#  $\ell$-adic Tautological Systems

Lei Fu, An Huang, Bong Lian, Shing-Tung Yau, Dingxin Zhang, Xinwen Zhu  Yau
Mathematical Sciences Center, Tsinghua University, Beijing 100084, P. R. China
<EMAIL_ADDRESS>Department of Mathematics, Brandeis University,
Waltham, MA 02454, USA<EMAIL_ADDRESS>Department of Mathematics,
Brandeis University, Waltham, MA 02454, USA<EMAIL_ADDRESS>Department of
Mathematics, Harvard University, Cambridge, MA 02138, USA
<EMAIL_ADDRESS>Yau Mathematical Sciences Center, Tsinghua University,
Beijing 100084, China<EMAIL_ADDRESS>Department of Mathematics,
California Institute of Technology, Pasadena, CA 91125, USA<EMAIL_ADDRESS>
###### Abstract.

Tautological systems was introduced in [13] as the system of differential
equations satisfied by period integrals of hyperplane sections of some complex
projective homogenous varieties. We introduce the $\ell$-adic tautological
systems for the case where the ground field is of characteristic $p$.

Key words: multiplicative sheaf; Deligne-Fourier transform.

Mathematics Subject Classification: 14F20.

We thank Quentin Guignard and Haoyu Hu for helpful discussions. The main part
of this work was done while Lei Fu visited the Center of Mathematical Sciences
and Applications (CMSA) at Harvard University in 2015. He would like to thank
CMSA for the hospitality. The research of Lei Fu is supported by NSFC
11531008.

## Introduction

Let $G$ be an algebraic group over $\mathbb{C}$, $\mathfrak{g}$ the Lie
algebra of $G$, $\beta:\mathfrak{g}\to\mathbb{C}$ a character, $V$ a finite
dimensional complex representation of $G$, and $\hat{X}$ a closed subscheme of
$V$ invariant under the action of $G$. For any $\theta\in\mathfrak{g}$, let
$L_{\theta}$ be the vector field on $V$ defined by

$L_{\theta}(x)=\frac{d}{dt}\Big{|}_{t=0}\Big{(}\exp(t\theta)\cdot x\Big{)}$

for any $x\in V$. Similarly, we can define a vector field $L_{\theta}$ on the
dual space $V^{\vee}$ provided with the contragredient action by $G$. Fix a
basis $\\{x_{1},\ldots,x_{N}\\}$ on $V$. Let $\\{\xi_{1},\ldots,\xi_{N}\\}$ be
the dual basis on the dual space $V^{\vee}$, and let $I(\hat{X})$ be the ideal
of of $\mathbb{C}[\xi_{1},\ldots,\xi_{N}]$ consisting of functions vanishing
on $\hat{X}$. The _tautological system_ $\tau(G,\beta,V,\hat{X})$ is the
following system of differential equations for functions
$f(x_{1},\ldots,x_{N})$ on $V^{\vee}$:

$\displaystyle L_{\theta}f=\beta(\theta)f,$ $\displaystyle
g\Big{(}\frac{\partial}{\partial x_{1}},\ldots,\frac{\partial}{\partial
x_{N}}\Big{)}f=0,$

where the first equation is taken for all $\theta\in\mathfrak{g}$, and the
second equation is taken for all $g(\xi_{1},\ldots,\xi_{N})\in I(\hat{X})$. In
[13], it is proved that under some conditions, the tautological systems are
satisfied by period integrals of hyperplane sections of some complex
projective homogenous varieties. The $D$-module corresponding to this systems
is introduced in [8]. As $f(x_{1},\ldots,x_{N})$ is a function on $V^{\vee}$,
the vector field $L_{\theta}$ in the first equation is the vector field on
$V^{\vee}$ for the contragredient action of $G$ on $V^{\vee}$. This equation
can be written as

$\frac{d}{dt}f(\exp(t\theta)\cdot\xi)=\beta(\theta)f(\exp(t\theta)\cdot\xi)$

for any $\theta\in\mathfrak{g}$ and $\xi\in V^{\vee}$. Solving this ordinary
differential equation, we get

$f(\exp(t\theta)\cdot\xi)=e^{\beta(\theta)t}f(\xi),$

that is, $f$ is homogeneous of weight $\beta$ with respect to the group action
of $G$ on $V^{\vee}$. Suppose $G$ is connected. Then $e^{\theta}$
($\theta\in\mathfrak{g}$) generate a dense subgroup of $G$. Suppose
furthermore that $f$ is continuous. Then for any $\xi\in V^{\vee}$, the
restriction of $f$ to the closure of the orbit $G\xi$ is uniquely determined
by the value $f(\xi)$.

Let $\rho:\mathfrak{g}\to\mathfrak{gl}(V)$ be the Lie algebra homomorphism
defined by the representation $V$ of $G$, and write

$\rho(\theta)(x_{i})=\sum_{j=1}^{N}\theta_{ij}x_{j}.$

Then the equation $L_{\theta}f=\beta(\theta)f$ can be written as

$\sum_{i,j}\theta_{ji}x_{i}\frac{\partial f}{\partial
x_{j}}+\beta(\theta)f=0.$

Define formally the Fourier transform $\hat{f}$ of $f$ by

$\hat{f}(\xi_{1},\ldots,\xi_{N})=\int
f(x_{1},\ldots,x_{N})e^{-(x_{1}\xi_{1}+\cdots+x_{N}\xi_{N})}dx_{1}\cdots
dx_{N}.$

Formally we have

$\widehat{(x_{i}f)}=-\frac{\partial}{\partial\xi_{i}}\hat{f},\quad\widehat{\Big{(}\frac{\partial}{\partial
x_{i}}f\Big{)}}=\xi_{i}\hat{f}.$

Taking the Fourier transform of the tautological system, we get

$\displaystyle\sum_{i,j}\theta_{ji}\xi_{j}\frac{\partial\hat{f}}{\partial\xi_{i}}+(\mathrm{Tr}(\rho(\theta))-\beta(\theta))\hat{f}=0\hbox{
for all }\theta\in\mathfrak{g},$ $\displaystyle
g(\xi_{1},\ldots,\xi_{N})\hat{f}=0\hbox{ for all }g\in I(\hat{X}).$

The first system of equations is just

$L_{\theta}\hat{f}=(\beta(\theta)-\mathrm{Tr}(\rho(\theta)))\hat{f}\hbox{ for
all }\theta\in\mathfrak{g},$

and it means that $\hat{f}$ is homogeneous of weight $\beta-\mathrm{Tr}(\rho)$
with respect to the action of $G$ on $V$. The second system of equations means
that $\hat{f}$ is supported in $\hat{X}$. For a rigorous treatment via the
Fourier transform, see [4, §3]. By the Fourier inversion formula, to get a
solution for the tautological system, we may start with a function on $V$
supported on $\hat{X}$ and homogenous of weight $\beta-\mathrm{Tr}(\rho)$ with
respect to the action of $G$, and take its inverse Fourier transform. This is
the method that we are going to use. In this paper, we work out a theory of
tautological systems when the ground field is of characteristic $p$, a problem
posed in [13, §10]. In characteristic $p$, differential equations or
$D$-modules do not behave well. Instead, we work with $\ell$-adic local
systems, and more generally objects in the derived category of
$\overline{\mathbb{Q}}_{\ell}$-sheaves. The idea is that the tautological
system is the Deligne-Fourier transform of a homogeneous object.

######  Remark 0.1.

Let $G_{0}$ be an algebraic group acting on a smooth projective variety $X$ of
dimension $d$, $\mathcal{L}$ a very ample $G_{0}$-linearized invertible sheaf,
and $V=\Gamma(X,\mathcal{L})^{\vee}$. Then $G_{0}$ acts on $V$. Let
$G=G_{0}\times\mathbb{G}_{m}$, let $\mathbb{G}_{m}$ act trivially on $X$, and
act on $V$ by scalar multiplication. We have a $G$-equivariant embedding
$X\to\mathbb{P}(V)$. Let $\hat{X}$ be the cone over $X$, and let
$\beta:\mathfrak{g}_{0}\oplus\mathbb{C}\to\mathbb{C}$ be a character, where
$\mathfrak{g}_{0}$ is the Lie algebra of $G_{0}$. The tautological system
$\tau(G,\beta,V,\hat{X})$

$\displaystyle
L_{\theta}f=\beta(\theta)f\quad(\theta\in\mathfrak{g}_{0}\oplus\mathbb{C}),$
$\displaystyle g(\frac{\partial}{\partial
x_{1}},\ldots,\frac{\partial}{\partial x_{1}})f=0\quad(g\hbox{ lies in the
ideal sheaf of }\hat{X}),$

is studied in detail [9, 13]. Under certain conditions, period integrals for
the family of hyperplane sections of $X$ are solutions of this system. More
precisely, let $B$ be the Zariski open subset of
$V^{\vee}=\Gamma(X,\mathcal{L})$ such that for any $\phi\in B$, the
hypersurface $Y_{\phi}=\\{\phi=0\\}$ of $X$ is smooth, let

$\mathcal{Y}=\\{(x,\phi)\in X\times B|\phi(x)=0\\},$

let $\pi:\mathcal{Y}\to B$ be the projection, let $\omega_{X}$ and
$\omega_{Y_{\phi}}$ be the invertible sheaves of top degree holomorphic forms
on $X$ and $Y_{\phi}$, respectively, and let $i:Y_{\phi}\to X$ be the close
immersion. By the adjunction formula, we have

$i^{\ast}(\mathcal{L}+\omega_{X})\cong\omega_{Y_{\phi}}.$

We thus have a restriction map

$\mathbf{R}_{\phi}:H^{0}(X,\mathcal{L}+\omega_{X})\to
H^{0}(Y_{\phi},\omega_{Y_{\phi}}).$

Let $\tau\in H^{0}(X,\mathcal{L}+\omega_{X})$ be an eigenvector for the action
of $\mathfrak{g}_{0}$, that is, there exists a character
$\beta_{0}:\mathfrak{g}_{0}\to\mathbb{C}$ with the property

$\theta\cdot\tau=\beta_{0}(\theta)\tau$

for any $\theta\in\mathfrak{g}_{0}$. Let
$\beta:\mathfrak{g}_{0}\oplus\mathbb{C}\to\mathbb{C}$ be the character defined
by $\beta(\theta,\lambda)=\beta_{0}(\theta)+\lambda$, and let
$\gamma_{\phi}\in H_{d-1}(Y_{\phi},\mathbb{Z})$ be locally constant homology
$(d-1)$-cycles. Then by [13, Theorem 8.8], the period integral

$\int_{\gamma_{\phi}}\mathbf{R}_{\phi}(\tau)$

as a function of $\phi$ is a solution of the tautological system
$\tau(G,\beta,V,\hat{X})$, provided that there exists a principal
$G$-equivariant $H$-bundle $M\to X$ with a CY-structure and a character
$\chi:H\to\mathbb{C}^{\ast}$ such that $\mathcal{L}$ is the invertible sheaf
corresponding to the line bundle $M\times_{H}\mathbb{C}\to X$, where $H$ acts
on the factor $\mathbb{C}$ through the character $\chi$. Here a CY-structure
is a nowhere vanishing holomorphic form $\omega_{M}$ on $M$ of top degree
which is an eigenvector of the action of $H$, that is
$h\cdot\omega_{M}=\chi_{M}(h)\omega_{M}$ for some character
$\chi_{M}:H\to\mathbb{C}^{\ast}$.

We first construct the homogenous objects. Let $k$ be a field, and let $\ell$
be a prime number distinct from the characteristic of $k$. For any $k$-scheme
$S$ of finite type, let $D_{c}^{b}(S,\overline{\mathbb{Q}}_{\ell})$ be the
derived category of $\overline{\mathbb{Q}}_{\ell}$-sheaves on $S$. Let
$G^{\prime}$ be a commutative algebraic group defined over $k$, and let

$m:G^{\prime}\times_{k}G^{\prime}\to G^{\prime},\quad
p_{1}:G^{\prime}\times_{k}G^{\prime}\to G^{\prime},\quad
p_{2}:G^{\prime}\times_{k}G^{\prime}\to G^{\prime}$

be the multiplication on $G^{\prime}$ and the projections. A _multiplicative
sheaf_ on $G^{\prime}$ is a rank $1$ lisse
$\overline{\mathbb{Q}}_{\ell}$-sheaf $\mathcal{L}$ together with an
isomorphism

$\theta:p_{1}^{*}\mathcal{L}\otimes
p_{2}^{*}\mathcal{L}\stackrel{{\scriptstyle\cong}}{{\to}}m^{*}\mathcal{L}$

such that the following conditions hold:

(1) Symmetry: Let $\sigma:G^{\prime}\times_{k}G^{\prime}\to
G^{\prime}\times_{k}G^{\prime}$ and
$\sigma^{\prime}:p_{1}^{*}\mathcal{L}\otimes p_{2}^{*}\mathcal{L}\to
p_{2}^{*}\mathcal{L}\otimes p_{1}^{*}\mathcal{L}$ be the morphisms defined by
permuting factors. The following diagram commutes:

$\begin{array}[]{ccccc}p_{1}^{*}\mathcal{L}\otimes
p_{2}^{*}\mathcal{L}&&\stackrel{{\scriptstyle\theta}}{{\to}}&&m^{*}\mathcal{L}\\\
{\scriptstyle\sigma^{\prime}}\downarrow&&&&\uparrow{\scriptstyle\cong}\\\
p_{2}^{*}\mathcal{L}\otimes
p_{1}^{*}\mathcal{L}&\cong&\sigma^{*}(p_{1}^{*}\mathcal{L}\otimes
p_{2}^{*}\mathcal{L})&\stackrel{{\scriptstyle\sigma^{*}(\theta)}}{{\to}}&\sigma^{*}m^{*}\mathcal{L}\end{array}$

(2) Associativity: Let
$q_{i}:G^{\prime}\times_{k}G^{\prime}\times_{k}G^{\prime}\to G^{\prime}$
(resp. $q_{ij}:G^{\prime}\times_{k}G^{\prime}\times_{k}G^{\prime}\to
G^{\prime}\times_{k}G^{\prime}$) be the projections to the $i$-th factor
(resp. $(i,j)$-th factor) for any $1\leq i\leq 3$ (resp. $1\leq i<j\leq 3$),
and let $m_{3}:G^{\prime}\times_{k}G^{\prime}\times_{k}G^{\prime}\to
G^{\prime}$ be the multiplication. The following diagram commutes:

$\begin{array}[]{ccc}q_{1}^{*}\mathcal{L}\otimes q_{2}^{*}\mathcal{L}\otimes
q_{3}^{*}\mathcal{L}&\stackrel{{\scriptstyle
q_{12}^{*}(\theta)\otimes\mathrm{id}}}{{\to}}&q_{12}^{*}m^{*}\mathcal{L}\otimes
q_{3}^{*}\mathcal{L}\\\ \scriptstyle{\mathrm{id}\otimes
q_{23}^{*}(\theta)}\downarrow&&\downarrow\scriptstyle{(mq_{12}\times
q_{3})^{*}(\theta)}\\\ q_{1}^{*}\mathcal{L}\otimes
q_{23}^{*}m^{*}\mathcal{L}&\stackrel{{\scriptstyle(q_{1}\times
mq_{23})^{*}(\theta)}}{{\to}}&m_{3}^{*}\mathcal{L}.\end{array}$

The isomorphism $\theta$ induces an isomorphism

$p_{1}^{*}\mathcal{L}\otimes p_{2}^{*}\mathcal{L}\otimes
m^{*}\mathcal{L}^{-1}\cong\overline{\mathbb{Q}}_{\ell}.$

Restricting this isomorphism to $(e,e)$, where $e:\mathrm{Spec}\,k\to
G^{\prime}$ is the unit section of $G^{\prime}$, we get an isomorphism

$e^{*}\mathcal{L}\cong\overline{\mathbb{Q}}_{\ell}.$

We can obtain multiplicative $\overline{\mathbb{Q}}_{\ell}$-sheaf on
$G^{\prime}$ is as follows: Let

$0\to N_{i}\to G^{\prime}_{i}\to G^{\prime}\to 0\quad(i\in I)$

be the inverse system of all extensions of $G^{\prime}$ by finite discrete
abelian groups $N_{i}$. Note that each $G^{\prime}_{i}\to G^{\prime}$ is a
Galois étale isogeny with Galois group $N_{i}$. We may regard the projective
system $\\{G^{\prime}_{i}\\}_{i\in I}$ as the universal covering space
classifying étale isogenies of $G^{\prime}$. Set

$\pi_{1}^{\mathrm{isogeny}}(G^{\prime})=\varprojlim_{i\in I}N_{i}.$

It is a quotient of the étale fundamental group $\pi_{1}(G^{\prime})$ of
$G^{\prime}$, and coincides with $\pi_{1}(G^{\prime})$ if $k$ is algebraically
closed of characteristic $0$. Let
$\beta:\pi_{1}^{\mathrm{isogeny}}(G^{\prime})\to\overline{\mathbb{Q}}_{\ell}^{*}$
be a character. It induces a character on the étale fundamental group of
$G^{\prime}$. Denote by $\mathcal{L}_{\beta}$ the corresponding
$\overline{\mathbb{Q}}_{\ell}$-sheaf associated to this Galois representation.
Then $\mathcal{L}_{\beta}$ is multiplicative. Any multiplicative sheaf on
$G^{\prime}$ is of this form. See [14, 2.3.10] and [6, 2.9] for a proof.

Suppose $k=\mathbb{F}_{q}$ is a finite field with $q$ elements of
characteristic $p$. Let $F$ be the Frobenius morphism, that is, $F$ is
identity on the underlying topological space of $G^{\prime}$, and $F$ maps any
section of $\mathcal{O}_{G}$ to its $q$-th power. Let

$L:G^{\prime}\to G^{\prime},\quad x\mapsto F(x)\cdot x^{-1}$

be the Lang isogeny. Its kernel is the group $G^{\prime}(\mathbb{F}_{q})$ of
$\mathbb{F}_{q}$-points of $G^{\prime}$, and it is a Galois étale covering
with Galois group $G^{\prime}(\mathbb{F}_{q})$. By [16, VI §1 Proposition 6],
any Galois isogeny over $G^{\prime}$ is a quotient of the Lang isogeny. It
follows that

$\pi_{1}^{\mathrm{isogeny}}(G^{\prime})\cong G^{\prime}(\mathbb{F}_{q}).$

Let $\beta:G^{\prime}(\mathbb{F}_{q})\to\overline{\mathbb{Q}}_{\ell}^{\ast}$
be a character. The multiplicative sheaf $\mathcal{L}_{\beta}$ is called the
_Lang sheaf_ associated to $\beta$. For any $x\in G^{\prime}(\mathbb{F}_{q})$,
we have

$(\mathrm{Frob}_{x},\mathcal{L}_{\beta,\bar{x}})=\beta(x^{\prime}),$

where $\mathrm{Frob}_{x}$ is the geometric Frobenius element in
$\pi_{1}(G^{\prime})$.

######  Remark 0.2.

Let $G^{\prime}$ be a connected commutative complex algebraic group, let
$\mathfrak{g}^{\prime}$ be its Lie algebra, and let
$\beta:\mathfrak{g}^{\prime}\to\mathbb{C}$ be a character. The exponential map

$\mathfrak{g}^{\prime}\to G^{\prime},\quad\theta\mapsto\exp(\theta)$

is an epimorphism. It is the universal covering space of $G^{\prime}$, and its
kernel is isomorphic to the fundamental group $\pi_{1}(G^{\prime})$ of
$G^{\prime}$. Restricting $e^{\beta}$ to this kernel, we get a character
$e^{\beta}:\pi_{1}(G^{\prime})\to\mathbb{C}^{*}$. The local system
$\mathcal{L}_{e^{\beta}}$ associated to this character is a multiplicative
sheaf. For any holomorphic function $f$ on $G^{\prime}$ and any invariant
vector field $\theta\in\mathfrak{g}$ on $G^{\prime}$, let

$\nabla_{\theta}(f)=\theta(f)-\beta(\theta)f.$

Then $\nabla$ is an integrable connection on the trivial vector bundle over
$G^{\prime}$. Horizontal sections are those holomorphic functions $f$ such
that

$\theta(f)=\beta(\theta)f,$

that is, $f$ are homogenous of weight $\beta$. The multiplicative local system
$\mathcal{L}_{e^{\beta}}$ is isomorphic to the local system of horizontal
sections of this connection.

From now on, we assume $k$ is a perfect field of characteristic $p$ unless we
state otherwise. Let $G$ be an algebraic group defined over $k$, and let
$G\to\mathrm{GL}(V)$ be a representation, where $V$ is a finite dimensional
vector space over $k$. By abuse of notation, we also denote by $V$ the affine
$k$-scheme $\mathrm{Spec}(\mathrm{Sym}^{\cdot}(V^{\vee}))$. Let $v$ be a
nonzero vector in $V$, and let $Q$ be the connected component of the
stabilizer of $v$. Consider the morphism

$\iota:G/Q\to V,\quad gQ\mapsto gv.$

It is quasi-finite. Let $G^{\prime}=G/Q[G,G]$, which is a commutative
algebraic group. Fix a multiplicative sheaf $\mathcal{L}_{\beta}$ on
$G^{\prime}$. Denote its inverse image by the canonical morphism $G/Q\to
G^{\prime}$ also by $\mathcal{L}_{\beta}$. Note that
$\iota_{!}\mathcal{L}_{\beta}$ and $R\iota_{\ast}\mathcal{L}_{\beta}$ are
homogeneous objects in $D_{c}^{b}(V,\overline{\mathbb{Q}}_{\ell})$ supported
in the $G$-invariant subset $\overline{Gv}$. By homogeneity, we mean that

$\mu^{\ast}\iota_{!}\mathcal{L}_{\beta}\cong\mathcal{L}^{\prime}_{\beta}\boxtimes\iota_{!}\mathcal{L}_{\beta},\quad\mu^{\ast}R\iota_{\ast}\mathcal{L}_{\beta}\cong\mathcal{L}^{\prime}_{\beta}\boxtimes
R\iota_{\ast}\mathcal{L}_{\beta},$

where $\mu:G\times V\to V$ is the action of $G$ on $V$, and
$\mathcal{L}^{\prime}_{\beta}$ is the inverse image of $\mathcal{L}_{\beta}$
by the morphism $G\to G/Q$. (Confer the proof of Lemma 1.1). Fixed a
nontrivial additive character
$\psi:\mathbb{F}_{p}\to\overline{\mathbb{Q}}_{\ell}^{\ast}$, and let
$\mathscr{F}_{\psi}:D_{c}^{b}(V,\overline{\mathbb{Q}}_{\ell})\to
D_{c}^{b}(V^{\vee},\overline{\mathbb{Q}}_{\ell})$ be the Deligne-Fourier
transform defined by $\psi$. We define the _$\ell$ -adic tautological sheaves_
to be

$\displaystyle\mathcal{T}_{!}(G,\beta,V,v,\psi)$ $\displaystyle=$
$\displaystyle\mathscr{F}_{\psi}(\iota_{!}\mathcal{L}_{\beta}[\mathrm{dim}\,G/Q]),$
$\displaystyle\mathcal{T}_{\ast}(G,\beta,V,v,\psi)$ $\displaystyle=$
$\displaystyle\mathscr{F}_{\psi}(R\iota_{\ast}\mathcal{L}_{\beta}[\mathrm{dim}\,G/Q]).$

Recall that for any vector bundle $E$ of rank $r$ over a $k$-scheme $S$ of
finite type, the Deligne-Fourier transform is the functor

$\mathscr{F}_{\psi}:D_{c}^{b}(E,\overline{\mathbb{Q}}_{\ell})\to
D_{c}^{b}(E^{\vee},\overline{\mathbb{Q}}_{\ell}),\quad K\mapsto
R\mathrm{pr}^{\vee}_{!}(\mathrm{pr}^{\ast}K\otimes\langle\,,\,\rangle^{\ast}\mathcal{L}_{\psi})[r],$

where $E^{\vee}\to S$ is the dual vector bundle of $E$,
$\mathrm{pr}:E\times_{k}E^{\vee}\to E$ and
$\mathrm{pr}^{\vee}:E\times_{k}E^{\vee}\to E^{\vee}$ are the projections, and
$\langle\,,\,\rangle:E\times_{k}E^{\vee}\to\mathbb{A}_{S}^{1}$ is the pairing,
and $\mathcal{L}_{\psi}$ is the Artin-Schreier sheaf on $\mathbb{A}^{1}$
associated to the nontrivial additive character
$\psi:\mathbb{F}_{p}\to\overline{\mathbb{Q}}_{\ell}^{\ast}$, that is, the
pulling back to $\mathbb{A}_{k}^{1}$ of the Lang sheaf on the algebraic group
$\mathbb{A}_{\mathbb{F}_{p}}^{1}$ associated to the character
$\psi:\mathbb{A}^{1}(\mathbb{F}_{p})\to\overline{\mathbb{Q}}_{\ell}^{\ast}$.
See [12] for properties of the Fourier transform. For any $k$-scheme
$a:S\to\mathrm{Spec}\,k$ of finite type, denote by
$D_{S}:D_{c}^{b}(S,\overline{\mathbb{Q}}_{\ell})\to
D_{c}^{b}(S,\overline{\mathbb{Q}}_{\ell})$ the Verdier dual functor
$D_{S}=R\mathcal{H}om(\cdot,Ra^{!}\overline{\mathbb{Q}}_{\ell})$.

######  Proposition 0.3.

Let $n=\mathrm{dim}\,G/Q$ and $N=\mathrm{dim}\,V$.

(i) We have
$D_{V^{\vee}}(\mathcal{T}_{\ast}(G,\beta,V,v,\psi))\cong\mathcal{T}_{!}(G,\beta^{-1},V,v,\psi^{-1})(n+N)$

(ii) In the case where $k$ is a finite field,
$\mathcal{T}_{!}(G,\beta,V,v,\psi)$ (resp.
$D_{V^{\vee}}(\mathcal{T}_{\ast}(G,\beta,V,v,\psi))$ ) is mixed of weights
$\leq n+N$ (resp. $\leq-(n+N)$).

###### Proof.

(i) By [12, 1.3.2.2], we have

$\displaystyle D_{V^{\vee}}(\mathcal{T}_{\ast}(G,\beta,V,v,\psi))$
$\displaystyle=$ $\displaystyle
D_{V^{\vee}}(\mathscr{F}_{\psi}(R\iota_{\ast}\mathcal{L}_{\beta}[n]))$
$\displaystyle\cong$
$\displaystyle\mathscr{F}_{\psi^{-1}}(D_{V}(R\iota_{\ast}\mathcal{L}_{\beta}[n]))(N)$
$\displaystyle\cong$
$\displaystyle\mathscr{F}_{\psi^{-1}}(\iota_{!}D_{G/Q}(\mathcal{L}_{\beta}[n]))(N)$
$\displaystyle\cong$
$\displaystyle\mathscr{F}_{\psi^{-1}}(\iota_{!}\mathcal{L}_{\beta^{-1}}[n])(n+N)$
$\displaystyle\cong$
$\displaystyle\mathcal{T}_{!}(G,\beta^{-1},V,v,\psi^{-1})(n+N).$

(ii) $\iota_{!}\mathcal{L}_{\beta}[n]$ is mixed of weights $\leq n$. Its
Deligne-Fourier transform $\mathcal{T}_{!}(G,\beta,V,v)$ is mixed of weights
$\leq n+N$ by [1, Théorème 3.3.1]. Then by (i),
$D_{V^{\vee}}(\mathcal{T}_{\ast}(G,\beta,V,v,\psi))$ is mixed of weights
$\leq-(n+N)$. ∎

######  Example 0.4.

Let $(w_{ij})$ be an $n\times N$ matrix of rank $n$ with integer entries. Take
$G$ to be the torus $G=\mathbb{G}_{m,k}^{n}$. Consider the representation $V$
of $G$ defined by the action

$\displaystyle\mathbb{G}_{m,k}^{n}\times_{k}\mathbb{A}_{k}^{N}$
$\displaystyle\to$ $\displaystyle\mathbb{A}_{k}^{N},$
$\displaystyle\Big{(}(t_{1},\ldots,t_{n}),(x_{1},\ldots,x_{N})\Big{)}$
$\displaystyle\mapsto$ $\displaystyle(t_{1}^{w_{11}}\cdots
t_{n}^{w_{n1}}x_{1},\ldots,t_{1}^{w_{1N}}\cdots t_{n}^{w_{nN}}x_{N}).$

The connected component of the stabilizer $v=(1,\ldots,1)\in V$ is trivial.
Let $\hat{X}$ the the closure of the orbit of $v$. Then $\hat{X}$ is the
closed subscheme of $\mathbb{A}_{k}^{N}$ defined by
$\prod_{a_{j}>0}{\xi_{j}}^{a_{j}}-\prod_{a_{j}<0}{\xi_{j}}^{-a_{j}}=0$, where
$(a_{1},\ldots,a_{N})\in\mathbb{Z}^{N}$ goes over the family of integral
linear relations

$\sum_{j=1}^{N}a_{j}{\mathbf{w}}_{j}=0$

among ${\mathbf{w}}_{1},\ldots,{\mathbf{w}}_{N}$ with
${\mathbf{w}}_{j}=(w_{1j},\ldots,w_{nj})\in\mathbb{Z}^{n}$.

Suppose $k$ is the complex field $\mathbb{C}$. Let $\beta$ the character

$\beta:\mathbb{C}^{n}\to\mathbb{C},\quad(\lambda_{1},\ldots,\lambda_{n})\mapsto\gamma_{1}\lambda_{1}+\cdots+\gamma_{n}\lambda_{n},$

where $\gamma_{1},\ldots,\gamma_{n}\in\mathbb{C}$. The tautological system
$\tau(G,\beta,V,\hat{X})$ is exactly the Gelfand-Kapranov-Zelevinsky (GKZ)
hypergeometric system

$\displaystyle\sum_{j=1}^{N}w_{ij}x_{j}\frac{\partial f}{\partial
x_{j}}+\gamma_{i}f=0\quad(i=1,\ldots,n),$
$\displaystyle\prod_{a_{j}>0}\left(\frac{\partial}{\partial
x_{j}}\right)^{a_{j}}f=\prod_{a_{j}<0}\left(\frac{\partial}{\partial
x_{j}}\right)^{-a_{j}}f$

introduced in [7]. It is the system of differential equations satisfied by the
exponential integral (period integral)

$f(x_{1},\ldots,x_{n})=\int_{\sigma}t_{1}^{\gamma_{1}}\cdots
t_{n}^{\gamma_{n}}e^{\sum_{j=1}^{N}x_{j}t_{1}^{w_{1j}}\cdots
t_{n}^{w_{nj}}}\frac{dt_{1}}{t_{1}}\cdots\frac{dt_{n}}{t_{n}},$

where $\sigma$ is any real $n$-dimensional cycles in $(\mathbb{C}^{*})^{n}.$

Suppose $k=\mathbb{F}_{q}$ is a finite field with $q$ elements. Let
$\chi_{1},\ldots,\chi_{n}:\mathbb{F}_{q}^{\ast}\to\overline{\mathbb{Q}}_{\ell}^{\ast}$
be multiplicative characters. They define a character

$\beta:{\mathbb{G}}_{m}^{n}(\mathbb{F}_{q})\to\overline{\mathbb{Q}}_{\ell}^{\ast},\quad(t_{1},\ldots,t_{n})\mapsto\chi_{1}(t_{1})\cdots\chi_{n}(t_{n}).$

The the tautological system $\mathcal{T}_{!}(G,\beta,V,v,\psi)$ is the GKZ
hypergeometric sheaf studied in [5]. For any $\mathbb{F}_{q}$-rational point
$x=(x_{1},\ldots,x_{N})$ of $V^{\vee}$, we have

$\mathrm{Tr}(\mathrm{Frob}_{x},\mathcal{T}_{!}(G,\beta,V,v,\psi)_{\bar{x}})=\sum_{t_{1},\ldots,t_{n}\in\mathbb{F}_{q}^{*}}\chi_{1}(t_{1})\cdots\chi_{n}(t_{n})\psi\Big{(}\mathrm{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{p}}\Big{(}\sum_{j=1}^{N}x_{j}t_{1}^{w_{1j}}\cdots
t_{n}^{w_{nj}}\Big{)}\Big{)}.$

The exponential sum on the righthand side is the arithmetic counterpart of the
above exponential integral.

We will give more examples of tautological systems in §2. In the following, we
focus on the tautological systems constructed from a quasi-projective
homogenous variety embedded equivariantly in $\mathbb{P}(V)$. More precisely,
let $G_{0}$ be an algebraic group, let $G_{0}\to\mathrm{GL}(V)$ be a
representation, and let $G=G_{0}\times\mathbb{G}_{m}$. Let $G$ act on $V$ so
that the factor $\mathbb{G}_{m}$ acts on $V$ by scalar multiplication. For any
nonzero vector $x\in V$, denote by $[x]$ the corresponding point in the
projective space $\mathbb{P}(V)=\mathrm{Proj}(\mathrm{Sym}(V^{\vee}))$, and
denote by $\mathbb{L}_{[x]}$ the line in $V$ spanned by $x$. Let $\mathbb{L}$
be the tautological line bundle over
$\mathbb{P}(V)=\mathrm{Proj}(\mathrm{Sym}(V^{\vee}))$. The fiber of
$\mathbb{L}$ at $[x]$ can be identified with the line $\mathbb{L}_{[x]}$. Note
that $G$ acts on $\mathbb{P}(V)$ and on $\mathbb{L}$, and the factor
$\mathbb{G}_{m}$ acts trivially on $\mathbb{P}(V)$ and acts by scalar
multiplication on fibers of $\mathbb{L}$. Let $\mathbb{L}^{\vee}$ be the dual
line bundle of $\mathbb{L}$. It is provided with the contragrediant action by
$G$. The fiber of $\mathbb{L}^{\vee}$ over $[x]$ consists of linear
functionals on the line $\mathbb{L}_{[x]}$. The canonical morphisms
$\mathbb{L}\to\mathbb{P}(V)$ and $\mathbb{L}^{\vee}\to\mathbb{P}(V)$ are
$G$-equivariant.

We have a canonical embedding

$i:\mathbb{L}\to V\times_{k}\mathbb{P}(V)$

of $\mathbb{L}$ into the trivial vector bundle
$V\times_{k}\mathbb{P}(V)\to\mathbb{P}(V)$. The fiber of this morphism over
$[x]\in\mathbb{P}(V)$ is the inclusion of the line $\mathbb{L}_{[x]}$ into
$V$. The transpose of $i$ is the evaluation morphism

$\mathrm{ev}:V^{\vee}\times_{k}\mathbb{P}(V)\to\mathbb{L}^{\vee}.$

For any $\phi\in V^{\vee}$, the fiber of $\mathrm{ev}$ over $[x]$ maps a point
$(\phi,[x])$ in $V^{\vee}\times_{k}\mathbb{P}(V)$ to the restriction
$\phi|_{\mathbb{L}_{[x]}}\in\mathbb{L}^{\vee}_{[x]}$ of $\phi$ to the subspace
$\mathbb{L}_{[x]}$. Let ${\mathbb{L}}^{\circ}$ be the complement of the zero
section of $\mathbb{L}$. We have an isomorphism

$V-\\{0\\}\stackrel{{\scriptstyle\cong}}{{\to}}{\mathbb{L}}^{\circ}$

which maps every nonzero vector $x$ to the point $x$ considered as an element
in the line $\mathbb{L}_{[x]}$. ${\mathbb{L}}^{\circ}$ is invariant under the
action of $G$.

Let $v\in V$ be a nonzero vector vector, let $Q\subset G$ be the connected
component of the stabilizer of $v$, and let $P=Q\mathbb{G}_{m}$. Note that $P$
stabilizes $[v]\in\mathbb{P}(V)$. Regard $v$ as an element in
$\mathbb{L}_{[v]}$. Let $v^{\ast}$ be the element in
$(\mathbb{L}^{\vee})_{[v]}$ define by $v^{\ast}(v)=1$. Note that $Q$ is also
the connected component of the stabilizer of $v^{\ast}$. Let

$\mathbb{L}_{G/P}=\mathbb{L}\times_{\mathbb{P}^{n}(V)}G/P,\quad\mathbb{L}^{\vee}_{G/P}=\mathbb{L}^{\vee}\times_{\mathbb{P}^{n}(V)}G/P,\quad
i_{G/P}:\mathbb{L}_{G/P}\to
V\times_{k}G/P,\quad\mathrm{ev}_{G/P}:V^{\vee}\times_{k}G/P\to\mathbb{L}^{\vee}_{G/P}$

the base changes of $\mathbb{L}$, $\mathbb{L}^{\vee}$, $i$ and $\mathrm{ev}$
respectively with respect to the morphism

$G/P\to\mathbb{P}(V),\quad gP\mapsto g[v].$

Let $\kappa:G/Q\to\mathbb{L}_{G/P}^{\circ}$ (resp.
$\kappa^{\vee}:G/Q\to\mathbb{L}_{G/P}^{\vee\circ}$) be the morphism $gQ\mapsto
gv$ (resp. $gQ\mapsto gv^{\ast}$), where we regard $v$ (resp. $v^{\ast}$) as
an element in the fiber $\mathbb{L}_{G/P,eP}\cong\mathbb{L}_{[v]}$ (resp.
${\mathbb{L}}^{\vee}_{G/P,eP}\cong{\mathbb{L}}^{\vee}_{[v]}$).

######  Lemma 0.5.

$\kappa:G/Q\to\mathbb{L}^{\circ}_{G/P}$ and
$\kappa^{\vee}:G/Q\to\mathbb{L}_{G/P}^{\vee\circ}$ are isomorphisms.

###### Proof.

Note that $Q$ is exactly the stabilizer of $(v,eP)$ considered as a point in
$\mathbb{L}^{\circ}_{G/P}$ with respect to the action of $G$ on
$\mathbb{L}^{\circ}_{G/P}$. Moreover, the action of $G$ on
$\mathbb{L}^{\circ}_{G/P}$ is transitive. We claim that the morphism

$G\to\mathbb{L}^{\circ}_{G/P},\quad g\mapsto g\cdot(v,eP)$

is smooth. It suffices to show it induces an epimorphism on tangent spaces

$T_{e}G\to T_{(v,eP)}(\mathbb{L}^{\circ}_{G/P}).$

The tangent space $T_{(v,eP)}(\mathbb{L}^{\circ}_{G/P})$ is isomorphic to the
direct sum of $T_{eP}(G/P)$ and the one dimensional space corresponding to the
fiber of the line bundle $\mathbb{L}_{G/P}$ over $eP$. The canonical
homomorphism $T_{e}G\to T_{e}(G/P)$ is surjective. The tangent vectors along
the direct factor $\mathbb{G}_{m}$ of $G$ are mapped surjectively onto the
above one dimensional subspace. This proves our claim. By [16, 5.5.4 and
5.5.5], $\mathbb{L}^{\circ}_{G/P}$ is isomorphic to $G/Q$. Similarly,
$\mathbb{L}_{G/P}^{\vee\circ}$ is also isomorphic to $G/Q$. ∎

We use the isomorphism in the above lemma to identify $G/Q$ with
$\mathbb{L}^{\circ}_{G/P}$ (resp. $\mathbb{L}_{G/P}^{\vee\circ}$). By abuse of
notation, we denote the sheaf $\kappa_{!}\mathcal{L}_{\beta}$ (resp.
$\kappa^{\vee}_{!}\mathcal{L}_{\beta}$) on $\mathbb{L}^{\circ}_{G/P}$ (resp.
$\mathbb{L}_{G/P}^{\vee\circ}$) also by $\mathcal{L}_{\beta}$. Let
$j_{\mathbb{L}_{G/P}^{\circ}}:\mathbb{L}_{G/P}^{\circ}\hookrightarrow\mathbb{L}_{G/P}$
and
$j_{\mathbb{L}_{G/P}^{\vee\circ}}:\mathbb{L}_{G/P}^{\vee\circ}\hookrightarrow\mathbb{L}_{G/P}^{\vee}$
be the open immersions of the complements of zero sections, and let
$\mathrm{pr}:V^{\vee}\times_{k}{G/P}\to V^{\vee}$ be the projection. Let
$\mathcal{L}_{\beta}|_{\mathbb{G}_{m}}$ (resp.
$\mathcal{L}_{\psi}|_{\mathbb{G}_{m}}$) be the inverse image of
$\mathcal{L}_{\beta}$ (resp. $\mathcal{L}_{\psi}$) under the canonical
morphism $\mathbb{G}_{m}\to G^{\prime}=G/Q[G,G]$ (resp.
$\mathbb{G}_{m}\to\mathbb{A}^{1}$). Let $G_{!}(\beta,\psi)$ and
$G_{*}(\beta,\psi)$ be the sheaves on $\mathrm{Spec}\,k$ defined by the
following Galois representations:

$\displaystyle G_{!}(\beta,\psi)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cl}H^{1}_{c}(\mathbb{G}_{m,\bar{k}},\mathcal{L}_{\beta}|_{\mathbb{G}_{m}}\otimes\mathcal{L}_{\psi}|_{\mathbb{G}_{m}})&\hbox{if
}\mathcal{L}_{\beta}|_{\mathbb{G}_{m,\bar{k}}}\hbox{ is nontrivial},\\\
\overline{\mathbb{Q}}_{\ell}&\hbox{otherwise},\end{array}\right.$
$\displaystyle G_{*}(\beta,\psi)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cl}H^{1}(\mathbb{G}_{m,\bar{k}},\mathcal{L}_{\beta}|_{\mathbb{G}_{m}}\otimes\mathcal{L}_{\psi}|_{\mathbb{G}_{m}})&\hbox{if
}\mathcal{L}_{\beta}|_{\mathbb{G}_{m,\bar{k}}}\hbox{ is nontrivial},\\\
\overline{\mathbb{Q}}_{\ell}(-1)&\hbox{otherwise},\end{array}\right.$

By Poincaré duality, we have

$G_{*}(\beta,\psi)\cong G_{!}(\beta,\psi)^{\vee}(-1).$

Actually if $\mathcal{L}_{\beta}|_{\mathbb{G}_{m,\bar{k}}}$ is nontrivial,
then

$H^{1}_{c}(\mathbb{G}_{m,\bar{k}},\mathcal{L}_{\beta}|_{\mathbb{G}_{m}}\otimes\mathcal{L}_{\psi}|_{\mathbb{G}_{m}})\cong
H^{1}(\mathbb{G}_{m,\bar{k}},\mathcal{L}_{\beta}|_{\mathbb{G}_{m}}\otimes\mathcal{L}_{\psi}|_{\mathbb{G}_{m}})$

and hence $G_{!}(\beta,\psi)=G_{*}(\beta,\psi)$ in this case. We will show
that $G_{\ast}(\beta,\psi)$ and $G_{!}(\beta,\psi)$ are of rank $1$. (Confer
Lemma 1.7). Denote the inverse images of $G_{!}(\beta,\psi)$ and
$G_{*}(\beta,\psi)$ on any $k$-scheme by the same notation. In the case where
$k=\mathbb{F}_{q}$ is a finite field with $q$ elements and
$\mathcal{L}_{\beta}|_{\mathbb{G}_{m,\bar{k}}}$ is nontrivial, by the
Grothendieck trace formula, $G_{\ast}(\beta,\psi)$ and $G_{\ast}(\beta,\psi)$
is defined by the Galois representation

$\mathrm{Gal}(\overline{\mathbb{F}}_{q}/\mathbb{F}_{q})\to\overline{\mathbb{Q}}_{\ell}^{\ast},\quad\mathrm{Frob}_{q}\mapsto-\sum_{t\in\mathbb{F}_{q}^{\ast}}\beta(t)\psi(t),$

where $\mathrm{Frob}_{q}$ is the geometric Frobenius element in
$\mathrm{Gal}(\overline{\mathbb{F}}_{q}/\mathbb{F}_{q})$, and the righthand
side is nothing but the Gauss sum.

Let $U=\mathrm{ev}_{G/P}^{-1}(\mathbb{L}_{G/P}^{\vee\circ})$ and let $H$ be
the complement of $U$ in $V^{\vee}\times_{k}G/P$. Then $U$ (resp. $H$) is an
open (resp. closed) subset of $V^{\vee}\times{G/P}$, and a rational point
$(\phi,gP)$ of $V^{\vee}\times G/P$ lies in $U$ (resp. $H$) if and only if
$\phi(g(v))\not=0$ (resp. $\phi(g(v))=0$). For any rational point $\phi$ of
$V^{\vee}$, the fiber $H_{\phi}$ of $H$ over $\phi$ is the hyperplane section
$\\{gP\in G/P|\phi(gv)=0\\}$ in $G/P$, and the fiber $U_{\phi}$ of $U$ over
$\phi$ is the complement in $G/P$ of the hyperplane section. Fix notations by
the following commutative diagram:

$\begin{array}[]{rccrl}G/Q\cong\mathbb{L}_{G/P}^{\vee\circ}&\stackrel{{\scriptstyle\mathrm{ev}^{\prime}_{G/P}}}{{\leftarrow}}&U&&H\\\
{\scriptstyle
j_{\mathbb{L}_{G/P}^{\vee\circ}}}\downarrow&&\downarrow{\scriptstyle
j^{\prime}}&{\scriptstyle i^{\prime}}\swarrow&\\\
\mathbb{L}^{\vee}_{G/P}&\stackrel{{\scriptstyle\mathrm{ev}_{G/P}}}{{\leftarrow}}&V^{\vee}\times_{k}(G/P)&\stackrel{{\scriptstyle\mathrm{pr}}}{{\to}}&V^{\vee}\\\
&\searrow&\downarrow&&\downarrow\\\
&&G/P&\to&\mathrm{Spec}\,\mathbb{F}_{q}\end{array}$

######  Theorem 0.6.

Notation as above. Suppose $Q$ is geometrically connected.

(i) We have

$\displaystyle\mathcal{T}_{!}(G,\beta,V,v,\psi)$ $\displaystyle\cong$
$\displaystyle
R\mathrm{pr}_{!}\mathrm{ev}_{G/P}^{\ast}Rj_{\mathbb{L}_{G/P}^{\vee\circ},*}\mathcal{L}_{\beta}[n+N-1]\otimes
G_{!}(\beta,\psi),$ $\displaystyle\mathcal{T}_{\ast}(G,\beta,V,v,\psi)$
$\displaystyle\cong$ $\displaystyle
R\mathrm{pr}_{\ast}\mathrm{ev}_{G/P}^{\ast}j_{\mathbb{L}_{G/P}^{\vee\circ},!}\mathcal{L}_{\beta}[n+N-1]\otimes
G_{*}(\beta,\psi)$ $\displaystyle\cong$ $\displaystyle
R\mathrm{pr}_{\ast}j^{\prime}_{!}\mathrm{ev}_{G/P}^{\prime\ast}\mathcal{L}_{\beta}[n+N-1]\otimes
G_{*}(\beta,\psi).$

(ii) Suppose furthermore that there exists a multiplicative sheaf
$\mathcal{L}_{\beta_{0}}$ on $G/P[G,G]$ such that $\mathcal{L}_{\beta}$ is
isomorphic to its inverse image under the canonical morphism $G/Q[G,G]\to
G/P[G,G]$. Then we have

$\displaystyle\mathcal{T}_{\ast}(G,\beta,V,v,\psi)$ $\displaystyle\cong$
$\displaystyle
R\mathrm{pr}_{\ast}j^{\prime}_{!}(\mathcal{L}_{\beta_{0}}|_{U})(-1)[n+N-1],$

and we have a distinguished triangle

$R\mathrm{pr}_{!}Ri^{\prime}_{*}(\mathcal{L}_{\beta_{0}}|_{H})(-1)[n+N-3]\to
R\mathrm{pr}_{!}(\mathcal{L}_{\beta_{0}}|_{V^{\vee}\times_{k}G/P})[n+N-1]\to\mathcal{T}_{!}(G,\beta,V,v,\psi)\to,$

where $\mathcal{L}_{\beta_{0}}|_{U}$, $\mathcal{L}_{\beta_{0}}|_{H}$ and
$(\mathcal{L}_{\beta_{0}}|_{V^{\vee}\times_{k}G/P})$ are the inverse images of
the sheaf $\mathcal{L}_{\beta_{0}}$ on $G/P$.

Combined with the proper base change theorem, we have the following corollary
which shows that tautological systems parameterize the cohomology of
$U_{\phi}$.

######  Corollary 0.7.

Suppose $G/P$ is proper and $Q$ is geometrically connected.

(i) For any point $\phi$ in $V^{\vee}$, we have

$\mathcal{T}_{\ast}(G,\beta,V,v,\psi)_{\bar{\phi}}=R\Gamma_{c}(U_{\bar{\phi}},(\mathrm{ev}_{G/P}^{\prime\ast}\mathcal{L}_{\beta})|_{U_{\bar{\phi}}})[n+N-1]\otimes
G_{*}(\beta,\psi).$

(ii) Suppose that there exists a multiplicative sheaf
$\mathcal{L}_{\beta_{0}}$ on $G/P[G,G]$ such that $\mathcal{L}_{\beta}$ is
isomorphic to its inverse image under the canonical morphism $G/Q[G,G]\to
G/P[G,G]$. Then any point $\phi$ in $V^{\vee}$, we have

$\mathcal{T}_{\ast}(G,\beta,V,v,\psi)_{\bar{\phi}}=R\Gamma_{c}(U_{\bar{\phi}},\mathcal{L}_{\beta_{0}}|_{U_{\bar{\phi}}})(-1)[n+N-1].$

###### Proof.

Follows from the formula for $\mathcal{T}_{\ast}(G,\beta,V,v,\psi)$ in Theorem
0.6 and the proper base change theorem. ∎

######  Remark 0.8.

Suppose $k=\mathbb{F}_{q}$ is a finite field. For any $\mathbb{F}_{q}$-point
$x$ of $G/P$, the fiber $\pi^{-1}(x)$ of the projection $\pi:G\to G/P$ is a
principal $P$-homogenous space. By [16, VI §1 Corollary 1], $\pi^{-1}(x)$ has
a $\mathbb{F}_{q}$-point. Thus any $\mathbb{F}_{q}$-point of $G/P$ is the
image of a $\mathbb{F}_{q}$-point of $G$. For any $\mathbb{F}_{q}$-point $g$
of $G$, denote its image in $G/P$ by $gP$. Suppose $\mathcal{L}_{\beta}$ is
the Lang sheaf defined by a character
$\beta:(G/Q[G,G])(\mathbb{F}_{q})\to\overline{\mathbb{Q}}_{\ell}^{*}$. Then by
the Grothendieck trace formula and Corollary 0.7, for any
$\mathbb{F}_{q}$-point $\phi$ of $V^{\vee}$, we have

$\displaystyle\mathrm{Tr}(\mathrm{Frob}_{\phi},\mathcal{T}_{\ast}(G,\beta,V,v,\psi)_{\bar{\phi}})=(-1)^{n+N-1}G_{*}(\beta,\psi)\sum_{x\in(G/P)(\mathbb{F}_{q}),\;\phi(g_{x}v)\not=0}\beta(\phi(g_{x}v)^{-1}\cdot
g_{x}),$

where for any $\mathbb{F}_{q}$-point $x$ of $G/P$, $g_{x}$ is a
$\mathbb{F}_{q}$-point of $G$ in the fiber $\pi^{-1}(x)$, $\phi(g_{x}v)^{-1}$
is considered as a $\mathbb{F}_{q}$-point of $\mathbb{G}_{m}\subset G$,
$\beta(\phi(g_{x}v)^{-1}\cdot g_{x})$ is the value of $\beta$ at the image of
the $\mathbb{F}_{q}$-point $\phi(g_{x}v)^{-1}\cdot g_{x}$ in $G/Q[G,G]$, and
$G_{*}(\beta,\psi)$ is the Gauss sum. Note that the value
$\beta(\phi(g_{x}v)^{-1}\cdot g_{x})$ is independent of the choice of the
$\mathbb{F}_{q}$-points $g_{x}$ in the fiber $\pi^{-1}(x)$. If there exists a
character
$\beta_{0}:(G/P[G,G])(\mathbb{F}_{q})\to\overline{\mathbb{Q}}_{\ell}^{\ast}$
such that $\beta$ is the composite

$(G/Q[G,G])(\mathbb{F}_{q})\to(G/P[G,G])(\mathbb{F}_{q})\stackrel{{\scriptstyle\beta_{0}}}{{\to}}\overline{\mathbb{Q}}_{\ell}^{\ast}.$

Then

$\displaystyle\mathrm{Tr}(\mathrm{Frob}_{\phi},\mathcal{T}_{\ast}(G,\beta,V,v,\psi)_{\bar{\phi}})$
$\displaystyle=$
$\displaystyle(-1)^{n+N-1}q\sum_{x\in(G/P)(\mathbb{F}_{q}),\;\phi(g_{x}v)\not=0}\beta_{0}(g_{x}),$
$\displaystyle\mathrm{Tr}(\mathrm{Frob}_{\phi},\mathcal{T}_{!}(G,\beta,V,v,\psi)_{\bar{\phi}})$
$\displaystyle=$
$\displaystyle(-1)^{n+N-1}\sum_{x\in(G/P)(\mathbb{F}_{q})}\beta_{0}(g_{x})-(-1)^{n+N-1}q\sum_{x\in(G/P)(\mathbb{F}_{q}),\;\phi(g_{x}v)=0}\beta_{0}(g_{x}).$

The second equation follows from the distinguished triangle in Theorem 0.6
(ii).

##  1\. Proof of Theorem 0.6

Let $\kappa_{0}:G/Q\to V-\\{0\\}$ be the morphism $gQ\mapsto gv$, let
$j_{0}:V-\\{0\\}\hookrightarrow V$ be the open immersion, and let $\pi:V\times
G/P\to V$ be the projection. The following diagram commutes:

$\begin{array}[]{rcccl}&&G/Q&&\\\
&\stackrel{{\scriptstyle\kappa}}{{\underset{\cong}{\swarrow}}}&&\stackrel{{\scriptstyle\kappa_{0}}}{{\searrow}}&\\\
\mathbb{L}_{G/P}^{\circ}&&&&V-\\{0\\}\\\ {\scriptstyle
j_{\mathbb{L}_{G/P}^{\circ}}}\downarrow&&&&\downarrow{\scriptstyle j_{0}}\\\
\mathbb{L}_{G/P}&\stackrel{{\scriptstyle
i_{G/P}}}{{\to}}&V\times_{k}(G/P)&\stackrel{{\scriptstyle\pi}}{{\to}}&V\\\
&\searrow&\downarrow&&\downarrow\\\
&&G/P&\to&\mathrm{Spec}\,\mathbb{F}_{q}\end{array}$

We have $\iota=j_{0}\kappa_{0}$, and hence

$\displaystyle\mathcal{T}_{!}(G,\beta,V,v,\psi)=\mathscr{F}_{\psi}(j_{0!}\kappa_{0!}\mathcal{L}_{\beta}[n])\cong\mathscr{F}_{\psi}(R\pi_{!}Ri_{G/P,!}j_{\mathbb{L}_{G/P}^{\circ},!}\mathcal{L}_{\beta}[n]).$

By [12, 1.2.3.5 and 1.2.2.4], we have

$\mathscr{F}_{\psi}R\pi_{!}\cong
R\mathrm{pr}_{!}\mathscr{F}_{\psi,V\times_{k}(G/P)},\quad\mathscr{F}_{\psi,V\times_{k}(G/P)}Ri_{G/P,!}\cong\mathrm{ev}_{G/P}^{\ast}\mathscr{F}_{\psi,\mathbb{L}_{G/P}}[N-1],$

where $\mathscr{F}_{\psi,V\times_{k}(G/P)}$ is the Deligne-Fourier transform
for the trivial vector bundle $V\times_{k}(G/P)\to G/P$ and
$\mathscr{F}_{\psi,\mathbb{L}_{G/P}}$ is the Deligne-Fourier transform for the
vector bundle $\mathbb{L}_{G/P}\to G/P$. We thus have

$\mathcal{T}_{!}(G,\beta,V,v,\psi)\cong\mathscr{F}_{\psi}(R\pi_{!}Ri_{G/P,!}j_{\mathbb{L}_{G/P}^{\circ},!}\mathcal{L}_{\beta}[n])\cong
R\mathrm{pr}_{!}\mathrm{ev}_{G/P}^{\ast}\mathscr{F}_{\psi,\mathbb{L}_{G/P}}(j_{\mathbb{L}_{G/P}^{\circ},!}\mathcal{L}_{\beta})[n+N-1].$

The assertion for $\mathcal{T}_{!}(G,\beta,V,v)$ in Theorem 0.6 (i) then
follows from Lemma 1.8 below.

Using [12, 1.3.1.1], the smooth base change theorem, and the same proof as
[12, 1.2.3.5], one can show

$\mathscr{F}_{\psi}R\pi_{\ast}=R\mathrm{pr}_{\ast}\mathscr{F}_{\psi,V\times_{k}(G/P)}.$

Since $i_{G/P}$ is a closed immersion we have $Ri_{G/P,\ast}=i_{G/P,!}$, and
hence by [12, 1.2.2.4], we have

$\mathscr{F}_{\psi,V\times_{k}(G/P)}Ri_{G/P,\ast}\cong\mathrm{ev}_{G/P}^{\ast}\mathscr{F}_{\psi,\mathbb{L}_{G/P}}[N-1].$

So we have

$\displaystyle\mathcal{T}_{\ast}(G,\beta,V,v,\psi)$ $\displaystyle\cong$
$\displaystyle\mathscr{F}_{\psi}(Rj_{0\ast}R\kappa_{0\ast}\mathcal{L}_{\beta}[n])$
$\displaystyle\cong$
$\displaystyle\mathscr{F}_{\psi}(R\pi_{\ast}Ri_{G/P,\ast}Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[n])$
$\displaystyle\cong$ $\displaystyle
R\mathrm{pr}_{\ast}\mathrm{ev}_{G/P}^{\ast}\mathscr{F}_{\psi,\mathbb{L}_{G/P}}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta})[n+N-1].$

The assertion for $\mathcal{T}_{\ast}(G,\beta,V,v,\psi)$ in Theorem 0.6 (i)
also follows from Lemma 1.8 below and the proper base change theorem .

Suppose furthermore that there exists a multiplicative sheaf
$\mathcal{L}_{\beta_{0}}$ on $G/P[G,G]$ such that $\mathcal{L}_{\beta}$ is
isomorphic to its inverse image under the canonical morphism $G/Q[G,G]\to
G/P[G,G]$. Then we have

$G_{*}(\beta,\psi)\cong\overline{\mathbb{Q}}_{\ell},\quad
G_{!}(\beta,\psi)\cong\overline{\mathbb{Q}}_{\ell}(-1).$

Moreover, by (i) and the proper base change theorem, we have

$\displaystyle\mathcal{T}_{\ast}(G,\beta,V,v,\psi)$ $\displaystyle\cong$
$\displaystyle
R\mathrm{pr}_{\ast}\mathrm{ev}_{G/P}^{\ast}j_{\mathbb{L}_{G/P}^{\vee\circ},!}(\mathcal{L}_{\beta_{0}}|_{\mathbb{L}_{G/P}^{\vee\circ}})(-1)[n+N-1]$
$\displaystyle\cong$ $\displaystyle
R\mathrm{pr}_{\ast}j^{\prime}_{!}\mathrm{ev}_{G/P}^{\prime\ast}(\mathcal{L}_{\beta_{0}}|_{\mathbb{L}_{G/P}^{\vee\circ}})(-1)[n+N-1]$
$\displaystyle\cong$ $\displaystyle
R\mathrm{pr}_{\ast}j^{\prime}_{!}(\mathcal{L}_{\beta_{0}}|_{U^{\prime}})(-1)[n+N-1]..$

Let $i:G/P\to\mathbb{L}_{G/P}^{\vee}$ be the zero section. We have

$Ri^{!}(\mathcal{L}_{\beta_{0}}|_{\mathbb{L}_{G/P}^{\vee}})\cong\mathcal{L}_{\beta_{0}}(-1)[-2]$

and hence we have a distinguished triangle

$i_{*}\mathcal{L}_{\beta_{0}}(-1)[-2]\to\mathcal{L}_{\beta_{0}}|_{\mathbb{L}_{G/P}^{\vee}}\to
Rj_{\mathbb{L}_{G/P}^{\vee\circ},*}(\mathcal{L}_{\beta_{0}}|_{\mathbb{L}_{G/P}^{\vee\circ}})\to.$

Applying $\mathrm{ev}^{*}_{G/P}$, we get a distinguished triangle

$i^{\prime}_{*}(\mathcal{L}_{\beta_{0}}|_{H})(-1)[-2]\to(\mathcal{L}_{\beta_{0}})|_{V^{\vee}\times_{k}G/P}\to\mathrm{ev}_{G/P}^{\ast}Rj_{\mathbb{L}_{G/P}^{\vee\circ},*}(\mathcal{L}_{\beta_{0}}|_{\mathbb{L}_{G/P}^{\vee\circ}})\to.$

Applying $R\mathrm{pr}_{!}$, we get the distinguished triangle

$R\mathrm{pr}_{!}Ri^{\prime}_{*}(\mathcal{L}_{\beta_{0}}|_{H})(-1)[n+N-3]\to
R\mathrm{pr}_{!}(\mathcal{L}_{\beta_{0}}|_{V^{\vee}\times_{k}G/P})[n+N-1]\to\mathcal{T}_{!}(G,\beta,V,v,\psi)\to.$

In the rest of this section, we prove Lemma 1.8 which gives formulas for
$\mathscr{F}_{\psi,\mathbb{L}_{G/P}}(j_{\mathbb{L}_{G/P}^{\circ},!}\mathcal{L}_{\beta})$
and
$\mathscr{F}_{\psi,\mathbb{L}_{G/P}}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta})$

######  Lemma 1.1.

Let $m:G\times_{k}\mathbb{L}_{G/P}\to\mathbb{L}_{G/P}$ be the morphism
$(g,x)\mapsto gx$. Denote the inverse image of $\mathcal{L}_{\beta}$ by the
morphism $G\to G/Q$ by $\mathcal{L}^{\prime}_{\beta}$. We have

$\displaystyle
m^{\ast}j_{\mathbb{L}_{G/P}^{\circ},!}\mathcal{L}_{\beta}\cong\mathcal{L}^{\prime}_{\beta}\boxtimes
j_{\mathbb{L}_{G/P}^{\circ},!}\mathcal{L}_{\beta},\quad
m^{\ast}Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}\cong\mathcal{L}^{\prime}_{\beta}\boxtimes
Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}.$

###### Proof.

We prove the second statement. The proof for the first statement is similar.

Let $m^{\prime}:G\times_{k}G/Q\to G/Q$ be the morphism $(g,x)\mapsto gx$. We
have

$m^{\prime\ast}\mathcal{L}_{\beta}\cong\mathcal{L}^{\prime}_{\beta}\boxtimes\mathcal{L}_{\beta}.$

Let $\phi:G\times_{k}\mathbb{L}_{G/P}\to G\times_{k}\mathbb{L}_{G/P}$ be the
isomorphism defined by $(g,x)\mapsto(g,gx)$ and fix notation by the following
commutative diagram:

$\begin{array}[]{ccrcrcr}G&\stackrel{{\scriptstyle\pi^{\prime}_{1}}}{{\leftarrow}}&G\times_{k}G/Q&\stackrel{{\scriptstyle\phi^{\prime}}}{{\to}}&G\times_{k}G/Q&\stackrel{{\scriptstyle\pi_{2}^{\prime}}}{{\to}}&G/Q\\\
\parallel&&{\scriptstyle\mathrm{id}_{G}\times(j_{\mathbb{L}_{G/P}^{\circ}}\kappa)}\downarrow&&{\scriptstyle\mathrm{id}_{G}\times(j_{\mathbb{L}_{G/P}^{\circ}}\kappa)}\downarrow&&{\scriptstyle
j_{\mathbb{L}_{G/P}^{\circ}}\kappa}\downarrow\\\
G&\stackrel{{\scriptstyle\pi_{1}}}{{\leftarrow}}&G\times_{k}\mathbb{L}_{G/P}&\stackrel{{\scriptstyle\phi}}{{\to}}&G\times_{k}\mathbb{L}_{G/P}&\stackrel{{\scriptstyle\pi_{2}}}{{\to}}&\mathbb{L}_{G/P}.\end{array}$

We have $m=\pi_{2}\phi$. It follows that $m$ is smooth. By the smooth base
change theorem and the projection formula, we have

$\displaystyle
m^{\ast}R(j_{\mathbb{L}_{G/P}^{\circ}}\kappa)_{\ast}\mathcal{L}_{\beta}$
$\displaystyle\cong$ $\displaystyle
R(\mathrm{id}_{G}\times(j_{\mathbb{L}_{G/P}^{\circ}}\kappa))_{\ast}m^{\prime\ast}\mathcal{L}_{\beta}$
$\displaystyle\cong$ $\displaystyle
R(\mathrm{id}_{G}\times(j_{\mathbb{L}_{G/P}^{\circ}}\kappa))_{\ast}(\mathcal{L}^{\prime}_{\beta}\boxtimes\mathcal{L}_{\beta})$
$\displaystyle\cong$ $\displaystyle
R(\mathrm{id}_{G}\times(j_{\mathbb{L}_{G/P}^{\circ}}\kappa))_{\ast}((\mathrm{id}_{G}\times(j_{\mathbb{L}_{G/P}^{\circ}}\kappa))^{\ast}\pi_{1}^{\ast}\mathcal{L}^{\prime}_{\beta}\otimes\pi^{\prime\ast}_{2}\mathcal{L}_{\beta})$
$\displaystyle\cong$
$\displaystyle\pi_{1}^{\ast}\mathcal{L}^{\prime}_{\beta}\otimes
R(\mathrm{id}_{G}\times(j_{\mathbb{L}_{G/P}^{\circ}}\kappa))_{\ast}\pi^{\prime\ast}_{2}\mathcal{L}_{\beta}$
$\displaystyle\cong$
$\displaystyle\pi_{1}^{\ast}\mathcal{L}^{\prime}_{\beta}\otimes\pi_{2}^{\ast}R(j_{\mathbb{L}_{G/P}^{\circ}}\kappa)_{\ast}\mathcal{L}_{\beta}.$

This proves the second assertion. ∎

######  Lemma 1.2.

Let $S$ and $T$ be $k$-schemes of finite type, $E\to S$ and $F\to T$ vector
bundles over $S$ and $T$ of ranks $r_{1}$ and $r_{2}$, respectively,
$\phi:S\stackrel{{\scriptstyle\cong}}{{\to}}T$ an isomorphism, and $g:E\to
F\times_{T}S$ a morphism of vector bundles. Let $f:E\to F$ be the composite of
the morphism $g$ and the projection $\pi:F\times_{T}S\to F$. Let
$f^{\vee}:F^{\vee}\to E^{\vee}$ be the transpose of $f$. Then for any object
$K\in\mathrm{ob}\,D_{c}^{b}(E,\overline{\mathbb{Q}}_{\ell})$, we have

$\mathscr{F}_{\psi}(Rf_{!}K)\cong
f^{\vee\ast}\mathscr{F}_{\psi}(K)[r_{2}-r_{1}].$

Let’s describe the transpose $f^{\vee}:F^{\vee}\to E^{\vee}$. We have
commutative diagrams

$\begin{array}[]{ccccccc}E&\stackrel{{\scriptstyle
f}}{{\to}}&F&&F^{\vee}&\stackrel{{\scriptstyle f^{\vee}}}{{\to}}&E^{\vee}\\\
\downarrow&&\downarrow&&\downarrow&&\downarrow\\\
S&\stackrel{{\scriptstyle\phi}}{{\to}}&T,&&T&\stackrel{{\scriptstyle\phi^{-1}}}{{\to}}&S.\end{array}$

Let $\mathcal{E}$ and $\mathcal{E}^{\vee}$ be the $\mathcal{O}_{S}$-modules of
sections of $E\to S$ and $E^{\vee}\to S$, respectively, and let $\mathcal{F}$
and $\mathcal{F}^{\vee}$ be the $\mathcal{O}_{T}$-modules of sections of $F\to
T$ and $F^{\vee}\to T$, respectively. The morphism $f:E\to F$ is completely
characterized by the homomorphism of $\mathcal{O}_{T}$-modules
$f^{\sharp}:\phi_{\ast}\mathcal{E}\to\mathcal{F}.$ For any open subset $V$ of
$T$, let $U=\phi^{-1}(V)$. The homomorphism
$f^{\sharp}_{V}:\mathcal{E}(U)\to\mathcal{F}(V)$ maps a section $s:U\to E$ of
the vector bundle $E\to S$ to the section $fs\phi^{-1}:V\to F$ of the vector
bundle $F\to T$. Similarly, the morphism $f^{\vee}:F^{\vee}\to E^{\vee}$ is
completely characterized by the homomorphism of $\mathcal{O}_{S}$-modules
$f^{\vee\sharp}:\phi^{-1}_{\ast}\mathcal{F}^{\vee}\to\mathcal{E}^{\vee},$ and
the homomorphism
$f^{\vee\sharp}_{U}:\mathcal{F}^{\vee}(V)\to\mathcal{E}^{\vee}(U)$ maps a
section $t^{\ast}:V\to F^{\vee}$ of the vector bundle $F^{\vee}\to T$ to the
section $f^{\vee}t^{\ast}\phi:U\to E^{\vee}$ of the vector bundle $E^{\vee}\to
S$. For any $s\in\mathcal{E}(U)$ and $t^{\ast}\in\mathcal{F}^{\vee}(V)$, we
have

$\phi_{V}^{\natural}(\langle f^{\sharp}_{V}(s),t^{\ast}\rangle)=\langle
s,f^{\vee\sharp}_{U}(t^{\ast})\rangle,$

where $\langle\,,\,\rangle$ denotes the pairings
$\mathcal{E}\times\mathcal{E}^{\vee}\to\mathcal{O}_{S}$ and
$\mathcal{F}\times\mathcal{F}^{\vee}\to\mathcal{O}_{T}$, and
$\phi^{\natural}:\mathcal{O}_{T}\to\phi_{\ast}\mathcal{O}_{S}$ is the morphism
on structure sheaves coming from the morphism $\phi:S\to T$.

###### Proof of Lemma 1.2.

Let $g^{\vee}:F^{\vee}\times_{T}S\to E^{\vee}$ be the transpose of $g$, and
let $\pi^{\vee}:F^{\vee}\times_{T}S\to F^{\vee}$ be the projection. Note that
$\pi^{\vee}$ is an isomorphism and $g^{\vee}=f^{\vee}\pi^{\vee}$. By [12,
1.2.3.5 and 1.2.2.4], we have

$\displaystyle\mathscr{F}_{\psi}(Rf_{!}K)$ $\displaystyle\cong$
$\displaystyle\mathscr{F}_{\psi}(R\pi_{!}Rg_{!}K)$ $\displaystyle\cong$
$\displaystyle R\pi^{\vee}_{!}\mathscr{F}_{\psi}(Rg_{!}K)$
$\displaystyle\cong$ $\displaystyle
R\pi^{\vee}_{!}g^{\vee\ast}\mathscr{F}_{\psi}(K)[r_{2}-r_{1}]$
$\displaystyle\cong$ $\displaystyle
f^{\vee\ast}\mathscr{F}_{\psi}(K)[r_{2}-r_{1}].$

∎

######  Lemma 1.3.

Let $X$ be an $k$-scheme of finite type, $E\to X$ a vector bundle, and $G$ an
algebraic group over $k$ acting on $X$ and acting equivariantly and linearly
on the vector bundle $E\to X$. Let $m:G\times_{k}E\to E$ be the action of $G$
on $E$, and let $m^{\vee}:G\times_{k}E^{\vee}\to E^{\vee}$ be the
contragredient action of $G$ on $E^{\vee}$. For any
$K\in\mathrm{ob}\,D_{c}^{b}(E,\overline{\mathbb{Q}}_{\ell})$, we have

$m^{\vee\ast}\mathscr{F}_{\psi}(K)\cong\mathscr{F}_{\psi}(m^{\ast}K),$

where the righthand side is the Deligne-Fourier transform for the vector
bundle $G\times_{k}E\to G\times_{k}X$.

###### Proof.

Let $f:G\times_{k}E\to G\times_{k}E$ be the isomorphism
$(g,x)\mapsto(g,g^{-1}x)$, and let $\phi:G\times_{k}X\to G\times_{k}X$ be the
isomorphism defined in the same way. The transpose of $f$ as described in
Lemma 1.2 is exactly the morphism

$f^{\vee}:G\times_{k}E^{\vee}\to G\times_{k}E^{\vee},\quad(g,x)\mapsto(g,gx).$

Let $\pi:G\times_{k}E\to E$ and $\pi^{\vee}:G\times_{k}E^{\vee}\to E^{\vee}$
be the projections. Since $m^{\vee}=\pi^{\vee}f^{\vee}$, we have

$m^{\vee\ast}\mathscr{F}_{\psi}(K)\cong
f^{\vee\ast}\pi^{\vee\ast}\mathscr{F}_{\psi}(K).$

Since the Deligne-Fourier transform commutes with base change ([12, 1.2.2.9]),
we have

$\pi^{\vee\ast}\mathscr{F}_{\psi}(K)\cong\mathscr{F}_{\psi}(\pi^{\ast}K).$

By Lemma 1.2, we have

$f^{\vee\ast}\mathscr{F}(\pi^{\ast}K)\cong\mathscr{F}_{\psi}(Rf_{!}\pi^{\ast}K).$

We thus have

$m^{\vee\ast}\mathscr{F}_{\psi}(K)\cong\mathscr{F}_{\psi}(Rf_{!}\pi^{\ast}K).$

Since $f$ is an isomorphism and $\pi f^{-1}=m$, we have

$Rf_{!}\pi^{\ast}K\cong m^{\ast}K.$

Our assertion follows. ∎

######  Lemma 1.4.

Let $S$ be an $k$-scheme of finite type, let $E\to S$ be a vector bundle,
$Y\to S$ a morphism of finite type, and
$K\in\mathrm{ob}\,D_{c}^{b}(E,\overline{\mathbb{Q}}_{\ell})$ and
$L\in\mathrm{ob}D_{c}^{b}(Y,\overline{\mathbb{Q}}_{\ell})$. Then we have

$\mathscr{F}_{\psi}(K\boxtimes L)\cong\mathscr{F}_{\psi}(K)\boxtimes L,$

where on the lefthand side the Deligne-Fourier transform is for the vector
bundle $E\times_{S}Y\to Y$, and on the righthand side it is for the vector
bundle $E\to S$.

###### Proof.

This follows from the definition of the Deligne-Fourier transform and the
Künneth formula. ∎

######  Lemma 1.5.

(i) Any multiplicative sheaf $\mathcal{L}$ on $\mathbb{G}_{m,k}$ is tamely
ramified at $0$ and $\infty$.

(ii) If a multiplicative sheaf on $\mathbb{G}_{m,k}$ is trivial on
$\mathbb{G}_{m,\bar{k}}$, then it is trivial.

###### Proof.

(i) This is proved in [6, Lemma 3.5]. For completeness, we include another
proof, which is communicated to me by Haoyu Hu. By base change, we may assume
$k$ is algebraically closed. Let $\bar{\eta}$ be a geometric generic point of
$\mathbb{G}_{m}$, let
$\rho:\pi_{1}(\mathbb{G}_{m},\bar{\eta})\to\overline{\mathbb{Q}}_{\ell}^{*}$
be the character defined by $\mathcal{L}$, and let $P$ be the wild inertia
subgroup at $0$ or at $\infty$. The group $P$ is a pro-$p$-group and $\rho(P)$
is finite. Let $p^{m}$ be the number of elements in $\rho(P)$. Then we have
$\rho^{p^{m}}|_{P}=1$. Let

$m_{n}:\mathbb{G}_{m,k}^{n}\to\mathbb{G}_{m,k}$

be the morphism defined by multiplication. We have

$m_{n}^{*}\mathcal{L}\cong\boxtimes^{n}\mathcal{L}.$

Pulling back this isomorphism by the diagonal morphism
$\mathbb{G}_{m,k}\to\mathbb{G}_{m,k}^{n}$, we get

$[n]^{*}\mathcal{L}\cong\mathcal{L}^{\otimes n},$

where $[n]:\mathbb{G}_{m,k}\to\mathbb{G}_{m,k}$ is the morphism $x\mapsto
x^{n}$. In particular, the character corresponding to the sheaf
$[p^{m}]^{*}\mathcal{L}$ is $\rho^{p^{m}}$. Since $\rho^{p^{m}}|_{P}=1$,
$[p^{m}]^{*}\mathcal{L}$ is tamely ramified at $0$ and $\infty$. But $[p^{m}]$
is a finite surjective radiciel morphism, and it has no effect on étale
topology. So $\mathcal{L}$ is tame at $0$ and $\infty$.

(ii) Let $\mathcal{L}$ be a multiplicative sheaf on $\mathbb{G}_{m,k}$ which
is trivial on $\mathbb{G}_{m,\bar{k}}$. Then $\mathcal{L}$ is isomorphic to
the inverse image of a sheaf on $\mathrm{Spec}\,k$. But the restriction of
$\mathcal{L}$ to the unit section $1:\mathrm{Spec}\,k\to\mathbb{G}_{m,k}$ is
trivial. So we have $\mathcal{L}\cong\overline{\mathbb{Q}}_{\ell}$. ∎

######  Lemma 1.6.

$Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}$ commutes with any base
change $Y\to G/P$.

###### Proof.

Note that $\mathbb{L}^{\circ}_{G/P}\to G/P$ is a principal homogeneous space
for the multiplicative group-scheme $\mathbb{G}_{m,G/P}$ over $G/P$, and
$\mathbb{L}_{G/P}\to G/P$ is the associated line bundle for the canonical
action of $\mathbb{G}_{m}$ on $\mathbb{A}^{1}$. Let $\\{U_{i}\\}$ be a Zariski
open covering of $G/P$ such that the line bundle $\mathbb{L}_{G/P}$ is trivial
over each $U_{i}$, and we compactify $\mathbb{L}_{U_{i}}^{\circ}$ to a
$\mathbb{P}^{1}$-bundle by adding the $0$ section and the $\infty$ section.
The sheaf $\mathcal{L}_{\beta}$ is homogeneous on $\mathbb{L}^{\circ}_{G/P}$
with respect to the $\mathbb{G}_{m,G/P}$-action. In particular, when
restricted to the generic point $\eta$ of $G/P$,
$\mathcal{L}_{\beta}|_{\mathbb{L}^{\circ}_{\eta}}$ is a multiplicative sheaf
on $\mathbb{G}_{m,\eta}$. By Lemma 1.5,
$\mathcal{L}_{\beta}|_{\mathbb{L}^{\circ}_{\eta}}$ is tamely ramified at $0$
and $\infty$. This implies that
$\mathcal{L}_{\beta}|_{\mathbb{L}_{U_{i}}^{\circ}}$ is tamely ramified at the
$0$ section and the $\infty$ section. Our assertion then follows from [3, XIII
2.1.10]. ∎

Let $G_{!}(\beta,\psi)$ and $G_{*}(\beta,\psi)$ be the sheaf on
$\mathrm{Spec}\,k$ defined in Theorem 0.6.

######  Lemma 1.7.

Regard $v^{\ast}$ as a $k$-point in the fiber of
${\mathbb{L}}^{\vee}_{G/P,eP}\cong{\mathbb{L}}^{\vee}_{[v]}$, and let
$0_{[v]}$ be the zero element in the line
${\mathbb{L}}^{\vee}_{G/P,eP}\cong{\mathbb{L}}^{\vee}_{[v]}$. We have

$\displaystyle\mathscr{F}_{\psi}(j_{\mathbb{L}_{G/P}^{\circ},!}\mathcal{L}_{\beta}[1])|_{v^{\ast}}$
$\displaystyle\cong$ $\displaystyle G_{!}(\beta,\psi)[1],$
$\displaystyle\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])|_{v^{\ast}}$
$\displaystyle\cong$ $\displaystyle G_{*}(\beta,\psi)[1],$
$\displaystyle\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])|_{0_{[v]}}$
$\displaystyle\cong$ $\displaystyle 0.$

Moreover $G_{!}(\beta,\psi)$ and $G_{*}(\beta,\psi)$ are of rank $1$.

###### Proof.

With respect to the morphism $\mathrm{Spec}\,k\to{G/P}$ defined by the
rational point $eP$, the base change of the morphism
$j_{\mathbb{L}_{G/P}^{\circ}}:\mathbb{L}_{G/P}^{\circ}\to\mathbb{L}_{G/P}$ can
be identified the canonical open immersion
$j:\mathbb{G}_{m,k}\hookrightarrow\mathbb{A}^{1}_{k}$. By Lemma 1.6, the base
change of $Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1]$ can be
identified with $Rj_{\ast}(\mathcal{L}_{\beta}|_{\mathbb{G}_{m,k}})[1]$. The
Deligne-Fourier transform and
$j_{\mathbb{L}_{G/P}^{\circ},!}\mathcal{L}_{\beta}[1]$ also commute with base
change ([12, 1.2.2.9]). So we are reduced to show

$\displaystyle\mathscr{F}_{\psi}(j_{!}(\mathcal{L}_{\beta}|_{\mathbb{G}_{m,k}})[1])|_{1}$
$\displaystyle\cong$ $\displaystyle G_{*}(\beta,\psi)[1],$
$\displaystyle\mathscr{F}_{\psi}(Rj_{\ast}(\mathcal{L}_{\beta}|_{\mathbb{G}_{m,k}})[1])|_{1}$
$\displaystyle\cong$ $\displaystyle G_{!}(\beta,\psi)[1],$
$\displaystyle\mathscr{F}_{\psi}(Rj_{\ast}(\mathcal{L}_{\beta}|_{\mathbb{G}_{m,k}})[1])|_{0}$
$\displaystyle\cong$ $\displaystyle 0,$

where the lefthand side are restrictions to the unit section
$1:\mathrm{Spec}\,k\to\mathbb{A}^{1}_{k}$ and the zero section
$0:\mathrm{Spec}\,k\to\mathbb{A}^{1}_{k}$.

For convenience, denote $\mathcal{L}_{\beta}|_{\mathbb{G}_{m,k}}$ by
$\mathcal{L}$. First consider the case where
$\mathcal{L}|_{\mathbb{G}_{m,\bar{k}}}$ is nontrivial. In this case, we claim
that $(Rj_{\ast}\mathcal{L})_{\bar{0}}=0$ so that we have
$j_{!}\mathcal{L}\cong Rj_{*}\mathcal{L}$. Indeed, by Lemma 1.5,
$\mathcal{L}|_{\mathbb{G}_{m,\bar{k}}}$ is a nontrivial multiplicative sheaf
on $\mathbb{G}_{m,\bar{k}}$. It is necessarily isomorphic to a nontrivial
Kummer sheaf $\mathcal{K}_{\chi}$ for a character
$\chi:\varprojlim_{(n,p)=1}\mu_{n}(\bar{k})\to\overline{\mathbb{Q}}_{\ell}^{*}$.
The invariant and coinvariant of the inertia subgroup $I_{0}$ at $0$ for the
representation of $\pi_{1}(\mathbb{G}_{m,\bar{k}})$ corresponding to
$\mathcal{K}_{\chi}$ are trivial. Our claim follows the calculation in [2,
Dualité 1.3]. So it suffices to prove

$\mathscr{F}_{\psi}(j_{!}\mathcal{L}[1])|_{1}\cong
G_{!}(\beta,\psi)[1],\quad\mathscr{F}_{\psi}(j_{!}\mathcal{L}[1])|_{0}\cong
0.$

By definition, we have

$\mathscr{F}_{\psi}(j_{!}\mathcal{L}[1])_{\bar{1}}\cong
R\Gamma_{c}(\mathbb{G}_{m,\bar{k}},\mathcal{L}\otimes\mathcal{L}_{\psi})[2].$

We have

$H_{c}^{0}(\mathbb{G}_{m,\bar{k}},\mathcal{L}\otimes\mathcal{L}_{\psi})=0$

since $\mathbb{G}_{m,\bar{k}}$ is an affine curve. We have

$H_{c}^{2}(\mathbb{G}_{m,\bar{k}},\mathcal{L}\otimes\mathcal{L}_{\psi})\cong
H^{0}(\mathbb{G}_{m,\bar{k}},\mathcal{L}^{-1}\otimes\mathcal{L}_{\psi^{-1}})(-1)=0$

by Poincaré duality and the fact that the invariant of the representation of
$\pi_{1}(\mathbb{G}_{m,\bar{k}})$ corresponding to
$\mathcal{L}\otimes\mathcal{L}_{\psi}$ is trivial. By the Grothendieck-Ogg-
Shafarevich formula and the fact that $\mathcal{L}\otimes\mathcal{L}_{\psi}$
is lisse on $\mathbb{G}_{m,\bar{k}}$, tame at $0$ and with Swan conductor $1$
at $\infty$, we have

$\chi_{c}(\mathbb{G}_{m,\bar{k}},\mathcal{L}\otimes\mathcal{L}_{\psi})=-1.$

So $H^{1}_{c}(\mathbb{G}_{m,\bar{k}},\mathcal{L}\otimes\mathcal{L}_{\psi})$ is
of rank $1$. Hence

$\mathscr{F}_{\psi}(j_{!}(\mathcal{L}_{\beta}|_{\mathbb{G}_{m,k}})[1])_{1}\cong
G_{!}(\beta,\psi)[1],$

and $G_{*}(\beta,\psi)$ is of rank 1. Similarly one can prove
$H^{1}_{c}(\mathbb{G}_{m,\bar{k}},\mathcal{L})=0$ for all $i$. Hence
$R\Gamma_{c}(\mathbb{G}_{m,\bar{k}},\mathcal{L})=0$. So we have

$\mathscr{F}_{\psi}(j_{!}\mathcal{L}[1])_{\bar{0}}\cong
R\Gamma_{c}(\mathbb{G}_{m,\bar{k}},\mathcal{L})[2]=0.$

Next, suppose $\mathcal{L}_{\mathbb{G}_{m,\bar{k}}}$ is trivial. By Lemma 1.5
(ii), we have $\mathcal{L}\cong\overline{\mathbb{Q}}_{\ell}$. So we have

$\mathscr{F}_{\psi}(j_{!}\mathcal{L}[1])_{\bar{1}}\cong
R\Gamma_{c}(\mathbb{G}_{m,\bar{k}},\mathcal{L}_{\psi})[2].$

We have a distinguished triangle

$R\Gamma_{c}(\mathbb{G}_{m,\bar{k}},\mathcal{L}_{\psi})\to
R\Gamma_{c}(\mathbb{A}^{1}_{\bar{k}},\mathcal{L}_{\psi})\to\mathcal{L}_{\psi,\bar{0}}\to.$

By [2, Sommes trig. 2,7], we have

$R\Gamma_{c}(\mathbb{A}^{1}_{\bar{k}},\mathcal{L}_{\psi})=0,\quad\mathcal{L}_{\psi}|_{0}\cong\overline{\mathbb{Q}}_{\ell}.$

So we have

$R\Gamma_{c}(\mathbb{G}_{m,\bar{k}},\mathcal{L}_{\psi})[2]\cong\overline{\mathbb{Q}}_{\ell}[1].$

Therefore

$\mathscr{F}_{\psi}(j_{!}\mathcal{L}[1])_{1}\cong\overline{\mathbb{Q}}_{\ell}[1]\cong
G_{!}(\beta,\psi)[1].$

Denote by $D$ the Verdier dual functor on $\mathbb{A}^{1}$. We have

$\displaystyle\mathscr{F}_{\psi}(Rj_{*}\overline{\mathbb{Q}}_{\ell}[1])$
$\displaystyle\cong$
$\displaystyle\mathscr{F}_{\psi}(D(j_{!}\overline{\mathbb{Q}}_{\ell}[1]))(-1)$
$\displaystyle\cong$ $\displaystyle
D(\mathscr{F}_{\psi^{-1}}(j_{!}\overline{\mathbb{Q}}_{\ell}[1]))(-2).$

By [12, 2.3.1.3], we have

$\mathscr{F}_{\psi^{-1}}(j_{!}\overline{\mathbb{Q}}_{\ell}[1])|_{\mathbb{A}^{1}_{k}-\\{0\\}}\cong\mathcal{F}[1]$

for a lisse sheaf $\mathcal{F}$ on $\mathbb{A}_{k}^{1}-\\{0\\}$, and by the
formula that we have already proved, we have
$\mathcal{F}|_{1}\cong\overline{\mathbb{Q}}_{\ell}$. So

$\mathscr{F}_{\psi}(Rj_{*}\mathcal{L}[1])_{1}\cong\mathscr{F}_{\psi}(Rj_{*}\overline{\mathbb{Q}}_{\ell}[1])_{1}\cong\mathrm{Hom}(\mathcal{F}_{\bar{1}},\overline{\mathbb{Q}}_{\ell})[1](-1)\cong\overline{\mathbb{Q}}_{\ell}(-1)[1]=G_{*}(\beta,\psi)[1].$

We have

$\displaystyle\mathscr{F}_{\psi}(Rj_{*}\mathcal{L}[1])_{\bar{0}}\cong\mathscr{F}_{\psi}(Rj_{*}\overline{\mathbb{Q}}_{\ell})_{\bar{0}}\cong
R\Gamma_{c}(\mathbb{A}^{1}_{\bar{k}},Rj_{*}\overline{\mathbb{Q}}_{\ell}).$

Let $J:\mathbb{G}_{m,k}\hookrightarrow\mathbb{P}^{1}_{k}$ be the open
immersion. We have a distinguished triangle

$\begin{array}[]{ccccc}R\Gamma_{c}(\mathbb{A}^{1}_{\bar{k}},Rj_{*}\overline{\mathbb{Q}}_{\ell})\to&R\Gamma(\mathbb{P}^{1}_{\bar{k}},RJ_{*}\overline{\mathbb{Q}}_{\ell})&\to&R\Gamma(\overline{\infty},RJ_{*}\overline{\mathbb{Q}}_{\ell})&\to\\\
&\wr\\!\\!\parallel&&\wr\\!\\!\parallel&\\\
&R\Gamma(\mathbb{G}_{m,\bar{k}},\overline{\mathbb{Q}}_{\ell})&&(RJ_{*}\overline{\mathbb{Q}}_{\ell})_{\overline{\infty}}&\end{array}$

We claim that

$R\Gamma(\mathbb{G}_{m,\bar{k}},\overline{\mathbb{Q}}_{\ell})\cong(RJ_{*}\overline{\mathbb{Q}}_{\ell})_{\overline{\infty}}.$

Therefore
$R\Gamma_{c}(\mathbb{A}^{1}_{\bar{k}},Rj_{*}\overline{\mathbb{Q}}_{\ell})=0$
and hence $\mathscr{F}_{\psi}(Rj_{*}\mathcal{L}[1])_{0}=0.$ To prove the
claim, let $\tilde{K}_{\infty}$ be the fraction field of the strict
henselization of $\mathbb{G}_{m}$ at $\infty$. We have

$(RJ_{*}\overline{\mathbb{Q}}_{\ell})_{\overline{\infty}}\cong
R\Gamma(\mathrm{Spec}\,\tilde{K}_{\infty},\overline{\mathbb{Q}}_{\ell}).$

It suffices to show

$H^{i}(\mathbb{G}_{m,\bar{k}},\mu_{n})\cong
H^{i}(\mathrm{Spec}\,\tilde{K}_{\infty},\mu_{n})$

for all $(n,p)=1$. We compute $H^{i}(\mathbb{G}_{m,\bar{k}},\mu_{n})$ and
$H^{i}(\mathrm{Spec}\,\tilde{K}_{\infty},\mu_{n})$ and compare them using the
long exact sequences of cohomology groups associated to the the Kummer short
exact sequence. ∎

######  Lemma 1.8.

We have

$\displaystyle\mathscr{F}_{\psi,\mathbb{L}_{G/P}}(j_{\mathbb{L}_{G/P}^{\circ},!}\mathcal{L}_{\beta}[1])$
$\displaystyle\cong$ $\displaystyle
Rj_{\mathbb{L}_{G/P}^{\vee\circ},\ast}\mathcal{L}_{\beta}[1]\otimes
G_{!}(\beta,\psi),$
$\displaystyle\mathscr{F}_{\psi,\mathbb{L}_{G/P}}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}R\mathcal{L}_{\beta}[1])$
$\displaystyle\cong$ $\displaystyle
j_{\mathbb{L}_{G/P}^{\vee\circ},!}\mathcal{L}_{\beta}[1]\otimes
G_{*}(\beta,\psi).$

###### Proof.

The first assertion can be deduced from the second one by taking Verdier dual.
Let’s prove the second assertion.

Let $m:G\times_{k}\mathbb{L}_{G/P}\to\mathbb{L}_{G/P}$ be the action of $G$ on
$\mathbb{L}_{G/P}$, let
$m^{\vee}:G\times_{k}\mathbb{L}_{G/P}^{\vee}\to\mathbb{L}_{G/P}^{\vee}$ be the
contragredient action, and let $\phi$ be the composite

$G\to
G/Q\stackrel{{\scriptstyle\kappa^{\vee}}}{{\underset{\cong}{\to}}}\mathbb{L}_{G/P}^{\vee\circ}.$

By Lemmas 1.1, 1.3, and 1.4, we have

$\displaystyle
m^{\vee\ast}\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])$
$\displaystyle\cong$
$\displaystyle\mathscr{F}_{\psi}(m^{\ast}Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])$
$\displaystyle\cong$
$\displaystyle\mathscr{F}_{\psi}(\mathcal{L}^{\prime}_{\beta}\boxtimes
Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])$
$\displaystyle\cong$
$\displaystyle\mathcal{L}^{\prime}_{\beta}\boxtimes\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])$
$\displaystyle\cong$
$\displaystyle\phi^{\ast}\mathcal{L}_{\beta}\boxtimes\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1]).$

Regard $v^{\ast}$ as a point in the fiber of
${\mathbb{L}}^{\vee}_{G/P,eP}\cong({\mathbb{L}}^{\vee})_{[v]}$. Restricting
the isomorphism

$m^{\vee\ast}\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])\cong\phi^{\ast}\mathcal{L}_{\beta}\boxtimes\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])$

to $G\times v^{\ast}$, we get

$\phi^{\ast}j^{\ast}_{\mathbb{L}_{G/P}^{\vee\circ}}\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])\cong\phi^{\ast}\mathcal{L}_{\beta}\otimes\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])|_{v^{\ast}}.$

By Lemma 1.7, we have

$\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])|_{v^{\ast}}\cong
G_{*}(\beta,\psi)[1].$

We thus have

$\phi^{\ast}j^{\ast}_{\mathbb{L}_{G/P}^{\vee\circ}}\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])\cong\phi^{\ast}(\mathcal{L}_{\beta}[1]\otimes
G_{*}(\beta,\psi)).$

The morphism $\phi$ is smooth. Locally with respect to étale topology, $\phi$
has sections. The above isomorphism then implies that
$j^{\ast}_{\mathbb{L}_{G/P}^{\vee\circ}}\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta})$
is a lisse sheaf on $\mathbb{L}_{G/P}^{\vee\circ}$. The fibers of $\phi$ are
geometrically connected. This implies that if
$V\to\mathbb{L}_{G/P}^{\vee\circ}$ is an étale covering space such that $V$ is
connected, then $G\times_{\mathbb{L}_{G/P}^{\vee\circ}}V$ is also connected.
So the homomorphism induced by $\phi$ on fundamental groups
$\pi_{1}(G)\to\pi_{1}(\mathbb{L}_{G/P}^{\vee\circ})$ is surjective. The
isomorphism
$\phi^{\ast}j^{\ast}_{\mathbb{L}_{G/P}^{\vee\circ}}\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])\cong\phi^{\ast}(\mathcal{L}_{\beta}[1]\otimes
G_{*}(\beta,\psi))$ thus implies that

$j^{\ast}_{\mathbb{L}_{G/P}^{\vee\circ}}\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])\cong\mathcal{L}_{\beta}[1]\otimes
G_{*}(\beta,\psi).$

By Lemma 1.7, we have

$\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])|_{0_{[v]}}=0.$

Restricting the isomorphism
$m^{\vee\ast}\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])\cong\phi^{\ast}\mathcal{L}_{\beta}\boxtimes\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])$
to $G\times 0_{[v]}$, we see the restriction of
$\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])$
to the zero section of ${\mathbb{L}}_{G/P}^{\vee}$ vanishes. It follows that

$\mathscr{F}_{\psi}(Rj_{\mathbb{L}_{G/P}^{\circ},\ast}\mathcal{L}_{\beta}[1])\cong
j_{\mathbb{L}_{G/P}^{\vee\circ},!}\mathcal{L}_{\beta}[1]\otimes
G_{*}(\beta,\psi).$

This proves our assertion. ∎

##  2\. Tautological systems associated to a family of representations

Let $G$ be an algebraic group $G$ over a perfect field $k$, let
$\rho_{\lambda}:G\to\mathbb{\mathrm{missing}}{GL}(V_{\lambda})$
($\lambda\in\Lambda)$ be a finite family of representations so that the
morphism

$\iota:G\to\prod_{\lambda\in\Lambda}\mathrm{End}(V_{\lambda}),\quad
g\mapsto(\rho_{\lambda}(g))$

is quasi-finite, and let

$V=\prod_{\lambda\in\Lambda}\mathrm{End}(V_{\lambda}).$

We have an action of $G$ on $V$ given by

$(g,(A_{\lambda})_{\lambda\in\Lambda})\mapsto(\rho_{\lambda}(g)A_{\lambda})_{\lambda\in\Lambda}$

for any points $g$ in $G$ and $(A_{\lambda})_{\lambda\in\Lambda}$ in $V$. Take
$v$ to be the $k$-point $v=(\mathrm{id}_{V_{\lambda}})_{\lambda\in\Lambda}$ of
$V$. Then the connected component of the stabilizer of $v$ is trivial. We have
a perfect pairing

$\langle\;,\;\rangle:V\times V\to
k,\quad((A_{\lambda})_{\lambda\in\Lambda},(B_{\lambda})_{\lambda\in\Lambda})\mapsto\sum_{\lambda\in\Lambda}\mathrm{Tr}(A_{\lambda}B_{\lambda}).$

We identify $V$ with $V^{\vee}$ through this pairing. Let
$\mathcal{L}_{\beta}$ be a multiplicative sheaf on the maximal abelian
quotient $G/[G,G]$ of $G$. By our assumption, $\iota$ is both quasi-finite and
affine. So $\iota_{!}\mathcal{L}_{\beta}[n]$ and
$R\iota_{*}\mathcal{L}_{\beta}[n]$ are perverse sheaves. Hence the $\ell$-adic
tautological systems
$\mathcal{T}_{!}(G,\beta,V,v,\psi)=\mathscr{F}_{\psi}(\iota_{!}\mathcal{L}_{\beta}[n])$
and
$\mathcal{T}_{*}(G,\beta,V,v,\psi)=\mathscr{F}_{\psi}(R\iota_{*}\mathcal{L}_{\beta}[n])$
on $V$ are perverse.

######  Lemma 2.1.

Notation as above. Let $F:G\times_{k}V\to\mathbb{A}_{k}^{1}$ be the morphism
defined by

$F(g,(A_{\lambda})_{\lambda\in\Lambda})=\sum_{\lambda\in\Lambda}\mathrm{Tr}(\rho_{\lambda}(g)A_{\lambda}),$

and let

$\pi_{1}:G\times_{k}V\to G,\quad\pi_{2}:G\times_{k}V\to V$

be the projections. Then we have

$\mathcal{T}_{!}(G,\beta,V,v,\psi)\cong
R\pi_{2!}(\pi_{1}^{\ast}\mathcal{L}_{\beta}\otimes F^{\ast}L_{\psi})[n+N],$

where $n=\mathrm{dim}\,G$ and $N=\mathrm{dim}\,V$.

###### Proof.

We have a commutative diagram

$\begin{array}[]{rclcc}G\times_{k}V&\stackrel{{\scriptstyle\iota\times{\mathrm{id_{V}}}}}{{\to}}&V\times_{k}V&\stackrel{{\scriptstyle\langle\;,\;\rangle}}{{\to}}&\mathbb{A}^{1}\\\
{\scriptstyle\pi_{1}}\downarrow&&\downarrow{\scriptstyle\mathrm{pr}_{1}}&&\\\
G&\stackrel{{\scriptstyle\iota}}{{\to}}&V.&&\end{array}$

By the proper base change theorem and the projection formula, we have

$\displaystyle\mathcal{T}_{!}(G,\beta,V,v,\psi)$ $\displaystyle\cong$
$\displaystyle
R\mathrm{pr}_{2!}\Big{(}\mathrm{pr}_{1}^{\ast}\iota_{!}\mathcal{L}_{\beta}\otimes\langle\;,\;\rangle^{\ast}\mathcal{L}_{\psi}\Big{)}[n+N]$
$\displaystyle\cong$ $\displaystyle
R\mathrm{pr}_{2!}\Big{(}(\iota\times\mathrm{id}_{V})_{!}\pi_{1}^{\ast}\mathcal{L}_{\beta}\otimes\langle\;,\;\rangle^{\ast}\mathcal{L}_{\psi}\Big{)}[n+N]$
$\displaystyle\cong$ $\displaystyle
R\big{(}\mathrm{pr}_{2}\circ(\iota\times\mathrm{id}_{V})\big{)}_{!}\Big{(}\pi_{1}^{\ast}\mathcal{L}_{\beta}\otimes(\langle\;,\;\rangle\circ\big{(}\iota\times\mathrm{id}_{V})\big{)}^{\ast}\mathcal{L}_{\psi}\Big{)}[n+N]$
$\displaystyle\cong$ $\displaystyle
R\pi_{2!}(\pi_{1}^{\ast}\mathcal{L}_{\beta}\otimes
F^{\ast}\mathcal{L}_{\psi})[n+N].$

∎

We call

$\mathrm{Hyp}_{\psi}(\Lambda,\beta)=R\pi_{2!}(\pi_{1}^{\ast}\mathcal{L}_{\beta}\otimes
F^{\ast}L_{\psi})[n+N]$

the _hypergeometric sheaf_ on $V$. Over the complex number field $\mathbb{C}$
and in the case where $G$ is a reductive group and $\Lambda$ is a finite
family of irreducible representations of $G$, this is introduced in [10]. The
GKZ hypergeometric sheaf in Example 0.4 is a special case for the group
$G=\mathbb{G}_{m}^{n}$ and the family of representations defined by the
character

$\mathbb{G}_{m}^{n}\to\mathbb{G}_{m},\quad(t_{1},\ldots,t_{n})\mapsto
t_{1}^{w_{1j}}\cdots t_{n}^{w_{nj}}\quad(j=1,\ldots,N).$

Suppose $k=\mathbb{F}_{q}$ is a finite field, and suppose
$\mathcal{L}_{\beta}$ is the Lang sheaf associated to a character

$\beta:(G/[G,G])(\mathbb{F}_{q})\to\overline{\mathbb{Q}}_{\ell}^{*}.$

Then for any $\mathbb{F}_{q}$-point $A=(A_{\lambda})_{\lambda\in\Lambda}$ of
$V$, we have

$\mathrm{Tr}(\mathrm{Frob}_{A},\mathrm{Hyp}_{\psi}(\beta,\Lambda)_{\bar{A}})=(-1)^{n+N}\sum_{g\in
G(\mathbb{F}_{q})}\beta(g)\psi\Big{(}\mathrm{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{p}}\Big{(}\sum_{\lambda\in\Lambda}\mathrm{Tr}(\rho_{\lambda}(g)A_{\lambda})\Big{)}\Big{)}$

by the Grothendieck trace formula.

Example 1. Let $G\subset\mathrm{GL}_{m}$ and let $\Lambda$ be the set
consisting of only the standard representation of
$\rho:G\hookrightarrow\mathrm{GL}_{m}$. Determinant on $\mathrm{GL}_{m}$
induces a homomorphism

$G/[G,G]\to\mathbb{G}_{m}.$

Suppose $k=\mathbb{F}_{q}$ is a finite field, and let
$\beta:\mathbb{G}_{m}(\mathbb{F}_{q})\to\overline{\mathbb{Q}}_{\ell}^{*}$ be a
multiplicative character. For any rational point $A=(a_{ij})$ in
$\mathfrak{gl}_{m}$, where $(a_{ij})$ is an $m\times m$ matrix with entries in
$\mathbb{F}_{q}$, we have

$\mathrm{Tr}(\mathrm{Frob}_{A},\mathrm{Hyp}_{\psi}(\beta,\Lambda)_{\bar{A}})=\sum_{(g_{ij})\in
G(\mathbb{F}_{q})}\beta(\mathrm{det}\big{(}g_{ij})\big{)}\psi\Big{(}\mathrm{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{p}}(\sum_{i,j}^{m}g_{ij}a_{ji})\Big{)}.$

Example 2 ([10, Example 6.1]). Let $L$ and $N$ be two 1-dimensional $k$-vector
spaces, and let $V$ be an $n$-dimensional $k$-vector space. Set

$G=\mathrm{GL}(N)\times_{k}\mathrm{GL}(L)\times_{k}\mathrm{GL}(V)\cong\mathbb{G}_{m}\times_{k}\mathbb{G}_{m}\times_{k}\mathrm{GL}_{n}.$

Let $\Lambda=\\{N\otimes_{k}N,N\otimes_{k}L,N\otimes_{k}V,L\otimes_{k}V\\}$.
In this case, we have

$V=\mathrm{End}(N\otimes_{k}N)\oplus\mathrm{End}(N\otimes_{k}L)\oplus\mathrm{End}(N\otimes_{k}V)\oplus\mathrm{End}(L\otimes_{k}V)$

which is of dimension $2(1+n^{2})$. The stabilizer in $G$ of the point
$v=(\mathrm{id}_{N\otimes_{k}N},\mathrm{id}_{N\otimes_{k}L},\mathrm{id}_{N\otimes_{k}V},\mathrm{id}_{L\otimes_{k}V})$
in $V$ is trivial. Let $\iota:G\to V$ be the morphism

$\iota:\mathbb{G}_{m}\times_{k}\mathbb{G}_{m}\times_{k}\mathrm{GL}_{n}\to\mathbb{A}^{1}\times_{k}\mathbb{A}^{1}\times_{k}\mathfrak{gl}_{n}\times_{k}\mathfrak{gl}_{n},\quad(s,t,g)\mapsto(s^{2},st,sg,tg).$

Its image can be identified with the orbit of $v$. Note that $\bar{k}$-points
in the image of $\iota$ can be described by

$\\{(u,v,X,Y)\in\bar{k}\times\bar{k}\times\mathfrak{gl}_{n}(\bar{k})\times\mathfrak{gl}_{n}(\bar{k}):\;uY-
vX=0\\}.$

We have

$G/[G,G]\cong\mathbb{G}_{m}\times_{k}\mathbb{G}_{m}\times_{k}\mathrm{GL}_{n}/[\mathrm{GL}_{n},\mathrm{GL}_{n}]\cong\mathbb{G}_{m}\times_{k}\mathbb{G}_{m}\times_{k}\mathbb{G}_{m}.$

Suppose $k=\mathbb{F}_{q}$ is a finite field with $q$ elements. Let
$\chi_{1},\chi_{2},\chi_{3}:\mathbb{F}_{q}^{*}\to\overline{\mathbb{Q}}_{\ell}^{*}$
be multiplicative characters and let
$\beta:(G/[G,G])(\mathbb{F}_{q})\to\overline{\mathbb{Q}}_{\ell}^{*}$ be the
character

$(s,t,g)\mapsto\chi_{1}(s)\chi_{2}(g)\chi_{3}(\mathrm{det}(g)).$

For any $\mathbb{F}_{q}$-points $x=(a,b,C,D)$ of $V$, we have

$\mathrm{Tr}(\mathrm{Frob}_{x},(\tau_{!}(G,\beta,V,v,\psi)_{\bar{x}})=\sum_{s\in\mathbb{F}_{q}^{*},\,t\in\mathbb{F}_{q}^{*},\,g\in\mathrm{GL}_{n}(\mathbb{F}_{q})}\chi_{1}(s)\chi_{2}(t)\chi_{3}(\mathrm{det}(g))\psi\Big{(}s^{2}a+stb+s\mathrm{Tr}(gC)+t\mathrm{Tr}(gD)\Big{)}.$

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