sigma_{2})=\sigma_{2}$ has no effect.

The proof relies on several auxiliary spaces, which we find helpful to collect
into a large diagram

---
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{-}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[\mathcal{M}_{\infty,d}/\underline{D_{1/d}^{\times}}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{g}_{d}}$$\scriptstyle{\tilde{h}_{2}}$$\textstyle{[\mathbf{P}_{C}^{d-1}/\underline{D_{1/d}^{\times}}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{d}}$$\scriptstyle{g_{d}}$$\textstyle{B\underline{\mathrm{GL}_{d}(F)}}$$\textstyle{\mathcal{I}\mathrm{nj}(\mathcal{O}^{n-d},\mathcal{O}(1/n))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{n,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}}$$\scriptstyle{h_{2}}$$\textstyle{B\underline{D_{1/d}^{\times}}}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\underline{\mathrm{GL}_{n-d}(F)}}$

of small v-stacks, where every square is cartesian. Here
$\ast=\operatorname{Spd}C$ for brevity, and $W_{n,d}$ and $h_{1},h_{2}$ are as
discussed above. Let us explain the parts of this diagram we have not yet
discussed.555We will not need every object in this diagram, but the reader
might find it helpful to know that they exist. Let $U\subset P_{n-d,d}$ be the
unipotent radical, so $U\cdot\mathrm{GL}_{n-d}(F)$ and
$U\cdot\mathrm{GL}_{d}(F)$ are both closed normal subgroups of $P_{n-d,d}$
with intersection $U$. Then by definition,
$X=\mathcal{M}_{\infty}/\underline{U}$,
$X_{+}=\mathcal{M}_{\infty}/\underline{U\cdot\mathrm{GL}_{d}(F)}$,
$X_{-}=\mathcal{M}_{\infty}/\underline{U\cdot\mathrm{GL}_{n-d}(F)}$, and
$X_{n,d}=\mathcal{M}_{\infty}/\underline{P_{n-d,d}}$, whence the upper left
square is cartesian. By [JLH21, Proposition A.3.3], the space $X_{n,d}$ admits
an explicit alternative description: it is the moduli space sending any
perfectoid space $S/C$ to (isomorphism classes of) diagrams
$\mathcal{O}(1/n)\twoheadrightarrow\mathcal{E}\hookleftarrow\mathcal{F}$ of
vector bundles over $\mathcal{X}_{S^{\flat}}$ such that
$\mathcal{E}\simeq\mathcal{O}(1/d)$ and $\mathcal{F}\simeq\mathcal{O}^{d}$ at
all geometric points, and such that
$\mathrm{coker}(\mathcal{F}\to\mathcal{E})\simeq\iota_{\ast}W$ for some rank
one projective $\mathcal{O}_{S}$-module $W$, where
$\iota:S\to\mathcal{X}_{S^{\flat}}$ is the usual divisor. The map
$\tilde{h}_{2}$ corresponds to remembering only
$\mathcal{E}\hookleftarrow\mathcal{F}$, while $\tilde{g}_{d}$ corresponds to
remembering only $\mathcal{O}(1/n)\twoheadrightarrow\mathcal{E}$. The
cartesian square spanned by $g_{d}$ and $h_{2}$ is then a consequence of
[JLH21, Proposition A.3.4]. Note that $g_{d}$ is proper and $p$-bounded, since
it comes from a proper map of rigid analytic spaces pro-étale-locally on the
target. Note also that $X_{n,d}\to W_{n,d}$ is proper and surjective, while
$X_{n,d}\to\ast$ is proper by [JLH21, Proposition A.3.2], so $W_{n,d}\to\ast$
is proper as claimed in the discussion preceding Theorem 5.2.

######  Lemma 5.3.

For any
$M_{1}\in\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(\mathrm{GL}_{n-d}(F),\mathcal{O}_{C}/p)^{\varphi}$
and
$M_{2}\in\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(\mathrm{GL}_{d}(F),\mathcal{O}_{C}/p)^{\varphi}$,
there is a natural isomorphism

$\mathcal{J}_{n,{\scalebox{0.5}{$\square$}}}(\mathrm{Ind}_{P_{n-d,d}}^{\mathrm{GL}_{n}(F)}(M_{1}\boxtimes
M_{2}))\cong
R\Gamma_{{\scalebox{0.5}{$\square$}}}(X_{n,d},\tilde{g}_{d}^{\ast}h_{1}^{\ast}M_{1}\otimes\tilde{h}_{2}^{\ast}f_{d}^{\ast}M_{2})$

in
$\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(D^{\times},\mathcal{O}_{C}/p)^{\varphi}$.

###### Proof.

Consider the commutative diagram

${{X_{n,d}}}$${{\ast}}$${{[X_{n,d}/\underline{D}_{1/n}^{\times}]}}$${{[\mathbf{P}_{C}^{n-1}/\underline{D}_{1/n}^{\times}]}}$${B\underline{D}_{1/n}^{\times}}$${B\underline{P_{n-d,d}}}$${B\underline{\operatorname{GL}_{n}(F)}}$${B\underline{\operatorname{GL}_{n-d}(F)}\times
B\underline{\operatorname{GL}_{d}(F)}}$$\scriptstyle{\tilde{\alpha}}$$\scriptstyle{\alpha}$$\scriptstyle{\overline{\pi}_{\mathrm{GH}}}$$\scriptstyle{r}$$\scriptstyle{f_{n}}$$\scriptstyle{g_{n}}$$\scriptstyle{q}$$\scriptstyle{s}$

where the square and rectangle are both cartesian. By definition,
$\mathrm{Ind}_{P_{n-d,d}}^{\mathrm{GL}_{n}(F)}(M_{1}\boxtimes
M_{2})=q_{\ast}s^{\ast}(M_{1}\boxtimes M_{2})$ and
$\mathcal{J}_{n,{\scalebox{0.5}{$\square$}}}=g_{n\ast}f_{n}^{\ast}$, so

$\displaystyle\mathcal{J}_{n,{\scalebox{0.5}{$\square$}}}(\mathrm{Ind}_{P_{n-d,d}}^{\mathrm{GL}_{n}(F)}(M_{1}\boxtimes
M_{2}))$ $\displaystyle=g_{n\ast}f_{n}^{\ast}q_{\ast}s^{\ast}(M_{1}\boxtimes
M_{2}),$ which by proper base-change transforms to
$\displaystyle=g_{n*}(\overline{\pi}_{\mathrm{GH}})_{*}r^{*}s^{*}(M_{1}\boxtimes
M_{2})$
$\displaystyle=R\Gamma_{\scalebox{0.5}{$\square$}}(X_{n,d},r^{*}s^{*}(M_{1}\boxtimes
M_{2})),$

where the $R\Gamma_{\scalebox{0.5}{$\square$}}$ on the last line denotes the
$D_{1/n}^{\times}$-equivariant cohomology. To finish the proof it remains to
identify the $D_{1/n}^{\times}$-equivariant sheaf $r^{*}s^{*}(M_{1}\boxtimes
M_{2})$ on $X_{n,d}$. After forgetting the $D_{1/n}^{\times}$-equivariance of
the sheaf, i.e. after applying $\tilde{\alpha}^{*}$, we get

$\displaystyle\tilde{\alpha}^{*}r^{*}s^{*}(M_{1}\boxtimes M_{2})$
$\displaystyle=(s\circ r\circ\tilde{\alpha})^{*}(M_{1}\boxtimes
M_{2})=((h_{1}\circ\tilde{g}_{d})\times(f_{d}\circ\tilde{h}_{2}))^{*}(M_{1}\boxtimes
M_{2})$
$\displaystyle=\tilde{g}_{d}^{\ast}h_{1}^{\ast}M_{1}\otimes\tilde{h}_{2}^{\ast}f_{d}^{\ast}M_{2},$

using that $s\circ
r\circ\tilde{\alpha}=(h_{1}\circ\tilde{g}_{d})\times(f_{d}\circ\tilde{h}_{2})$.
This gives the result. ∎

###### Proof of Theorem 5.2.

By some straightforward manipulations with the Riemann-Hilbert correspondence,
we easily reduce to proving the solid version of the theorem. In other words,
we need to show that for any
$M_{1}\in\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(\mathrm{GL}_{n-d}(F),\mathcal{O}_{C}/p)^{\varphi}$
and
$M_{2}\in\operatorname{\mathcal{D}}^{{\mathrm{sm}},a}_{\scalebox{0.5}{$\square$}}(\mathrm{GL}_{d}(F),\mathcal{O}_{C}/p)^{\varphi}$,
there is a natural isomorphism

$\mathcal{J}_{n,{\scalebox{0.5}{$\square$}}}(\mathrm{Ind}_{P_{n-d,d}}^{\mathrm{GL}_{n}(F)}(M_{1}\boxtimes
M_{2}))\cong
R\Gamma_{{\scalebox{0.5}{$\square$}}}(W_{n,d},h_{1}^{\ast}M_{1}\otimes
h_{2}^{\ast}\mathcal{J}_{d,{\scalebox{0.5}{$\square$}}}(M_{2})).$

By the previous lemma, we already have a canonical isomorphism

$\mathcal{J}_{n,{\scalebox{0.5}{$\square$}}}(\mathrm{Ind}_{P_{n-d,d}}^{\mathrm{GL}_{n}(F)}(M_{1}\boxtimes
M_{2}))\cong
R\Gamma_{{\scalebox{0.5}{$\square$}}}(X_{n,d},\tilde{g}_{d}^{\ast}h_{1}^{\ast}M_{1}\otimes\tilde{h}_{2}^{\ast}f_{d}^{\ast}M_{2}).$

We then compute that

$\displaystyle
R\Gamma_{{\scalebox{0.5}{$\square$}}}(X_{n,d},\tilde{g}_{d}^{\ast}h_{1}^{\ast}M_{1}\otimes\tilde{h}_{2}^{\ast}f_{d}^{\ast}M_{2})$
$\displaystyle\cong
R\Gamma_{{\scalebox{0.5}{$\square$}}}(W_{n,d},\tilde{g}_{d\ast}(\tilde{g}_{d}^{\ast}h_{1}^{\ast}M_{1}\otimes\tilde{h}_{2}^{\ast}f_{d}^{\ast}M_{2}))$
$\displaystyle\cong
R\Gamma_{{\scalebox{0.5}{$\square$}}}(W_{n,d},h_{1}^{\ast}M_{1}\otimes\tilde{g}_{d\ast}\tilde{h}_{2}^{\ast}f_{d}^{\ast}M_{2}))$
$\displaystyle\cong
R\Gamma_{{\scalebox{0.5}{$\square$}}}(W_{n,d},h_{1}^{\ast}M_{1}\otimes
h_{2}^{\ast}g_{d\ast}f_{d}^{\ast}M_{2})).$

Here the first isomorphism is trivial, the second follows from the projection
formula, and the third follows from proper base change. Since
$g_{d\ast}f_{d}^{\ast}M_{2}=\mathcal{J}_{d,{\scalebox{0.5}{$\square$}}}(M_{2})$
by definition, this gives the desired result. ∎

By work of Johansson-Ludwig [JLH21, Theorem A], $W_{n,1}$ is a perfectoid
space for all $n$. The proof of [JLH21, Theorem A] is a highly nontrivial
argument using the $p$-adic geometry of global Shimura varieties. In the
special case $n=2$, the first author reproved this result by purely local
methods [JLH21, §A.4]. However, the arguments in [JLH21, §A.4] easily
generalize to show that $W_{n,n-1}$ is also a perfectoid space for all $n$. It
thus seems natural to make the following conjecture.

######  Conjecture 5.4.

For all integers $0<d<n$, $W_{n,d}$ is a perfectoid space.

This conjecture implies a vanishing result.

######  Proposition 5.5.

Suppose $W_{n,d}$ is a perfectoid space. Then for any admissible smooth
representations
$\sigma_{1}\in\mathrm{Rep}_{\mathbf{F}_{p}}^{\mathrm{sm}}(\mathrm{GL}_{n-d}(F))$
and
$\sigma_{2}\in\mathrm{Rep}_{\mathbf{F}_{p}}^{\mathrm{sm}}(\mathrm{GL}_{d}(F))$,
$\mathcal{J}_{n}^{i}(\mathrm{Ind}_{P_{n-d,d}}^{\mathrm{GL}_{n}(F)}(\sigma_{1}\boxtimes\sigma_{2}))=0$
for all $i>d+n-2$.

Note that $\mathcal{J}_{n}^{i}$ trivially vanishes in degrees $>2n-2$, while
this result gives vanishing in degrees $>d+n-2=2n-2-(n-d)$, so we improve on
the trivial bound by an interval of length $n-d$. When $d=1$, this specializes
to the main vanishing theorem in [JLH21] (using of course the perfectoidness
of $W_{n,1}$ proved in loc. cit.).

###### Proof.

By the primitive comparison theorem plus some small additional arguments, we
get an almost isomorphism

$\mathcal{J}_{n}^{i}(\mathrm{Ind}_{P_{n-d,d}}^{\mathrm{GL}_{n}(F)}(\sigma_{1}\boxtimes\sigma_{2}))\otimes\mathcal{O}_{C}/p\cong
H^{i}(W_{n,d},h_{1}^{\ast}\sigma_{1}\otimes
h_{2}^{\ast}\mathcal{J}_{d}(\sigma_{2})\otimes\mathcal{O}^{+}/p).$

It thus suffices to show that $H^{i}(W_{n,d},h_{1}^{\ast}\sigma_{1}\otimes
h_{2}^{\ast}\mathcal{J}_{d}(\sigma_{2})\otimes\mathcal{O}^{+}/p)$ is almost
zero in the claimed range of degrees. Recall that
$\mathcal{J}_{d}(\sigma_{2})$ is concentrated in degrees $\leq 2d-2$. Note
also that $W_{n,d}$ is perfectoid by assumption, so the functor
$H^{i}(W_{n,d},-\otimes\mathcal{O}^{+}/p)$ on overconvergent étale
$\mathbf{F}_{p}$-sheaves is almost zero for all $i>\dim W_{n,d}=n-d$, by the
usual almost equality of étale and analytic cohomology. The evident spectral
sequence then gives almost vanishing in degrees $>2d-2+n-d=d+n-2$, whence the
claim. ∎

When $n=3$, $W_{3,1}$ and $W_{3,2}$ are both perfectoid spaces. Combining this
observation with the previous proposition, the right-exactness of
$\mathcal{J}_{3}^{4}$, and the classification of irreducible admissible
representations of $\mathrm{GL}_{3}(F)$ [Her11, Theorem 1.1], we get the
following vanishing result.

######  Corollary 5.6.

Let $\pi$ be an irreducible admissible representation of $\mathrm{GL}_{3}(F)$
which is not supersingular, and whose underlying vector space is infinite-
dimensional. Then $\mathcal{J}^{4}(\pi)=0$.

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