Similarly, we consider a variant in Chow cohomology (§3.5):

$\mathrm{FT}^{\mathrm{arith}}\colon\operatorname{CH}^{*}({}^{\dagger}{V};\mathbf{Q})\rightarrow\operatorname{CH}^{*}({}^{\dagger}{\widehat{V}};\mathbf{Q})$

given by

$\alpha\mapsto(-1)^{d}\operatorname{pr}_{1!}(\operatorname{pr}_{0}^{*}(\alpha)\cup\operatorname{ev}^{*}\xi).$

####  9.5.3. Basic properties

The following properties of the homogeneous arithmetic Fourier transform
follow by the same arguments as in the non-homogeneous case (which are found
in [FYZ22, §8.2]).

######  Notation 9.5.1.

For $\alpha\in\operatorname{CH}_{i}({}^{\dagger}{V})$ and
$\beta\in\operatorname{CH}^{j}({}^{\dagger}{V})$, we write

$\langle\alpha,\beta\rangle\coloneqq\pi_{*}(\alpha\cap\beta)\in\operatorname{CH}_{i-j}(T)$

where $\pi\colon{}^{\dagger}{V}\rightarrow T$ is the projection.

######  Lemma 9.5.2 (Plancherel property).

Let $\alpha_{1}\in\operatorname{CH}_{i}({}^{\dagger}{V})$ and
$\beta_{2}\in\operatorname{CH}^{j}({}^{\dagger}{\widehat{V}})$. Then

$\langle\alpha_{1},\mathrm{FT}^{\mathrm{arith}}(\beta_{2})\rangle=\langle\mathrm{FT}^{\mathrm{arith}}(\alpha_{1}),\beta_{2}\rangle\in\operatorname{CH}_{i-j}(T).$

######  Lemma 9.5.3 (Involutivity).

We have
$\mathrm{FT}^{\mathrm{arith}}_{\widehat{V}}\circ\mathrm{FT}^{\mathrm{arith}}_{V}=q^{d}$
where $d$ is the rank of $V$.

###  9.6. Compatibility of motivic and arithmetic Fourier transforms

We establish the compatibility of the motivic Fourier transform with
arithmetic Fourier transform under the motivic sheaf-cycle correspondence.

####  9.6.1. Setup

Let $Y$ be a derived Artin stack locally of finite type over a field. Let
$p\colon E\to Y$ be a vector bundle222222i.e., a derived vector bundle of
amplitude $[0,0]$.. Suppose $c=(c_{0},c_{1})\colon C\to Y\times Y$ is a
correspondence of derived Artin stacks and we are given an isomorphism of
vector bundles over $C$

$\iota\colon c_{0}^{*}E\cong c_{1}^{*}E.$ (9.6.1)

Let $C_{E}$ be the total space of
$c_{0}^{*}E\stackrel{{\scriptstyle\iota}}{{\cong}}c_{1}^{*}E$. For
$i\in\\{0,1\\}$ we let $e_{i}\colon C_{E}\cong c_{i}^{*}E\to E$ be the
corresponding projection map. Then we have a map of correspondences

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}}$$\textstyle{C_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{C}}$$\scriptstyle{e_{0}}$$\scriptstyle{e_{1}}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}}$$\textstyle{Y}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{0}}$$\scriptstyle{c_{1}}$$\textstyle{Y}$
(9.6.2)

such that both squares are Cartesian.

The above data induces a correspondence $\widehat{e}\colon
C_{\widehat{E}}\to\widehat{E}\times\widehat{E}$ by passing to the dual vector
bundles.

Consider the Frobenius twisted correspondence map
$c^{(1)}=(\operatorname{Frob}\circ c_{0},c_{1})\colon C^{(1)}\rightarrow
Y\times Y$. Similarly, we define $C^{(1)}_{E}$ (a self-correspondence of $E$)
and $C^{(1)}_{\widehat{E}}$ (a self-correspondence of $\widehat{E}$). Recall
notation (§6.4.2)

$\operatorname{Sht}(C)\coloneqq\operatorname{Fix}(C^{(1)}),\quad\operatorname{Sht}(C_{E})\coloneqq\operatorname{Fix}(C^{(1)}_{E}),\quad\operatorname{Sht}(C_{\widehat{E}})\coloneqq\operatorname{Fix}(C^{(1)}_{\widehat{E}}).$
(9.6.3)

The projections

$\pi:\operatorname{Sht}(C_{E})\to\operatorname{Sht}(C),\quad\widehat{\pi}:\operatorname{Sht}(C_{\widehat{E}})\to\operatorname{Sht}(C)$
(9.6.4)

are étale $\mathbf{F}_{q}$-vector space bundles over $\operatorname{Sht}(C)$
that are dual to each other.

#### 9.6.2.

Let
$\mathcal{K}\in\mathcal{D}_{\mathrm{mot},\mathrm{gm}}({{}^{\dagger}{E}};\mathbf{Q})$
and
$\mathfrak{c}\in\mathrm{Corr}_{{}^{\dagger}{C_{E}}}(\mathcal{K},\mathcal{K}{{\scriptstyle\langle-i\rangle}})$.

On one hand, we can form the Frobenius-twisted trace (§6.4.3)

$\operatorname{Tr}^{\operatorname{Sht}}(\mathfrak{c})\coloneqq\operatorname{Tr}(\mathfrak{c}^{(1)})\in\operatorname{CH}_{i}({}^{\dagger}{\operatorname{Sht}(C_{E})})$
(9.6.5)

where $\mathfrak{c}^{(1)}$ is the cohomological correspondence

$\mathfrak{c}^{(1)}\colon e_{0}^{*}\operatorname{Frob}_{E}^{*}\mathcal{K}\cong
e_{0}^{*}\mathcal{K}\xrightarrow{\mathfrak{c}}e_{1}^{!}\mathcal{K}{\scriptstyle\langle-i\rangle}$
(9.6.6)

supported on ${}^{\dagger}{C_{E}^{(1)}}$ (see §6.4.2).

On the other hand, we can first apply the homogeneous motivic Fourier
transform to get a cohomological correspondence

$\operatorname{FT}_{C_{E}}(\mathfrak{c})\in\mathrm{Corr}_{{}^{\dagger}{C_{\widehat{E}}}}(\operatorname{FT}_{E}(\mathcal{K}),\operatorname{FT}_{E}(\mathcal{K}))$

given by the composite

$\widehat{e}_{0}^{*}\operatorname{FT}_{E}(\mathcal{K})\cong\operatorname{FT}_{C_{E}}(e_{0}^{*}\mathcal{K})\xrightarrow{\operatorname{FT}_{C_{E}}(\mathfrak{c})}\operatorname{FT}_{C_{E}}(e_{1}^{!}\mathcal{K}{\scriptstyle\langle-i\rangle})\cong\widehat{e}_{1}^{!}\operatorname{FT}_{E}(\mathcal{K}){{\scriptstyle\langle-i\rangle}},$
(9.6.7)

where we used the commutativity of the Fourier transform with base change
(§8.2.6). We can then (using that $\operatorname{FT}(\mathcal{K})$ is
geometric, thanks to Corollary 8.4.2) form the Frobenius-twisted trace

$\operatorname{Tr}^{\operatorname{Sht}}(\operatorname{FT}_{C_{E}}(\mathfrak{c}))\coloneqq\operatorname{Tr}(\operatorname{FT}_{C_{E}}(\mathfrak{c})^{(1)})\in\operatorname{CH}_{i}({}^{\dagger}{\operatorname{Sht}(C_{\widehat{E}})}).$
(9.6.8)

The two constructions (9.6.5) and (9.6.8) agree up to the arithmetic Fourier
transform:

######  Theorem 9.6.1.

In the above situation, we have

$\operatorname{Tr}^{\operatorname{Sht}}(\operatorname{FT}_{C_{E}}(\mathfrak{c}))=\operatorname{FT}^{\mathrm{arith}}_{\operatorname{Sht}(C_{E})}(\operatorname{Tr}^{\operatorname{Sht}}(\mathfrak{c}))\in\operatorname{CH}_{i}({}^{\dagger}{\operatorname{Sht}(C_{\widehat{E}})}).$
(9.6.9)

###### Proof.

The version for $\ell$-adic coefficients is [FYZ22, Theorem 8.3.2], and the
proof here is similar with a few modifications. The renormalized homogeneous
Fourier transform (resp. arithmetic homogeneous Fourier transform) is the
composition of three steps:

  1. (a)

pullback along
${}^{\dagger}{E}\leftarrow{}^{\dagger}{E}\times_{Y}{}^{\dagger}{\widehat{E}}$
(resp. pullback along
${}^{\dagger}{\operatorname{Sht}(E)}\leftarrow{}^{\dagger}{\operatorname{Sht}(E)}\times_{\operatorname{Sht}(C)}{}^{\dagger}{\operatorname{Sht}(\widehat{E})}$),

  2. (b)

tensor with $\mathscr{P}_{E}[r-1]$ (resp. multiply by
$(-1)^{r}\operatorname{ev}^{*}\xi$) where $r=\operatorname{\mathrm{rk}}(E)$,

  3. (c)

pushforward along
${}^{\dagger}{E}\times_{Y}{}^{\dagger}{\widehat{E}}\rightarrow{}^{\dagger}{\widehat{E}}$
(resp. pushforward along
${}^{\dagger}{\operatorname{Sht}(E)}\times_{\operatorname{Sht}(C)}{}^{\dagger}{\operatorname{Sht}(\widehat{E})}\rightarrow{}^{\dagger}{\operatorname{Sht}(\widehat{E})}$).

It suffices to show that each of these steps is compatible with the formation
of the trace. The first two are easy:

  * •

Since the pullback in (a) is smooth, the compatibility there follows from
Proposition 6.2.2.

  * •

The formation of trace takes the operation of tensoring with a self-
correspondence of a local system to the operation of multiplication by the
trace. Hence (b) follows from the computation that the Frobenius trace
function of $\mathscr{P}_{E}$ is $-\operatorname{ev}^{*}\xi$.

For (c), we need to show that pushforward through the projection
$\operatorname{pr}_{1}\colon{}^{\dagger}{E}\times_{Y}{}^{\dagger}{\widehat{E}}\rightarrow{}^{\dagger}{\widehat{E}}$
commutes with formation of trace. From the map of correspondences

${E}$${C_{E}}$${E}$${{}^{\dagger}{E}}$${{}^{\dagger}{C_{E}}}$${{}^{\dagger}{E}}$$\scriptstyle{f}$$\scriptstyle{e_{0}}$$\scriptstyle{e_{1}}$$\scriptstyle{f}$$\scriptstyle{f}$$\scriptstyle{{}^{\dagger}{e}_{0}}$$\scriptstyle{{}^{\dagger}{e}_{1}}$

we have maps of correspondences

$\mathrm{Corr}_{{}^{\dagger}{C_{E}}}(\mathcal{K},\mathcal{K}{\scriptstyle\langle-i\rangle})\xrightarrow{f^{*}}\mathrm{Corr}_{C_{E}}(f^{*}\mathcal{K},f^{*}\mathcal{K}{\scriptstyle\langle-i\rangle})$

and

$\mathrm{Corr}_{C_{E}}(\mathcal{K},\mathcal{K}{\scriptstyle\langle-i\rangle})\xrightarrow{f_{!}}\mathrm{Corr}_{{}^{\dagger}{C_{E}}}(f_{!}\mathcal{K},f_{!}\mathcal{K}{\scriptstyle\langle-i\rangle}).$

The endofunctor
$f_{!}f^{*}\colon\mathcal{D}_{\mathrm{mot}}({{}^{\dagger}{E}};\mathbf{Q})\rightarrow\mathcal{D}_{\mathrm{mot}}({{}^{\dagger}{E}};\mathbf{Q})$
is given by tensoring with $f_{!}\mathbf{Q}_{E}$. Since $f$ is a
$\mathbf{G}_{m}$-torsor, this implies that
$\operatorname{Tr}(f_{!}f^{*}\operatorname{\mathfrak{c}})=(q-1)\operatorname{Tr}(\operatorname{\mathfrak{c}})$,
which agrees with
$\operatorname{Fix}(f)_{*}\operatorname{Fix}(f)^{!}(\operatorname{Tr}(\operatorname{\mathfrak{c}}))$
since $\operatorname{Fix}(f)$ is a $\mathbf{G}_{m}(\mathbf{F}_{q})$-torsor.
Replacing $\operatorname{\mathfrak{c}}$ by $f^{*}\operatorname{\mathfrak{c}}$,
it therefore suffices to show that the projection $\operatorname{pr}_{1}\colon
E\times_{Y}\widehat{E}\rightarrow\widehat{E}$ commutes with formation of
trace. This follows from Lemma 9.6.2. ∎

######  Lemma 9.6.2.

In the situation of (9.6.2), let
$\mathcal{K}\in\mathcal{D}_{\mathrm{mot},\mathrm{gm}}({E};\mathbf{Q})$ and
$\mathfrak{c}\in\mathrm{Corr}_{C_{E}}(\mathcal{K},\mathcal{K}{\scriptstyle\langle-i\rangle})$.
Let
$\operatorname{Sht}(\operatorname{pr})\colon\operatorname{Sht}(C_{E})\to\operatorname{Sht}(C)$
be the induced map on fixed points of $C_{E}^{(1)}$ and $C^{(1)}$, which is an
étale $\mathbf{F}_{q}$-vector space bundle (in particular a finite morphism).
Then

$\operatorname{Tr}(\operatorname{pr}_{C!}(\mathfrak{c}))=\operatorname{Sht}(\operatorname{pr})_{*}(\operatorname{Tr}(\mathfrak{c}^{(1)}))\in\operatorname{CH}_{i}(\operatorname{Sht}(C)).$
(9.6.10)

###### Proof.

Note that this does not follow from Proposition 6.2.1 since
$\operatorname{pr}$ is far from proper. The analogous result for $\ell$-adic
sheaves is [FYZ22, Lemma 8.3.3], and the proof here is essentially the same:
compactify the map of correspondences, use the compatibility of trace and
pushforward on the compactification, and then show that the boundary
contribution vanishes. As some references need to be replaced, we will spell
out the argument.

We compactify the map of correspondences (9.6.2) to

$\textstyle{\overline{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\operatorname{pr}}}$$\textstyle{C_{\overline{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\operatorname{pr}}_{C}}$$\scriptstyle{\overline{e}_{0}}$$\scriptstyle{\overline{e}_{1}}$$\textstyle{\overline{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\operatorname{pr}}}$$\textstyle{Y}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{0}}$$\scriptstyle{c_{1}}$$\textstyle{Y}$
(9.6.11)

where $\overline{E}\coloneqq\mathbf{P}(E\oplus\mathcal{O})\to Y$ and
$C_{\overline{E}}\coloneqq\mathbf{P}(c_{0}^{*}E\oplus\mathcal{O})\stackrel{{\scriptstyle\iota}}{{\cong}}\mathbf{P}(c_{1}^{*}E\oplus\mathcal{O})\to
C$ is the pullback projective bundle over $C$. Then $C_{\overline{E}}$ is a
self-correspondence of $\overline{E}$ with a proper map to $C$. Let
$E_{\infty}\coloneqq\overline{E}-E$ be the divisor at infinity, which is
isomorphic to $\mathbf{P}(E)$. Similarly define $C_{E_{\infty}}\coloneqq
C_{\overline{E}}-C_{E}$, which is a self-correspondence of $E_{\infty}$. Let
$C^{(1)}_{\overline{E}}$ and $C^{(1)}_{E_{\infty}}$ be the twists by Frobenius
as in §6.4.2. Since the vertical maps in (9.6.11) are proper, Proposition
6.2.1 implies that $\overline{\operatorname{pr}}_{C!}$ is compatible with
formation of traces.

Let $j:E\hookrightarrow\overline{E}$ and $j_{C}:C_{E}\hookrightarrow
C_{\overline{E}}$ be the open inclusions. The map of correspondences

${E}$${C_{E}}$${E}$${\overline{E}}$${C_{\overline{E}}}$${\overline{E}}$$\scriptstyle{j}$$\scriptstyle{e_{0}}$$\scriptstyle{e_{1}}$$\scriptstyle{j_{C}}$$\scriptstyle{j}$$\scriptstyle{\overline{e}_{0}}$$\scriptstyle{\overline{e}_{1}}$

has both squares Cartesian, so it is left pushable. Therefore the pushforward
cohomological correspondence

$\overline{\mathfrak{c}}\coloneqq
j_{C!}(\mathfrak{c})\in\mathrm{Corr}_{C_{\overline{E}}}(j_{!}\mathcal{K},j_{!}\mathcal{K}{\scriptstyle\langle-i\rangle})$
(9.6.12)

is defined. It remains to show that
$\operatorname{Sht}(\operatorname{pr})_{*}\operatorname{Tr}(\overline{c})=\operatorname{Sht}(\operatorname{pr})_{*}\operatorname{Tr}(\mathfrak{c})$,
which amounts to the vanishing of the contribution from the boundary
correspondence. The rest of the argument is exactly the same as the
corresponding step in the proof of [FYZ22, Lemma 8.3.3], except using Theorem
7.5.1 instead of [Var07] to see that the contribution from the boundary
correspondence vanishes, because $C^{(1)}_{\overline{E}}$ is contracting near
$E_{\infty}$ by Example 7.2.2. ∎

##  10\. Generic modularity of higher theta series

In this section we will assemble the preceding theory to prove the main
result, Theorem 1.2.1.

###  10.1. Notation

We fix the following notation throughout the section.

We let $\nu\colon X^{\prime}\rightarrow X$ be an étale double cover of smooth
projective curves over a finite field $\mathbf{F}_{q}$ of characteristic
$p>2$, and $\sigma\colon X^{\prime}\rightarrow X^{\prime}$ be the non-trivial
automorphism over $X$. We let $\operatorname{Frob}$ denote the $q$-power
Frobenius.

For a torsion coherent sheaf $Q$ on a curve $X^{\prime}$ we let $D_{Q}$ be its
scheme-theoretic support, viewed as a divisor on $X^{\prime}$, and $|Q|\subset
X^{\prime}$ its set-theoretic support.

Let $n\in\mathbf{Z}_{\geq 1}$. The (smooth, classical, $1$-Artin) stack
$\operatorname{Bun}_{GU(n)}$ parametrizes triples
$(\mathcal{F},\mathfrak{L},h)$, where $\mathcal{F}$ is a vector bundle on
$X^{\prime}$ of rank $n$, $\mathfrak{L}$ is a line bundle on $X$, and
$h:\mathcal{F}\stackrel{{\scriptstyle\sim}}{{\to}}\sigma^{*}\mathcal{F}^{\vee}\otimes\nu^{*}\mathfrak{L}$
is an $\mathfrak{L}$-twisted Hermitian structure (i.e.,
$\sigma^{*}h^{\vee}=h$). The corresponding moduli space of shtukas is denoted
$\operatorname{Sht}_{GU(n)}$.

###  10.2. Higher theta series

We briefly summarize the construction of higher theta series on
$\operatorname{Sht}_{GU(n)}$, which can be found in [FYZ21b, §4]. Let
$m\in\mathbf{Z}_{\geq 1}$. The stack $\operatorname{Bun}_{GU^{-}(2m)}$
parametrizes triples $(\mathcal{G},\mathfrak{M},h)$, where $\mathcal{G}$ is a
vector bundle on $X^{\prime}$ of rank $2m$, $\mathfrak{M}$ is a line bundle on
$X$, and
$h:\mathcal{G}\stackrel{{\scriptstyle\sim}}{{\to}}\sigma^{*}\mathcal{G}^{\vee}\otimes\nu^{*}\mathfrak{M}$
is an $\mathfrak{M}$-twisted skew-Hermitian structure (i.e.,
$\sigma^{*}h^{\vee}=-h$). Alternatively, we can think of $h$ as an
$\mathcal{O}_{X^{\prime}}$-bilinear perfect pairing

$(\cdot,\cdot)_{h}:\mathcal{G}\times\sigma^{*}\mathcal{G}\to\nu^{*}(\mathfrak{M}\otimes\omega_{X})$
(10.2.1)

satisfying
$(\sigma^{*}\beta,\sigma^{*}\alpha)_{h}=-\sigma^{*}(\alpha,\beta)_{h}$ for
local sections $\alpha$ and $\beta$ of $\mathcal{G}$ respectively.

Let $\operatorname{Bun}_{\widetilde{P}_{m}}$ be the moduli stack of quadruples
$(\mathcal{G},\mathfrak{M},h,\mathcal{E})$ where
$(\mathcal{G},\mathfrak{M},h)\in\operatorname{Bun}_{GU^{-}(2m)}$, and
$\mathcal{E}\subset\mathcal{G}$ is a Lagrangian sub-bundle (i.e.,
$\mathcal{E}$ has rank $m$ and the composition
$\mathcal{E}\subset\mathcal{G}\xrightarrow{h}\sigma^{*}\mathcal{G}^{\vee}\otimes\nu^{*}\mathfrak{M}\to\sigma^{*}\mathcal{E}^{\vee}\otimes\nu^{*}\mathfrak{M}$
is zero). Thus $\widetilde{P}_{m}$ corresponds to the Siegel parabolic of
$GU^{-}(2m)$.

Assume now that $1\leq m\leq n$. In [FYZ21b, §4.6], we defined for each $r\geq
0$ and $m\leq n$ a _higher theta series_ , which is a function

$\widetilde{Z}^{r}_{m}\colon\operatorname{Bun}_{\widetilde{P}_{m}}(k)\to\operatorname{CH}_{r(n-m)}(\operatorname{Sht}^{r}_{GU(n)}).$

We repeat the brief recap of the definition of $\widetilde{Z}^{r}_{m}$ from
[FYZ22, §2]. Let
$(\mathcal{G},\mathfrak{M},h,\mathcal{E})\in\operatorname{Bun}_{\widetilde{P}_{m}}(k)$
and set $\mathfrak{L}\coloneqq\omega_{X}\otimes\mathfrak{M}$. Let
$\operatorname{Sht}^{r}_{U(n),\mathfrak{L}}\subset\operatorname{Sht}^{r}_{GU(n)}$
be the moduli stack of rank $n$ Hermitian shtukas

$\mathcal{F}_{\bullet}=((x_{i}),(\mathcal{F}_{i}),(f_{i}),\varphi:\mathcal{F}_{r}\stackrel{{\scriptstyle\sim}}{{\to}}{}^{\tau}\mathcal{F}_{0})$

on $X^{\prime}$ with $r$ legs and similitude line bundle $\mathfrak{L}$. For a
vector bundle $\mathcal{E}$ on $X^{\prime}$ of rank $m$, the special cycle
$\mathcal{Z}^{r}_{\mathcal{E},\mathfrak{L}}$ parametrizes a point
$\mathcal{F}_{\bullet}$ of $\operatorname{Sht}^{r}_{U(n),\mathfrak{L}}$, and
maps $t_{i}:\mathcal{E}\to\mathcal{F}_{i}$ for each $0\leqslant i\leqslant r$,
compatible with the shtuka structure on $\mathcal{F}_{\bullet}$. For a
Hermitian map
$a:\mathcal{E}\to\sigma^{*}\mathcal{E}^{\vee}\otimes\nu^{*}\mathfrak{L}$, let
$\mathcal{Z}^{r}_{\mathcal{E},\mathfrak{L}}(a)$ be the open-closed substack of
$\mathcal{Z}^{r}_{\mathcal{E},\mathfrak{L}}$ consisting of
$(\mathcal{F}_{\bullet},t_{\bullet})$ such that the Hermitian form on
$\mathcal{F}_{\bullet}$ induces the Hermitian map $a$ on $\mathcal{E}$ via
$t_{\bullet}$. Let
$\zeta:\mathcal{Z}^{r}_{\mathcal{E},\mathfrak{L}}(a)\to\operatorname{Sht}^{r}_{U(n),\mathfrak{L}}\subset\operatorname{Sht}^{r}_{GU(n)}$
be the map forgetting $t_{\bullet}$, which is known to be finite [FYZ21a,
Proposition 7.5] and unramified.

In [FYZ21b, Definition 4.8] there is constructed a virtual fundamental class
$[\mathcal{Z}^{r}_{\mathcal{E},\mathfrak{L}}(a)]\in\operatorname{CH}_{r(n-m)}(\mathcal{Z}^{r}_{\mathcal{E},\mathfrak{L}}(a))$,
although the interpretation as a derived fundamental class in [FYZ21b, §5,6]
will be more useful in the proofs. Pushing forward along $\zeta$, we get Chow
classes

$\zeta_{*}[\mathcal{Z}^{r}_{\mathcal{E},\mathfrak{L}}(a)]\in\operatorname{CH}_{r(n-m)}(\operatorname{Sht}^{r}_{U(n),\mathfrak{L}}).$

The value of $\widetilde{Z}^{r}_{m}$ on
$(\mathcal{G},\mathfrak{M},h,\mathcal{E})$ (recall
$\mathfrak{M}=\omega_{X}^{-1}\otimes\mathfrak{L}$), which we henceforth
abbreviate as $(\mathcal{G},\mathcal{E})$, is defined as

$\widetilde{Z}^{r}_{m}(\mathcal{G},\mathcal{E})=\chi(\det\mathcal{E})q^{n(\deg\mathcal{E}-\deg\mathfrak{L}-\deg\omega_{X})/2}\sum_{a\in\mathcal{A}_{\mathcal{E},\mathfrak{L}}(k)}\psi(\langle{e_{\mathcal{G},\mathcal{E}},a}\rangle)\zeta_{*}[\mathcal{Z}^{r}_{\mathcal{E},\mathfrak{L}}(a)],$
(10.2.2)

where:

  * •

$\chi:\operatorname{Pic}_{X^{\prime}}(k)\to\overline{\mathbf{Q}}^{\times}$ is
a character whose restriction to $\operatorname{Pic}_{X}(k)$ is the $n$th
power of the quadratic character $\operatorname{Pic}_{X}(k)\to\\{\pm 1\\}$
corresponding to the double cover $X^{\prime}/X$ by class field theory.

  * •

$\psi:\mathbf{F}_{q}\to\overline{\mathbf{Q}}^{\times}$ is a nontrivial
additive character.

  * •

The sum is indexed over $\mathcal{A}_{\mathcal{E},\mathfrak{L}}(k)$, the set
of Hermitian maps
$a:\mathcal{E}\to\sigma^{*}\mathcal{E}^{\vee}\otimes\nu^{*}\mathfrak{L}$.

  * •

Let $\mathcal{E}^{\prime}=\mathcal{G}/\mathcal{E}$. The pairing
$(\cdot,\cdot)_{h}$ in (10.2.1) induces a perfect pairing
$\mathcal{E}\times\sigma^{*}\mathcal{E}^{\prime}\to\nu^{*}\mathfrak{L}$, which
identifies $\mathcal{E}^{\prime}$ with
$\sigma^{*}\mathcal{E}^{*}\otimes\nu^{*}\mathfrak{L}$. We thus have a short
exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\sigma^{*}\mathcal{E}^{*}\otimes\nu^{*}\mathfrak{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

and
$e_{\mathcal{G},\mathcal{E}}\in\operatorname{Ext}^{1}(\sigma^{*}\mathcal{E}^{*}\otimes\nu^{*}\mathfrak{L},\mathcal{E})$
is its extension class.

  * •

The pairing $\langle{-,-}\rangle$ is the Serre duality pairing between
$\operatorname{Ext}^{1}(\sigma^{*}\mathcal{E}^{*}\otimes\nu^{*}\mathfrak{L},\mathcal{E})$
and
$\operatorname{Hom}(\mathcal{E},\sigma^{*}\mathcal{E}^{\vee}\otimes\nu^{*}\mathfrak{L})$.

###  10.3. The generic modularity theorem

The Modularity Conjecture of [FYZ21b] predicts that $\widetilde{Z}_{m}^{r}$
descends to a function on $\operatorname{Bun}_{GU^{-}(2m)}(k)$. As explained
just after [FYZ21b, Conjecture 4.15], this can be reformulated as the
assertion that $\widetilde{Z}^{r}_{m}(\mathcal{G},\mathcal{E})$ is actually
independent of the choice of Lagrangian sub-bundle
$\mathcal{E}\subset\mathcal{G}$. We will prove this statement after
restriction to the generic locus (cf. §1.2.2):

######  Theorem 10.3.1.

For any $\mathcal{G}\in\operatorname{Bun}_{GU^{-}(2m)}(k)$ and any Lagrangian
sub-bundles
$\widetilde{\mathcal{E}}_{1},\widetilde{\mathcal{E}}_{2}\subset\mathcal{G}$,
we have

$\widetilde{Z}_{m}^{r}(\mathcal{G},\widetilde{\mathcal{E}}_{1})=\widetilde{Z}_{m}^{r}(\mathcal{G},\widetilde{\mathcal{E}}_{2})\in\operatorname{CH}_{r(n-m)}(\operatorname{Sht}^{r}_{GU(n)}\times_{(X^{\prime})^{r}}\eta^{r}).$
(10.3.1)

where $\eta^{r}$ is as in §1.2.2.

####  10.3.1. Reduction to transverse Lagrangians

As explained in [FYZ22, §2.2], Theorem 10.3.1 is reduced to the case where the
sub-bundles $\widetilde{\mathcal{E}}_{1},\widetilde{\mathcal{E}}_{2}$ are
_transverse_ in the sense that their intersection within $\mathcal{G}$ is the
zero section. Henceforth we assume this to be the case.

####  10.3.2. Reduction to Harder–Narasimhan truncations

Given a Harder–Narasimhan polygon $\mu$ for $GU(n)$, we have a
Harder–Narasimhan truncation
$\operatorname{Sht}^{r,\leq\mu}_{GU(n)}\hookrightarrow\operatorname{Sht}^{r}_{GU(n)}$.
Because

$\operatorname{CH}_{*}(\operatorname{Sht}^{r}_{GU(n)}\times_{(X^{\prime})^{r}}\eta^{r})\cong\varprojlim_{\mu}\operatorname{CH}_{*}(\operatorname{Sht}^{r,\leq\mu}_{GU(n)}\times_{(X^{\prime})^{r}}\eta^{r}),$

it suffices to show (10.3.1) after restriction to this truncation. Henceforth
we fix a Harder–Narasimhan polygon $\mu$ and write
$S=\operatorname{Bun}_{GU(n)}^{\leqslant\mu}$ for the corresponding open
substack of $\operatorname{Bun}_{GU(n)}$. As in [FYZ22, §9.1.2], we write
$\operatorname{Hk}_{S}^{r}\coloneqq h_{0}^{-1}(S)\cap\ldots\cap h_{r}^{-1}(S)$
where
$h_{i}\colon\operatorname{Hk}_{GU(n)}^{r}\rightarrow\operatorname{Bun}_{GU(n)}$
are the “leg maps”.

For a space over $\operatorname{Bun}_{GU(n)}$ such as
$\operatorname{Sht}_{GU(n)}^{r}$ or the special cycles on it, we write
$(-)^{\leq\mu}$ for the pullback to $S$.

###  10.4. Transverse Lagrangians ansatz

Let $\mathcal{G}\in\operatorname{Bun}_{GU^{-}(2m)}(k)$. For notational
simplicity, we assume the similitude line bundle of $\mathcal{G}$ is trivial,
therefore the skew-Hermitian form reads
$h_{\mathcal{G}}:\mathcal{G}\stackrel{{\scriptstyle\sim}}{{\to}}\sigma^{*}\mathcal{G}^{*}$.
The general case has the same content.

Thanks to the transversality assumption from §10.3.1, we have a commutative
diagram (cf. [FYZ22, §2.3.3])

${\widetilde{\mathcal{E}}_{1}}$${\sigma^{*}\widetilde{\mathcal{E}}_{2}^{*}}$${Q_{2}}$${\mathcal{G}}$${\mathcal{G}^{\sharp}}$${Q}$${\widetilde{\mathcal{E}}_{2}}$${\sigma^{*}\widetilde{\mathcal{E}}_{1}^{*}}$${Q_{1}}$$\scriptstyle{b_{12}}$$\scriptstyle{i_{1}}$$\scriptstyle{\iota_{2}}$$\scriptstyle{b_{21}}$$\scriptstyle{i_{2}}$$\scriptstyle{\iota_{1}}$
(10.4.1)

where the horizontal rows and diagonal sequences are short exact sequences of
coherent sheaves on $X^{\prime}$. Here $b_{12}$ is the composition

$\widetilde{\mathcal{E}_{1}}\rightarrow\mathcal{G}\stackrel{{\scriptstyle\sim}}{{\to}}\sigma^{*}\mathcal{G}^{*}\rightarrow\sigma^{*}\widetilde{\mathcal{E}}_{2}^{*}.$

The map $b_{21}$ is defined similarly. The maps $\iota_{1}$ and $\iota_{2}$
are isomorphisms of torsion sheaves on $X^{\prime}$. As in [FYZ22, §2.3.3],
the duality between $Q_{1}$ and $Q_{2}$ equips $Q$ with two Hermitian
structures $h_{12}\colon Q\xrightarrow{\sim}\sigma^{*}Q^{*}$ and $h_{21}\colon
Q\xrightarrow{\sim}\sigma^{*}Q^{*}$, related by $h_{12}=-h_{21}$.

For each $i\in\\{1,2\\}$ let
$\mathcal{E}_{i}\hookrightarrow\widetilde{\mathcal{E}_{i}}$ be a sub-sheaf
with cokernel a torsion coherent sheaf $\mathcal{T}_{i}$ on $X^{\prime}$. Let
$\mathcal{T}^{*}_{i}=\operatorname{RHom}(\mathcal{T}_{i},\mathcal{O}_{X^{\prime}})[1]$
be its dual torsion sheaf on $X^{\prime}$. Therefore,
$\widetilde{\mathcal{E}_{i}}$ is the saturation of $\mathcal{E}_{i}$ in
$\mathcal{G}$, and we have the diagram below

${\mathcal{E}_{1}}$${\widetilde{\mathcal{E}_{1}}}$${\mathcal{G}}$${\sigma^{*}\widetilde{\mathcal{E}}_{2}^{*}}$${\sigma^{*}\mathcal{E}_{2}^{*}}$$\scriptstyle{\mathcal{T}_{1}}$$\scriptstyle{Q}$$\scriptstyle{\sigma^{*}\mathcal{T}_{2}^{*}}$
(10.4.2)

where the arrows are labeled by their cokernels.

####  10.4.1. Assumptions on $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$

We make the following assumptions:

  1. (a)

The supports $|Q|,|\mathcal{T}_{1}|,|\mathcal{T}_{2}|$ are disjoint after
mapping to $X$.

  2. (b)

For all $\mathcal{F}\in
S(\overline{k})=\operatorname{Bun}_{U(n)}^{\leq\mu}(\overline{k})$ we have for
$i=1,2$

$\operatorname{Ext}^{1}_{X_{\overline{k}}}(\mathcal{F}^{*},\mathcal{E}_{i}^{*})=0.$
(10.4.3)

These conditions can always be arranged, as discussed in [FYZ22, Remark
10.1.1]. Note that by the dualities in [FYZ22, (10.1.3)], (10.4.3) is
equivalent to

$\operatorname{Hom}_{X_{\overline{k}}}(\mathcal{F}^{*},\sigma^{*}\mathcal{E}_{i})=0$
(10.4.4)

for all $\mathcal{F}\in\operatorname{Bun}_{U(n)}^{\leq\mu}(\overline{k})$ and
$i=1,2$.

Let

$\displaystyle\widetilde{Q}_{1}\coloneqq
Q^{*}\oplus\mathcal{T}_{1}^{*}\oplus\sigma^{*}\mathcal{T}_{2},$ (10.4.5)
$\displaystyle\widetilde{Q}_{2}\coloneqq\sigma^{*}Q\oplus\sigma^{*}\mathcal{T}_{1}\oplus\mathcal{T}_{2}^{*}.$
(10.4.6)

From (10.4.2) and the disjointness assumption in §10.4.1, we have short exact
sequences

$\displaystyle
0\rightarrow\sigma^{*}\mathcal{E}_{2}\rightarrow\mathcal{E}_{1}^{*}\rightarrow\widetilde{Q}_{1}\rightarrow
0,$ (10.4.7) $\displaystyle
0\rightarrow\sigma^{*}\mathcal{E}_{1}\rightarrow\mathcal{E}_{2}^{*}\rightarrow\widetilde{Q}_{2}\rightarrow
0.$ (10.4.8)

###  10.5. Moduli spaces

We will use the same moduli spaces as in [FYZ22, §9.1, §9.2]:

  1. (a)

For $i\in\\{0,r\\}$: $U_{i},V_{i},W_{i}$, which are derived vector bundles
over $S$; and their respective dual derived vector bundles
$W_{i}^{\perp},\widehat{V}_{i},U_{i}^{\perp}$.

  2. (b)

For $i\in\\{0,r\\}$: $\widetilde{U}_{i},\widetilde{V}_{i},\widetilde{W}_{i}$,
which are derived vector bundles over $\operatorname{Hk}_{S}^{r}$ obtained by
pulling back $U_{i},V_{i},W_{i}$ respectively along $h_{i}$; and their
respective dual derived vector bundles
$\widetilde{W}_{i}^{\perp},\widehat{\widetilde{V}}_{i},\widetilde{U}_{i}^{\perp}$.

  3. (c)

The Hecke stacks
$\operatorname{Hk}_{U}^{\perp},\operatorname{Hk}_{U}^{\sharp},\operatorname{Hk}_{V}^{\flat},\operatorname{Hk}_{V}^{\sharp},\operatorname{Hk}_{W}^{\flat},\operatorname{Hk}_{W}^{\sharp}$,
which are derived vector bundles over $\operatorname{Hk}_{S}^{r}$; and their
respective dual derived vector bundles
$\operatorname{Hk}_{W^{\perp}}^{\sharp},\operatorname{Hk}_{W^{\perp}}^{\flat},\operatorname{Hk}_{\widehat{V}}^{\sharp},\operatorname{Hk}_{\widehat{V}}^{\flat},\operatorname{Hk}_{U^{\perp}}^{\sharp},\operatorname{Hk}_{U^{\perp}}^{\flat}$.

We give an informal summary of the definitions. Recall that for an animated
$\mathbf{F}_{q}$-algebra $R$, an $R$-point of $S$ is a Hermitian bundle
$\mathcal{F}$ on $X^{\prime}_{R}$. In these terms,

  * •

The fiber of $U_{i}$ over $\mathcal{F}\in S(R)$ is
$\operatorname{RHom}_{X_{R}^{\prime}}(\mathcal{F}^{*},\mathcal{E}_{1}^{*}\otimes
R)$.

  * •

The fiber of $V_{i}$ over $\mathcal{F}\in S(R)$ is
$\operatorname{RHom}_{X_{R}^{\prime}}(\mathcal{F}^{*},\widetilde{Q}_{1}\otimes
R)$.

  * •

The fiber of $W_{i}$ over $\mathcal{F}\in S(R)$ is
$\operatorname{RHom}_{X_{R}^{\prime}}(\mathcal{F}^{*},\sigma^{*}\mathcal{E}_{2}[1]\otimes
R)$.

  * •

The fiber of $U_{i}^{\perp}$ over $\mathcal{F}\in S(R)$ is
$\operatorname{RHom}_{X_{R}^{\prime}}(\mathcal{F}^{*},\mathcal{E}_{2}^{*}\otimes
R)$.

  * •

The fiber of $\widehat{V}_{i}$ over $\mathcal{F}\in S(R)$ is
$\operatorname{RHom}_{X_{R}^{\prime}}(\mathcal{F}^{*},\widetilde{Q}_{2}\otimes
R)$.

  * •

The fiber of $W_{i}^{\perp}$ over $\mathcal{F}\in S(R)$ is
$\operatorname{RHom}_{X_{R}^{\prime}}(\mathcal{F}^{*},\sigma^{*}\mathcal{E}_{1}[1]\otimes
R)$.

Note that the definitions of the six spaces above are, in fact, independent of
$i$. However, it the notation is useful for indexing purposes.

For an animated $\mathbf{F}_{q}$-algebra $R$, an $R$-point of
$\operatorname{Hk}_{S}^{r}$ is a sequence of modifications of Hermitian
bundles $(\mathcal{F}_{0}\dashrightarrow\ldots\dashrightarrow\mathcal{F}_{r})$
on $X_{R}^{\prime}$ that we abbreviate $\mathcal{F}_{\star}$. For
$i\in\\{0,r\\}$ and
$?\in\\{U_{i},V_{i},W_{i},U_{i}^{\perp},\widehat{V}_{i},W_{i}^{\perp}\\}$, the
fiber of $\widetilde{?}$ over
$(\mathcal{F}_{\star})\in\operatorname{Hk}_{S}^{r}(R)$ is obtained by
replacing $\mathcal{F}$ with $\mathcal{F}_{i}$ in the descriptions of $?(R)$
above.

From $\mathcal{F}_{\star}\in\operatorname{Hk}_{S}^{r}(R)$ we can define
perfect complexes $\mathcal{F}_{\bullet}^{\flat}$ and $F_{\bullet}^{\sharp}$
on $X_{R}^{\prime}$ as in [FYZ22, §9.1.3, §9.2.2]. For $?\in\\{U,V,W\\}$ the
fiber of $\operatorname{Hk}_{?}^{\flat}$ is obtained by replacing
$\mathcal{F}$ with $\mathcal{F}_{\bullet}^{\flat}$ in the description of
$?(R)$ above, while the fiber of $\operatorname{Hk}_{?}^{\sharp}$ over
$\mathcal{F}_{\star}\in\operatorname{Hk}_{S}^{r}(R)$ is obtained by replacing
$\mathcal{F}$ with $\mathcal{F}_{\bullet}^{\sharp}$ in the description of
$?(R)$ above.

######  Remark 10.5.1.

The vanishing assumption (10.4.3) implies that $U,V$ and $W$ are all classical
vector bundles over $S$, and we have a short exact sequence of classical
vector bundles over $S$

$0\to U_{i}\to V_{i}\to W_{i}\to 0.$ (10.5.1)

Similarly, we have a short exact sequence of classical vector bundles over $S$

$0\to U^{\bot}_{i}\to\widehat{V}_{i}\to W^{\bot}_{i}\to 0.$ (10.5.2)

Thus we have a pair of commutative diagrams:

${\operatorname{Hk}_{U}^{\flat}}$${\widetilde{U}_{0}}$${\widetilde{U}_{r}}$${U_{0}}$${\operatorname{Hk}_{U}^{\sharp}}$${U_{r}}$${\operatorname{Hk}_{V}^{\flat}}$${\widetilde{V}_{0}}$${\widetilde{V}_{r}}$${V_{0}}$${\operatorname{Hk}_{V}^{\sharp}}$${V_{r}}$${\operatorname{Hk}_{W}^{\flat}}$${\widetilde{W}_{0}}$${\widetilde{W}_{r}}$${W_{0}}$${\operatorname{Hk}_{W}^{\sharp}}$${W_{r}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\widetilde{a}_{0}}$$\scriptstyle{\widetilde{a}_{r}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,.5,0}\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}f}$$\scriptstyle{h_{0}^{U}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\widetilde{f}_{0}}$$\scriptstyle{\widetilde{a}_{r}^{\prime}}$$\scriptstyle{\widetilde{a}_{0}^{\prime}}$$\scriptstyle{\widetilde{f}_{r}}$$\scriptstyle{h_{r}^{U}}$$\scriptstyle{f_{0}}$$\scriptstyle{f^{\sharp}}$$\scriptstyle{f_{r}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21875,0.33203125,0.046875}\color[rgb]{0.21875,0.33203125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.21875,0.33203125,0.046875}\widetilde{b}_{0}}$$\scriptstyle{\widetilde{b}_{r}}$$\scriptstyle{g}$$\scriptstyle{\widetilde{g}_{0}}$$\scriptstyle{\widetilde{b}_{r}^{\prime}}$$\scriptstyle{h_{0}^{V}}$$\scriptstyle{\widetilde{b}_{0}^{\prime}}$$\scriptstyle{\widetilde{g}_{r}}$$\scriptstyle{h_{r}^{V}}$$\scriptstyle{g_{0}}$$\scriptstyle{g^{\sharp}}$$\scriptstyle{g_{r}}$$\scriptstyle{\widetilde{c}_{0}}$$\scriptstyle{\widetilde{c}_{r}}$$\scriptstyle{\widetilde{c}_{r}^{\prime}}$$\scriptstyle{h_{0}^{W}}$$\scriptstyle{\widetilde{c}_{0}^{\prime}}$$\scriptstyle{h_{r}^{W}}$
${\operatorname{Hk}_{U^{\perp}}^{\flat}}$${\widetilde{U}_{0}^{\perp}}$${\widetilde{U}_{r}^{\perp}}$${U_{0}^{\perp}}$${\operatorname{Hk}_{U^{\perp}}^{\sharp}}$${U_{r}^{\perp}}$${\operatorname{Hk}_{\widehat{V}}^{\flat}}$${\widehat{\widetilde{V}}_{0}}$${\widehat{\widetilde{V}}_{r}}$${\widehat{V}_{0}}$${\operatorname{Hk}_{\widehat{V}}^{\sharp}}$${\widehat{V}_{r}}$${\operatorname{Hk}_{W^{\perp}}^{\flat}}$${\widetilde{W}_{0}^{\perp}}$${\widetilde{W}_{r}^{\perp}}$${W_{0}^{\perp}}$${\operatorname{Hk}_{W^{\perp}}^{\sharp}}$${W_{r}^{\perp}}$$\scriptstyle{\widetilde{a}_{0}^{\perp}}$$\scriptstyle{\widetilde{a}_{r}^{\perp}}$$\scriptstyle{f^{\perp}}$$\scriptstyle{h_{0}^{U^{\bot}}}$$\scriptstyle{\widetilde{f}_{0}^{\perp}}$$\scriptstyle{(\widetilde{a}_{r}^{\prime})^{\perp}}$$\scriptstyle{(\widetilde{a}_{0}^{\prime})^{\perp}}$$\scriptstyle{\widetilde{f}_{r}^{\perp}}$$\scriptstyle{h_{r}^{U^{\bot}}}$$\scriptstyle{f_{0}^{\perp}}$$\scriptstyle{(f^{\sharp})^{\perp}}$$\scriptstyle{f_{r}^{\perp}}$$\scriptstyle{\widetilde{\beta}_{0}}$$\scriptstyle{\widetilde{\beta}_{r}}$$\scriptstyle{g^{\bot}}$$\scriptstyle{h_{0}^{\widehat{V}}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\widetilde{g}_{0}^{\perp}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0.21875,0.33203125,0.046875}\color[rgb]{0.21875,0.33203125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.21875,0.33203125,0.046875}\widetilde{\beta}^{\prime}_{r}}$$\scriptstyle{\widetilde{\beta}^{\prime}_{0}}$$\scriptstyle{\widetilde{g}_{r}^{\perp}}$$\scriptstyle{h_{r}^{\widehat{V}}}$$\scriptstyle{g_{0}^{\perp}}$$\scriptstyle{\definecolor[named]{.}{rgb}{1,.5,0}\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}(g^{\sharp})^{\perp}}$$\scriptstyle{g_{r}^{\perp}}$$\scriptstyle{\widetilde{c}_{0}^{\perp}}$$\scriptstyle{\widetilde{c}_{r}^{\perp}}$$\scriptstyle{h_{0}^{W^{\bot}}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\widetilde{c}_{r}^{\prime})^{\perp}}$$\scriptstyle{(\widetilde{c}_{0}^{\prime})^{\perp}}$$\scriptstyle{h_{r}^{W^{\bot}}}$
(10.5.3)

In each diagram:

  * •

The maps in the columns come from exact triangles of perfect complexes.

  * •

The three diamonds in the middle are derived Cartesian.

  * •

The four parallelograms on the left and right sides are derived Cartesian.

The diagram on the right is dual to the diagram on the left. The duality
exchanges $U$ with $W^{\perp}$, $V$ with $\widehat{V}$, and $W$ with
$U^{\perp}$, and exchanges $\flat$ and $\sharp$ superscripts. Sample examples
of dual morphisms are colored with the same color. By [FYZ22, §9.3], each of
these diagrams is globally presented, so that we may apply the results of §9.2
to them.

###  10.6. Calculation of motivic Fourier transforms

We refer to the diagrams in (10.5.3). By [FYZ22, Corollary 9.1.4], the map
$a_{r}$ is quasi-smooth, hence it has a relative fundamental class, which
defines as in §6.3 a cohomological correspondence

$\operatorname{\mathfrak{c}}_{U}=[a_{r}]\in\mathrm{Corr}_{{}^{\dagger}{\operatorname{Hk}^{\flat}_{U}}}(\mathbf{Q}_{U_{0}},\mathbf{Q}_{U_{r}}{\scriptstyle\langle-d(a_{r})\rangle}).$
(10.6.1)

Similarly, the relative fundamental class of $a^{\bot}_{r}$ defines a
cohomological correspondence

$\operatorname{\mathfrak{c}}_{U^{\bot}}=[a^{\bot}_{r}]\in\mathrm{Corr}_{{}^{\dagger}{\operatorname{Hk}^{\flat}_{U^{\bot}}}}(\mathbf{Q}_{U^{\bot}_{0}},\mathbf{Q}_{U^{\bot}_{r}}{\scriptstyle\langle-d(a^{\bot}_{r})\rangle}).$
(10.6.2)

By [FYZ22, Corollary 9.1.5] the pushforward of cohomological correspondences
(§4.3.1) along the morphism of correspondences
$f:\operatorname{Hk}^{\flat}_{U}\to\operatorname{Hk}^{\flat}_{V}$ is defined,
giving

$f_{!}(\operatorname{\mathfrak{c}}_{U})\in\mathrm{Corr}_{{}^{\dagger}{\operatorname{Hk}^{\flat}_{V}}}(f_{0!}\mathbf{Q}_{U_{0}},f_{r!}\mathbf{Q}_{U_{r}}{\scriptstyle\langle-d(a_{r})\rangle}).$
(10.6.3)

Similarly,

$f^{\bot}_{!}(\operatorname{\mathfrak{c}}_{U^{\bot}})\in\mathrm{Corr}_{{}^{\dagger}{\operatorname{Hk}^{\flat}_{\widehat{V}}}}(f^{\bot}_{0!}\mathbf{Q}_{U^{\bot}_{0}},f^{\bot}_{r!}\mathbf{Q}_{U^{\bot}_{r}}{\scriptstyle\langle-d(a^{\bot}_{r})\rangle})$
(10.6.4)

is defined.

For a $\mathbf{G}_{m}$-equivariant cohomological correspondence
$\operatorname{\mathfrak{c}}$ on a derived vector bundle $E$, we denote by
${}^{\dagger}{\operatorname{\mathfrak{c}}}$ the descended cohomological
correspondence on ${}^{\dagger}{E}$. Recall the notion of homogeneous Fourier
transform of cohomological correspondences from §9.4.1. We have (simplifying
some dimensions as in [FYZ22, §9.4])

$\mathrm{FT}^{\mathrm{ren}}_{\operatorname{Hk}^{\flat}_{V}}({}^{\dagger}{f}_{!}({}^{\dagger}{\operatorname{\mathfrak{c}}_{U}}))\in\mathrm{Corr}_{{}^{\dagger}{\operatorname{Hk}^{\flat}_{\widehat{V}}}}(\mathrm{FT}^{\mathrm{ren}}_{V_{0}}(f_{0!}\mathbf{Q}_{U_{0}}),\mathrm{FT}^{\mathrm{ren}}_{V_{r}}(f_{r!}\mathbf{Q}_{U_{r}}){\scriptstyle\langle
d(\widetilde{b}_{r})-d(a_{r})\rangle}).$ (10.6.5)

Since $U^{\bot}$ is the orthogonal complement of $U$ relative to $V$ (in the
derived sense), by §8.2.7 we have canonical isomorphisms for $i=0,r$:

$\displaystyle\mathrm{FT}^{\mathrm{ren}}_{V_{i}}(f_{i!}\mathbf{Q}_{U_{i}})\cong(f^{\bot}_{i})_{!}\mathbf{Q}_{U^{\bot}_{i}}{\scriptstyle[\operatorname{rank}(\mathcal{V})]}{\scriptstyle\langle-\operatorname{rank}(\mathcal{U})\rangle}.$

Note the shift and twist on the right side is the same for $i=0$ and $i=r$.
Therefore we may view (simplifying some dimensions as in [FYZ22, §9.4])

$\mathrm{FT}^{\mathrm{ren}}_{\operatorname{Hk}^{\flat}_{V}}({}^{\dagger}{f}_{!}({}^{\dagger}{\operatorname{\mathfrak{c}}_{U}}))\in\mathrm{Corr}_{{}^{\dagger}{\operatorname{Hk}^{\flat}_{\widehat{V}}}}((f^{\bot}_{0})_{!}\mathbf{Q}_{U^{\bot}_{0}},(f^{\bot}_{r})_{!}\mathbf{Q}_{U^{\bot}_{r}}{\scriptstyle\langle-d(a_{r}^{\bot})\rangle}).$

######  Theorem 10.6.1.

Recall the shift and twist notation from §6.5. Let $\pi_{i}\colon
U_{i}\rightarrow S$ be the bundle projection. Then we have

$\mathbb{T}_{[d(f_{0})+d(\pi_{0})](d(\pi_{0}))}\mathrm{FT}^{\mathrm{ren}}_{{}^{\dagger}{\operatorname{Hk}_{V}^{\flat}}}({}^{\dagger}{f}_{!}({}^{\dagger}{\operatorname{\mathfrak{c}}_{U}}))=({}^{\dagger}{f^{\perp}})_{!}({}^{\dagger}{\operatorname{\mathfrak{c}}_{U^{\perp}}})\in\mathrm{Corr}_{{}^{\dagger}{\operatorname{Hk}_{\widehat{V}}^{\flat}}}((f_{0}^{\perp})_{!}\mathbf{Q}_{U^{\bot}_{0}},(f_{r}^{\perp})_{!}\mathbf{Q}_{U^{\bot}_{r}}{\scriptstyle\langle-d(a_{r}^{\perp})\rangle}).$

###### Proof.

Let
$\mathfrak{s}\in\mathrm{Corr}_{{}^{\dagger}{\operatorname{Hk}^{r}_{S}}}(\mathbf{Q}_{S},\mathbf{Q}_{S}{\scriptstyle\langle-d(h_{r})\rangle})$
be the cohomological correspondence obtained from the relative fundamental
class of $h_{r}$ as in §6.3. Let

$\pi:\operatorname{Hk}^{\flat}_{U}\to\operatorname{Hk}_{S}^{r},\quad\pi^{\bot}:\operatorname{Hk}^{\flat}_{U^{\bot}}\to\operatorname{Hk}_{S}^{r},$

be the bundle projections viewed as maps of correspondences, and also recall
the maps of correspondences

$\displaystyle
z^{\perp}:\operatorname{Hk}^{r}_{S}\to\operatorname{Hk}^{\flat}_{W^{\perp}},\quad
g^{\perp}:\operatorname{Hk}_{\widehat{V}}^{\flat}\to\operatorname{Hk}_{W^{\perp}}^{\flat}.$

The proof is completed by the sequence of equalities of cohomological
correspondences

$\displaystyle\mathbb{T}_{[d(f_{0})+d(\pi_{0})](d(\pi_{0}))}\mathrm{FT}^{\mathrm{ren}}({}^{\dagger}{f}_{!}{}^{\dagger}{\operatorname{\mathfrak{c}}_{U}})$
$\displaystyle\stackrel{{\scriptstyle(1)}}{{=}}\mathbb{T}_{[d(f_{0})+d(\pi_{0})](d(\pi_{0}))}\mathrm{FT}^{\mathrm{ren}}({}^{\dagger}{f}_{!}{}^{\dagger}{\pi}^{*}{}^{\dagger}{\mathfrak{s}})\stackrel{{\scriptstyle(2)}}{{=}}({}^{\dagger}{g^{\perp}})^{*}({}^{\dagger}{z^{\perp}})_{!}\mathrm{FT}^{\mathrm{ren}}(\mathfrak{s})$
$\displaystyle\stackrel{{\scriptstyle(3)}}{{=}}({}^{\dagger}{g^{\perp}})^{*}({}^{\dagger}{z^{\perp}})_{!}\mathfrak{s}\stackrel{{\scriptstyle(4)}}{{=}}({}^{\dagger}{f^{\perp}})_{!}({}^{\dagger}{\pi^{\perp}})^{*}\mathfrak{s}\stackrel{{\scriptstyle(5)}}{{=}}({}^{\dagger}{f^{\perp}})_{!}{}^{\dagger}{\operatorname{\mathfrak{c}}_{U^{\perp}}}.$

Here:

  1. (1), (5)

follow from the $\mathbf{G}_{m}$-equivariant identifications

$\pi^{*}\mathfrak{s}=\operatorname{\mathfrak{c}}_{U},\quad(\pi^{\bot})^{*}\mathfrak{s}=\operatorname{\mathfrak{c}}_{U^{\bot}}$

which are proved exactly as in [FYZ22, Lemma 9.4.2].

  2. (2)

involves two applications of Proposition 9.4.1, namely

$\mathbb{T}_{[d(f_{0})]}\mathrm{FT}^{\mathrm{ren}}\circ{}^{\dagger}{f}_{!}=({}^{\dagger}{g}^{\perp})^{*}\circ\mathrm{FT}^{\mathrm{ren}},\quad\mathbb{T}_{[d(\pi_{0})](d(\pi_{0}))}\mathrm{FT}^{\mathrm{ren}}\circ{}^{\dagger}{\pi}^{*}={}^{\dagger}{z}^{\perp}_{!}\circ\mathrm{FT}^{\mathrm{ren}}.$

  3. (3)

is the trivial equality
$\mathfrak{s}=\mathrm{FT}^{\mathrm{ren}}_{\operatorname{Hk}_{S}^{r}}(\mathfrak{s})$,
where $\operatorname{Hk}_{S}^{r}$ is regarded as a trivial vector bundle over
itself.

  4. (4)

follows from Theorem 4.4.2. (Note that the hypotheses of Theorem 4.4.2 hold in
this situation by [FYZ22, Corollary 9.1.5].)

∎

###  10.7. Calculations with the homogeneous arithmetic Fourier transform

We denote

$\displaystyle X^{\circ}$ $\displaystyle\coloneqq$ $\displaystyle
X-\nu(|Q|\cup|\mathcal{T}_{1}|\cup|\mathcal{T}_{2}|)=X-\nu(|\widetilde{Q}_{1}|)=X-\nu(|\widetilde{Q}_{2}|);$
(10.7.1) $\displaystyle X^{\prime\circ}$ $\displaystyle\coloneqq$
$\displaystyle\nu^{-1}(X^{\circ}).$ (10.7.2)

For a stack $A$ over $X^{\circ}$, we denote

$A^{\circ}\coloneqq A\times_{X^{r}}(X^{\circ})^{r}.$ (10.7.3)

In particular,
$\operatorname{Hk}_{S}^{r,\circ}\subset\operatorname{Hk}_{S}^{r}$ denotes the
open substack where the legs are all disjoint from
$|\widetilde{Q}_{1}|\cup|\widetilde{Q}_{2}|$.

By [FYZ22, Lemma 10.13], for each $i$ the restriction
$\widetilde{b}_{i}^{\circ}\colon\operatorname{Hk}_{V}^{\flat,\circ}\rightarrow\widetilde{V}^{\circ}_{i}$
of $\widetilde{b}_{i}$ and the restriction
$\widetilde{b}^{\prime\circ}_{i}\colon\widetilde{V}^{\circ}_{i}\to\operatorname{Hk}_{V}^{\sharp,\circ}$
of $\widetilde{b}^{\prime}_{i}$ are isomorphisms. This implies [FYZ22,
Corollary 10.1.4] that the projection map
$\operatorname{Sht}^{r,\circ}_{V}\to\operatorname{Sht}^{r,\circ}_{S}=\operatorname{Sht}^{r,\leqslant\mu,\circ}_{U(n)}$
is an étale $\mathbf{F}_{q}$-vector space bundle. Hence the theory of the
homogeneous arithmetic Fourier transform (§9.5) applies to it.

####  10.7.1. Virtual fundamental classes

By [FYZ22, Remark 9.1.1] the spaces $U_{i}$ from §10.5 can be viewed as the
derived fiber of the derived Hitchin stack $\mathscr{M}_{H_{1},H_{2}}$ from
[FYZ21b, §5] over $\\{\mathcal{E}_{1}\\}\times
S\rightarrow\operatorname{Bun}_{\operatorname{GL}(m)^{\prime}}\times\operatorname{Bun}_{U(n)}$,
where $H_{1}=\operatorname{GL}(m)^{\prime}$ and $H_{2}=U(n)$. Similarly, by
[FYZ22, Remark 9.1.2] $\operatorname{Hk}_{U}^{\flat}$ is an open substack of
the derived fiber of the derived Hecke stack
$\operatorname{Hk}_{\mathscr{M}_{H_{1},H_{2}}}^{r}$ from [FYZ21b, §5] over
$\\{\mathcal{E}_{1}\\}\times
S\rightarrow\operatorname{Bun}_{\operatorname{GL}(m)^{\prime}}\times\operatorname{Bun}_{U(n)}$.
Therefore, the derived fibered product

${\operatorname{Sht}_{U}^{r}}$${\operatorname{Hk}_{U}^{\flat}}$${U_{0}}$${U_{0}\times
U_{r}}$$\scriptstyle{(a_{0},a_{r})}$$\scriptstyle{(\operatorname{Id},\operatorname{Frob})}$

is equipped with an open embedding in
$\operatorname{Sht}_{\mathscr{M}_{H_{1},H_{2}}}^{r}$, and in particular is of
virtual dimension $d(a_{r})$. We then have two natural cycles in
$\operatorname{CH}_{d(a_{r})}(\operatorname{Sht}_{U}^{r})$:

  1. (a)

The intrinsic derived fundamental class
$[\operatorname{Sht}_{U}^{r}]\in\operatorname{CH}_{d(a_{r})}(\operatorname{Sht}_{U}^{r})$.

  2. (b)

The trace of the cohomological correspondence
$\operatorname{\mathfrak{c}}_{U}$ (calculated using the canonical Weil
structure), denoted
$\operatorname{Tr}^{\operatorname{Sht}}(\operatorname{\mathfrak{c}}_{U})\in\operatorname{CH}_{d(a_{r})}(\operatorname{Sht}_{U}^{r})$
(cf. §6.4.2).

We assemble the earlier results to calculate the trace of our cohomological
correspondences. The assumptions (10.4.3) imply that the maps
$U_{i}\rightarrow S$ and $U_{i}^{\perp}\rightarrow S$ are smooth, hence
$U_{i}$ and $U_{i}^{\perp}$ are smooth. Then by Corollary 6.4.1 we have

$\operatorname{Tr}^{\operatorname{Sht}}(\operatorname{\mathfrak{c}}_{U})=[\operatorname{Sht}_{U}^{r}]\in\operatorname{CH}_{d(a_{r})}(\operatorname{Sht}_{U}^{r}).$
(10.7.4)

In particular, $\operatorname{Sht}_{U}^{r}$ is an open substack of
$\operatorname{Sht}_{\mathcal{M}_{\mathcal{E}_{1}}}^{r}$, so it is quasi-
smooth and $[\operatorname{Sht}_{U}^{r}]$ is the restriction of what was
called $[\mathcal{Z}_{\mathcal{E}_{1}}^{r}]$ in [FYZ21b].

Similarly, we have

$\operatorname{Tr}^{\operatorname{Sht}}(\operatorname{\mathfrak{c}}_{U^{\perp}})=[\operatorname{Sht}_{U^{\perp}}^{r}]\in\operatorname{CH}_{d(a_{r}^{\perp})}(\operatorname{Sht}_{U^{\perp}}^{r}),$
(10.7.5)

where $\operatorname{Sht}_{U^{\perp}}^{r}$ is defined by the derived Cartesian
square

${\operatorname{Sht}_{U^{\perp}}^{r}}$${\operatorname{Hk}_{U^{\perp}}^{\flat}}$${U_{0}^{\perp}}$${U_{0}^{\perp}\times
U_{r}^{\perp}}$$\scriptstyle{(a_{0}^{\perp},a_{r}^{\perp})}$$\scriptstyle{(\operatorname{Id},\operatorname{Frob})}$

Next, the assumptions (10.4.4) imply that the maps $f_{i}\colon
U_{i}\rightarrow V_{i}$,
$f\colon\operatorname{Hk}_{U}^{\flat}\rightarrow\operatorname{Hk}_{V}^{\flat}$,
$f_{i}^{\perp}\colon U_{i}^{\perp}\rightarrow\widehat{V}_{i}$, and
$f^{\perp}\colon\operatorname{Hk}_{U^{\perp}}^{\flat}\rightarrow\operatorname{Hk}_{\widehat{V}}^{\flat}$
are all closed embeddings. Then Proposition 6.2.1 applies to give

$\operatorname{Tr}^{\operatorname{Sht}}(f_{!}\operatorname{\mathfrak{c}}_{U})=\operatorname{Sht}(f)_{!}\operatorname{Tr}^{\operatorname{Sht}}(\operatorname{\mathfrak{c}}_{U})\stackrel{{\scriptstyle(\ref{eq:
trace of
cc_U})}}{{=}}\operatorname{Sht}(f)_{!}[\operatorname{Sht}_{U}^{r}]\in\operatorname{CH}_{d(a_{r})}(\operatorname{Sht}_{V}^{r}),$
(10.7.6)

where we write
$\operatorname{Sht}(f)\coloneqq\operatorname{Fix}(f^{(1)})\colon\operatorname{Sht}_{U}^{r}\rightarrow\operatorname{Sht}_{V}^{r}$
for the map induced by taking fixed points of the twisted cohomological
correspondence $\mathfrak{\operatorname{\mathfrak{c}}}_{U}^{(1)}$, and
similarly for other cohomological correspondences. We similarly have

$\operatorname{Tr}^{\operatorname{Sht}}(f_{!}^{\perp}\operatorname{\mathfrak{c}}_{U^{\perp}})=\operatorname{Sht}(f^{\perp})_{!}\operatorname{Tr}^{\operatorname{Sht}}(\operatorname{\mathfrak{c}}_{U^{\perp}})\stackrel{{\scriptstyle(\ref{eq:
trace of cc_U
perp})}}{{=}}\operatorname{Sht}(f^{\perp})_{!}[\operatorname{Sht}_{U^{\perp}}^{r}]\in\operatorname{CH}_{d(a_{r}^{\perp})}(\operatorname{Sht}_{\widehat{V}}^{r}).$
(10.7.7)

######  Notation 10.7.1.

For an $\mathbf{F}_{q}$-vector space stack $Y\rightarrow T$ as in §9.5.1 and a
class $\alpha\in\operatorname{CH}_{*}(Y)$ (resp. $\operatorname{CH}^{*}(Y)$),
we denote

${}^{\dagger}{\alpha}=\operatorname{Av}(\alpha)\coloneqq\frac{1}{q-1}\operatorname{pr}_{!}(\alpha)\in\operatorname{CH}_{*}({}^{\dagger}{Y})\quad\text{
(resp. $\operatorname{CH}^{*}({}^{\dagger}{Y})$)}$

where $\operatorname{pr}\colon Y\rightarrow{}^{\dagger}{Y}$ is the quotient
map.

######  Example 10.7.2 (Homogeneous cycles).

We say that $\alpha$ on $Y$ is _homogeneous_ if
$\alpha=\operatorname{pr}^{*}({}^{\dagger}{\alpha})$. Note that
$[\operatorname{Sht}^{r}_{U}]\in\operatorname{CH}_{d(a_{r})}(\operatorname{Sht}_{U}^{r})$
is homogeneous, hence
$\operatorname{Sht}(f)_{!}[\operatorname{Sht}^{r}_{U}]\in\operatorname{CH}_{d(a_{r})}(\operatorname{Sht}_{V}^{r})$
is homogeneous.

####  10.7.2. Arithmetic Fourier transform of cycles

Recall that
$\operatorname{Sht}^{r,\circ}_{V}\to\operatorname{Sht}^{r,\circ}_{S}$ is an
étale $\mathbf{F}_{q}$-vector space bundle. We now relate the cycle classes
(10.7.6) and (10.7.7) under the homogeneous arithmetic Fourier transform on
$\operatorname{Sht}^{r,\circ}_{V}$ as defined in §9.5.

######  Theorem 10.7.3.

We have

$\mathrm{FT}^{\mathrm{arith}}(\operatorname{Sht}(f)^{\circ}_{!}({}^{\dagger}{[\operatorname{Sht}_{U}^{r,\circ}]}))=(-1)^{d(U/S)+d(f_{0})}q^{d(U/S)}\cdot\operatorname{Sht}(f^{\perp})^{\circ}_{!}({}^{\dagger}{[\operatorname{Sht}_{U^{\perp}}^{r,\circ}]})\in\operatorname{CH}_{d(a_{r})}({}^{\dagger}{\operatorname{Sht}_{\widehat{V}}^{r,\circ}}).$

Here
$\operatorname{Sht}(f)^{\circ}:\operatorname{Sht}_{U}^{r,\circ}\to\operatorname{Sht}^{r,\circ}_{V}$
is the restriction of $\operatorname{Sht}(f)$, and similarly for
$\operatorname{Sht}(f^{\perp})^{\circ}$. We use the same notation for induced
maps on the homogeneous quotients ${}^{\dagger}{(-)}$. (We are using [FYZ22,
Remark 10.2.2] to match the degrees of homology.)

###### Proof.

We apply Theorem 9.6.1 with $E=V$, $C_{E}=\operatorname{Hk}_{V}^{\flat,\circ}$
and
$\operatorname{\mathfrak{c}}=(f_{!}\operatorname{\mathfrak{c}}_{U})|_{\operatorname{Hk}_{V}^{\flat,\circ}}$.
Then Theorem 9.6.1 tells us that

$\mathrm{FT}^{\mathrm{arith}}_{\operatorname{Sht}_{V}^{r,\circ}}\left(\operatorname{Tr}^{\operatorname{Sht}}({}^{\dagger}{f}_{!}{}^{\dagger}{\operatorname{\mathfrak{c}}_{U}})|_{{}^{\dagger}{\operatorname{Sht}^{r,\circ}_{V}}}\right)=\left(\operatorname{Tr}^{\operatorname{Sht}}\mathrm{FT}^{\mathrm{ren}}_{\operatorname{Hk}_{V}^{\flat}}({}^{\dagger}{f}_{!}{}^{\dagger}{\operatorname{\mathfrak{c}}_{U}})\right)|_{{}^{\dagger}{\operatorname{Sht}^{r,\circ}_{\widehat{V}}}}\in\operatorname{CH}_{d(a_{r})}({}^{\dagger}{\operatorname{Sht}_{\widehat{V}}^{r,\circ}}).$
(10.7.8)

By Theorem 10.6.1 we have

$\mathrm{FT}^{\mathrm{ren}}_{\operatorname{Hk}_{V}^{\flat}}({}^{\dagger}{f}_{!}{}^{\dagger}{\operatorname{\mathfrak{c}}_{U}})=\mathbb{T}_{[-d(U/S)-d(f_{0})](-d(U/S))}({}^{\dagger}{f}^{\perp})_{!}{}^{\dagger}{\operatorname{\mathfrak{c}}_{U^{\perp}}}.$

Putting this into (10.7.8) and then taking the trace, using (6.5.1), (10.7.6)
and (10.7.7), yields the result. ∎

####  10.7.3. Test functions

We introduce some notation for functions on $\operatorname{Sht}_{V}$ and
$\operatorname{Sht}_{\widehat{V}}$. The decompositions
$\widetilde{Q}_{1}\coloneqq
Q^{*}\oplus\mathcal{T}_{1}^{*}\oplus\sigma^{*}\mathcal{T}_{2}$ and
$\widetilde{Q}_{2}\coloneqq\sigma^{*}Q\oplus\sigma^{*}\mathcal{T}_{1}\oplus\mathcal{T}_{2}^{*}$
induce the following decompositions defined in [FYZ22, §10.2.1] (with the same
notation):

$\operatorname{Sht}_{V}^{r}=\operatorname{Sht}_{V^{(0)}}^{r}\times_{\operatorname{Sht}_{S}^{r}}\operatorname{Sht}_{V^{(1)}}^{r}\times_{\operatorname{Sht}_{S}^{r}}\operatorname{Sht}_{V^{(2)}}^{r},$
$\operatorname{Sht}_{\widehat{V}}^{r}=\operatorname{Sht}_{\widehat{V}^{(0)}}^{r}\times_{\operatorname{Sht}_{S}^{r}}\operatorname{Sht}_{\widehat{V}^{(1)}}^{r}\times_{\operatorname{Sht}_{S}^{r}}\operatorname{Sht}_{\widehat{V}^{(2)}}^{r}.$

We note that $\operatorname{Sht}_{\widehat{V}^{(i)}}^{r,\circ}$ is dual to
$\operatorname{Sht}_{V^{(i)}}^{r,\circ}$ as $\mathbf{F}_{q}$-vector spaces
over $\operatorname{Sht}^{r,\circ}_{S}$ in the sense of §9.5.1.

We denote
$\mathfrak{q}_{12}\colon\operatorname{Sht}^{r}_{V^{(0)}}\rightarrow\underline{\mathbf{F}_{q}}$
and
$\mathfrak{q}_{21}\colon\operatorname{Sht}^{r}_{V^{(0)}}\rightarrow\underline{\mathbf{F}_{q}}$
the two quadratic forms induced by the Hermitian structures $h_{12}$ and
$h_{21}$ on $Q$ from [FYZ22, §2.3.3], respectively. Namely,
$\mathfrak{q}_{12}$ is the composition

$\mathfrak{q}_{12}:\operatorname{Sht}^{r}_{V^{(0)}}\xrightarrow{(\operatorname{Id},h_{12})}\operatorname{Sht}^{r}_{V^{(0)}}\times_{\operatorname{Sht}^{r}_{S}}\operatorname{Sht}^{r}_{\widehat{V}^{(0)}}\to\underline{\mathbf{F}_{q}},$
(10.7.9)

and similarly for $\mathfrak{q}_{21}$. They are related by
$\mathfrak{q}_{12}=-\mathfrak{q}_{21}$. Recall the additive character
$\psi\colon\mathbf{F}_{q}\rightarrow\overline{\mathbf{Q}}^{\times}$ from
§10.2.

  * •

We let $\mathfrak{q}_{12}^{*}\psi$ be the pullback of $\psi$ to
$\operatorname{Sht}_{V^{(0)}}^{r}$ via $\mathfrak{q}_{12}$, and similarly for
$\mathfrak{q}_{21}$. Abusing notation, we will also use the same notation
$\mathfrak{q}^{*}_{12}\psi$ to denote its pullback to
$\operatorname{Sht}_{V}^{r}$ and to
$\operatorname{Sht}^{r,\circ}_{\widehat{V}}$. The meaning will be clear from
context.

  * •

We let $\delta_{\operatorname{Sht}_{V^{(i)}}^{r,\circ}}$ be the indicator
function of the zero-section of the étale $\mathbf{F}_{q}$-vector space bundle
$\operatorname{Sht}_{V^{(i)}}^{r,\circ}\rightarrow\operatorname{Sht}_{S}^{r,\circ}$.
Abusing notation, we will also use this same notation to denote its pullback
to $\operatorname{Sht}_{V}^{r,\circ}$.

  * •

We let $\mathbbm{1}_{\operatorname{Sht}_{V^{(i)}}^{r,\circ}}$ be the constant
function of $\operatorname{Sht}_{V^{(i)}}^{r,\circ}$ with value $1$. Abusing
notation, we will also use this same notation to denote its pullback to
$\operatorname{Sht}_{V}^{r,\circ}$.

  * •

We use similar notation on $\operatorname{Sht}_{\widehat{V}}^{r,\circ}$ and
${}^{\dagger}{\operatorname{Sht}_{\widehat{V}}^{r,\circ}}$.

######  Lemma 10.7.4.

Let $d^{(i)}$ be the rank of $\operatorname{Sht}_{V^{(i)}}^{r,\circ}$ as an
étale $\mathbf{F}_{q}$-vector space bundle over
$\operatorname{Sht}_{S}^{r,\circ}$. Let $d=d^{(0)}+d^{(1)}+d^{(2)}$ be the
rank of $\operatorname{Sht}_{V}^{r,\circ}$ as an étale $\mathbf{F}_{q}$-vector
space bundle over $\operatorname{Sht}_{S}^{r,\circ}$. Then we have

$\displaystyle\mathrm{FT}^{\mathrm{arith}}_{\operatorname{Sht}_{V}^{r,\circ}}\left(\operatorname{Av}({(\mathfrak{q}_{12}^{*}\psi)}\cdot{\delta_{\operatorname{Sht}^{r,\circ}_{V^{(1)}}}}\cdot{\mathbbm{1}_{\operatorname{Sht}^{r,\circ}_{V^{(2)}}}})\right)=(-1)^{d}q^{d^{(2)}+\frac{1}{2}d^{(0)}}\eta_{F^{\prime}/F}(D_{Q})^{n}\cdot\operatorname{Av}((\mathfrak{q}_{12}^{*}[-1]^{*}\psi)\cdot\mathbbm{1}_{\operatorname{Sht}^{r,\circ}_{V^{(1)}}}\cdot\delta_{\operatorname{Sht}^{r,\circ}_{V^{(2)}}})$

as functions on ${}^{\dagger}{\operatorname{Sht}_{\widehat{V}}^{r,\circ}}$.

###### Proof.

This follows from [FYZ22, Corollary 10.2.4] by applying
$\operatorname{Av}(-)$. ∎

######  Lemma 10.7.5.

For $i=1,2$, let $\mathcal{A}_{\widetilde{\mathcal{E}}_{i}}$ be the Hitchin
base as in [FYZ21b, §3.3]. For
$a\in\mathcal{A}_{\widetilde{\mathcal{E}}_{i}}(k)$, recall that
$\mathcal{Z}^{r,\leq\mu,\circ}_{\widetilde{\mathcal{E}_{i}}}(a)\coloneqq\mathcal{Z}^{r}_{\widetilde{\mathcal{E}}_{i}}(a)\times_{\operatorname{Sht}^{r}_{U(n)}}\operatorname{Sht}^{r,\circ}_{S}$.

(1) We have an equality in
$\operatorname{CH}_{d(a_{r})}({}^{\dagger}{\operatorname{Sht}_{V}^{r,\circ}})$:

$(\operatorname{Sht}(f)^{\circ}_{!}{}^{\dagger}{[\operatorname{Sht}_{U}^{r,\circ}]})\cdot\operatorname{Av}({(\mathfrak{q}_{12}^{*}\psi)}\cdot{\delta_{\operatorname{Sht}_{V^{(1)}}^{r,\circ}}}\cdot{\mathbbm{1}_{\operatorname{Sht}_{V^{(2)}}^{r,\circ}}})=\operatorname{Av}\left(\sum_{a\in\mathcal{A}_{\widetilde{\mathcal{E}}_{1}}(k)}\psi(\langle
e_{\mathcal{G},\mathcal{E}_{1}},a\rangle)\operatorname{Sht}(f)^{\circ}_{!}[\mathcal{Z}^{r,\leq\mu,\circ}_{\widetilde{\mathcal{E}}_{1}}(a)]\right).$

(2) We have an equality in
$\operatorname{CH}_{d(a_{r})}({}^{\dagger}{\operatorname{Sht}_{\widehat{V}}^{r,\circ}})$:

$(\operatorname{Sht}(f^{\perp})^{\circ}_{!}{}^{\dagger}{[\operatorname{Sht}_{U^{\perp}}^{r,\circ}]})\cdot\operatorname{Av}({(\mathfrak{q}_{21}^{*}\psi)}\cdot{\mathbbm{1}_{\operatorname{Sht}_{\widehat{V}^{(1)}}^{r,\circ}}}\cdot{\delta_{\operatorname{Sht}_{\widehat{V}^{(2)}}^{r,\circ}}})=\operatorname{Av}\left(\sum_{a\in\mathcal{A}_{\widetilde{\mathcal{E}}_{2}}(k)}\psi(\langle
e_{\mathcal{G},\mathcal{E}_{2}},a\rangle)\operatorname{Sht}(f^{\perp})^{\circ}_{!}[\mathcal{Z}^{r,\leq\mu,\circ}_{\widetilde{\mathcal{E}}_{2}}(a)]\right).$

###### Proof.

This follows from [FYZ22, Lemma 10.2.7] by applying $\operatorname{Av}(-)$. ∎

###  10.8. Proof of Theorem 10.3.1

We will now complete the proof of Theorem 10.3.1. Let $d,d^{(i)}$, for
$i\in\\{0,1,2\\}$, be be the rank of $\operatorname{Sht}_{V^{(i)}}^{r,\circ}$
as an étale $\mathbf{F}_{q}$-vector space bundle over
$\operatorname{Sht}_{S}^{r,\circ}$. Note that $d^{(i)}$ is also the rank of
$V^{(i)}$ as a vector bundle over $S$.

From [FYZ22, (10.3.1)] we may rewrite the higher theta series in the following
way, using the non-homogeneous variant of Notation 9.5.1,

$\widetilde{Z}_{m}^{r}(\widetilde{\mathcal{E}}_{1},\mathcal{G})|_{\operatorname{Sht}^{r,\circ}_{S}}=\chi(\det\widetilde{\mathcal{E}}_{1})q^{n(\deg\widetilde{\mathcal{E}}_{1}-\deg\omega_{X})/2}\langle\operatorname{Sht}(f)^{\circ}_{!}[\operatorname{Sht}_{U}^{r,\circ}],\mathfrak{q}_{12}^{*}\psi\cdot\delta_{\operatorname{Sht}_{V^{(1)}}^{r,\circ}}\cdot\mathbbm{1}_{\operatorname{Sht}_{V^{(2)}}^{r,\circ}}\rangle$
(10.8.1)

and

$\widetilde{Z}_{m}^{r}(\widetilde{\mathcal{E}}_{2},\mathcal{G})|_{\operatorname{Sht}^{r,\circ}_{S}}=\chi(\det\widetilde{\mathcal{E}}_{2})q^{n(\deg\widetilde{\mathcal{E}}_{2}-\deg\omega_{X})/2}\langle\operatorname{Sht}(f^{\perp})^{\circ}_{!}[\operatorname{Sht}_{U^{\perp}}^{r,\circ}],\mathfrak{q}_{21}^{*}\psi\cdot\mathbbm{1}_{\operatorname{Sht}_{\widehat{V}^{(1)}}^{r,\circ}}\cdot\delta_{\operatorname{Sht}_{\widehat{V}^{(2)}}^{r,\circ}}\rangle.$
(10.8.2)

Since $\operatorname{Sht}(f)^{\circ}_{!}[\operatorname{Sht}_{U}^{r,\circ}]$
and
$\operatorname{Sht}(f^{\perp})^{\circ}_{!}[\operatorname{Sht}_{U^{\perp}}^{r,\circ}]$
are homogeneous (cf. Example 10.7.2), it suffices to show that

$\displaystyle\chi(\det\widetilde{\mathcal{E}}_{1})q^{n(\deg\widetilde{\mathcal{E}}_{1}-\deg\omega_{X})/2}\langle\operatorname{Sht}(f)^{\circ}_{!}\
{}^{\dagger}{[\operatorname{Sht}_{U}^{r,\circ}]},\operatorname{Av}(\mathfrak{q}_{12}^{*}\psi\cdot{\delta_{\operatorname{Sht}_{V^{(1)}}^{r,\circ}}}\cdot{\mathbbm{1}_{\operatorname{Sht}_{V^{(2)}}^{r,\circ}}})\rangle$
$\displaystyle=$
$\displaystyle\chi(\det\widetilde{\mathcal{E}}_{2})q^{n(\deg\widetilde{\mathcal{E}}_{2}-\deg\omega_{X})/2}\langle\operatorname{Sht}(f^{\perp})^{\circ}_{!}\
{}^{\dagger}{[\operatorname{Sht}_{U^{\perp}}^{r,\circ}]},\operatorname{Av}(\mathfrak{q}_{21}^{*}\psi\cdot{\mathbbm{1}_{\operatorname{Sht}_{\widehat{V}^{(1)}}^{r,\circ}}}\cdot{\delta_{\operatorname{Sht}_{\widehat{V}^{(2)}}^{r,\circ}}})\rangle.$
(10.8.3)

Let $d=d^{(0)}+d^{(1)}+d^{(2)}$ be the rank of
$\operatorname{Sht}_{V}^{r,\circ}$ as an $\mathbf{F}_{q}$-vector space over
$\operatorname{Sht}_{S}^{r,\circ}$. By the Plancherel formula from Lemma 9.5.2
and the involutivity of $\mathrm{FT}^{\mathrm{arith}}$ from Lemma 9.5.3, we
have

$\displaystyle\langle\operatorname{Sht}(f)^{\circ}_{!}{}^{\dagger}{[\operatorname{Sht}_{U}^{r,\circ}]},\operatorname{Av}(\mathfrak{q}_{12}^{*}\psi\cdot{\delta_{\operatorname{Sht}_{V^{(1)}}^{r,\circ}}}\cdot{\mathbbm{1}_{\operatorname{Sht}_{V^{(2)}}^{r,\circ}}})\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{q^{d}}\langle\mathrm{FT}^{\mathrm{arith}}(\operatorname{Sht}(f)^{\circ}_{!}{}^{\dagger}{[\operatorname{Sht}_{U}^{r,\circ}]}),\mathrm{FT}^{\mathrm{arith}}(\operatorname{Av}(\mathfrak{q}_{12}^{*}\psi\cdot{\delta_{\operatorname{Sht}_{V^{(1)}}^{r,\circ}}}\cdot{\mathbbm{1}_{\operatorname{Sht}_{V^{(2)}}^{r,\circ}}}))\rangle.$

Using Theorem 10.7.3 and Lemma 10.7.4, we rewrite the RHS of (10.8) as

$\displaystyle\frac{1}{q^{d}}q^{d(U/S)}(-1)^{d(U/S)+d(f_{0})}(-1)^{d}q^{d^{(2)}+\frac{1}{2}d^{(0)}}\eta_{F^{\prime}/F}(D_{Q})^{n}$
(10.8.5) $\displaystyle\hskip
28.45274pt\cdot\langle\operatorname{Sht}(f^{\perp})^{\circ}_{!}{}^{\dagger}{[\operatorname{Sht}_{U^{\perp}}^{r,\circ}]},\operatorname{Av}(\mathfrak{q}_{12}^{*}[-1]^{*}\psi\cdot{\mathbbm{1}_{\operatorname{Sht}_{\widehat{V}^{(1)}}^{r,\circ}}}\cdot{\delta_{\operatorname{Sht}_{\widehat{V}^{(2)}}^{r,\circ}}})\rangle.$

Since $\mathfrak{q}_{12}=-\mathfrak{q}_{21}$, we have
$\mathfrak{q}_{12}^{*}[-1]^{*}=\mathfrak{q}_{21}^{*}$. Then clearly (10.8.5)
agrees with the RHS of (10.8) up to sign and exponent of $q$. These signs and
the exponents of $q$ on either side are matched exactly as in [FYZ22, §10.3].

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