# On the Siegel-Weil Theorem for Loop Groups (I)

Howard Garland and Yongchang Zhu The second author’s research is supported by
Hong Kong Research Grant Council earmark grant number HKUST 604507

##  1 Introduction

The notion of an snt-module and our extension (1.14) of the Siegel-Weil
Theorem (also see Theorem 8.1, §8) grew out of our work on the Siegel-Weil
Theorem for arithmetic quotients of loop groups, which we prove in Part II of
this paper ([3]). In fact (1.14) is a vital step in our proof of the Siegel-
Weil theorem in the loop case, and as far as we know, it also seems to be a
new result for automorphic forms on certain, finite-dimensional, non-reductive
groups. At the same time, the theory of automorphic forms on arithmetic
quotients of loop groups, specifically a loop version of Godement’s criterion
(Theorem 5.3, Part II [3]) is used to prove the convergence theorem (Theorem
3.3) we need for the Eisenstein series ${\rm Et}$ defined in (3.10). At the
moment, even though the statement of Theorem 3.3 only involves finite-
dimensional groups, the proof using loop groups is the only one we have!

To state our main result of this part, we first recall the Siegel-Weil theorem
proved in [7].

Let $F$ be a number field, ${\bf A}$ be the ring of adeles of $F$, and $M$ be
a symplectic space over $F$ with symplectic pairing $\langle,\rangle$. Let
$(V,(\,,\,))$ be a finite dimensional vector space over $F$ with the non-
degenerate, symmetric, bilinear form $(\,,\,)$. The space $M\otimes_{F}V$ has
the symplectic form given by

$\langle u_{1}\otimes v_{1},u_{2}\otimes v_{2}\rangle=\langle
u_{1},u_{2}\rangle(v_{1},v_{2}).$

The group $Sp_{M}$ and the orthogonal group $G$ of $V$ form a dual pair in the
symplectic group $Sp_{2N}$ of $M\otimes_{F}V$ (where $2N={\rm dim}\,M\,{\rm
dim}\,V$). Let

$M=M_{-}\oplus M_{+}$ (1.1)

be a direct sum of Lagrangian subspaces of $M$, then

$M\otimes V=M_{-}\otimes V\oplus M_{+}\otimes V$

is a direct sum of Lagrangian subspaces of $M\otimes V$. The Hilbert space
$L^{2}((M_{-}\otimes V)_{\bf A})$ is an irreducible unitary representation of
the metaplectic group $\widehat{Sp}_{2N}({\bf A})$, which is called the Weil
representation. The dense subspace ${\cal S}((M_{-}\otimes V)_{\bf A}))$,
formed by the Schwartz-Bruhat functions on $(M_{-}\otimes V)_{\bf A}$, is
invariant under the action of $\widehat{Sp}_{2N}({\bf A})$. The theta
functional

$\theta:{\cal S}((M_{-}\otimes V)_{\bf A})\to{\mathbb{C}}$

given by

$\theta(\phi)=\sum_{r\in M_{-}\otimes V}\phi(r)$

is $Sp_{2N}(F)$-invariant.

The group $\widehat{Sp}_{2N}({\bf A})$ contains the commuting pair of groups
$\widehat{Sp}_{M}({\bf A})$ and $G({\bf A})$, where $\widehat{Sp}_{M}({\bf
A})$ is the preimage of $Sp_{M}({\bf A})$ in $\widehat{Sp}_{2N}({\bf A})$.

For a given $\phi\in{\cal S}((M_{-}\otimes V)_{\bf A})$, there are two simple
ways to construct a function on $Sp_{M}(F)\backslash\widehat{Sp}_{M}({\bf
A})$: one is

${\rm I}(\phi,g)\stackrel{{\scriptstyle\rm def}}{{=}}\int_{G(F)\backslash
G({\bf A})}\theta(\pi(g,h)\phi)dh,\,\,\,\,\,\,g\in\widehat{Sp}_{M}({\bf A}),$
(1.2)

where $\pi$ is the Weil representation and $dh$ is the Haar measure on $G({\bf
A})$ such that the volume of $G(F)\backslash G({\bf A})$ is $1$. For the
second way, we first consider the function $g\mapsto(\pi(g)\phi)(0)$, which is
$P({\bf A})$-invariant (where $P$ is the Siegel parabolic subgroup of $Sp_{M}$
that fixes $M_{+}$). We then form the Eisenstein series

${\rm E}(\phi,g)\stackrel{{\scriptstyle\rm def}}{{=}}\sum_{r\in P(F)\backslash
Sp_{M}(F)}(rg\cdot\phi)(0).$ (1.3)

This gives a function on $Sp_{M}(F)\backslash\widehat{Sp}_{M}({\bf A})$. When
${\rm dim}\,V>{\rm dim}\,M+2$, both (1.2) and (1.3) converge, and the the
Siegel-Weil formula asserts that the above two constructions are equal, i.e.

${\rm I}(\phi,g)={\rm E}({\phi},g).$ (1.4)

Weil [7] proved such an identity in more generality for dual pairs constructed
from semisimple algebras with involutions. Generalizations of this formula to
non-convergent cases can be found in [5].

Put ${\rm I}({\phi})\stackrel{{\scriptstyle\rm def}}{{=}}{\rm I}({\phi},e)$
and ${\rm E}({\phi})\stackrel{{\scriptstyle\rm def}}{{=}}{\rm E}({\phi},e)$,
we have

${\rm I}({\phi})={\rm E}(\phi)$ (1.5)

Since ${\rm I}({\phi},g)={\rm I}({g\cdot\phi})$ and ${\rm E}(\phi,g)={\rm
E}({g\cdot\phi})$, we see that (1.4) and (1.5) are equivalent.

Since the cosets $P(F)\backslash Sp_{M}(F)$ are in one-to-one correspondence
with the set $Gr(M)$ of Lagrangian subspaces of $M$, via the map $Pg\mapsto
M_{+}g$ (we assume the symplectic group acts on $M$ from the right). The
Eisenstein series (1.3) can also be written as a summation over $Gr(M)$ as
follows. Let $\pi_{-}:M\to M_{-}$ denote the projection with respect to the
decomposition (1.1). For $r\in P(F)\backslash Sp_{M}(F)$, let $U=M_{+}r$ .
Then the symplectic pairing $\pi_{-}(U)\times M_{+}\to F$ factors through a
non-degenerate pairing $\pi_{-}(U)\times M_{+}/(M_{+}\cap U)\to F$. For each
$v\in\pi_{-}(U)$, let $\tilde{v}\in U$ be a lifting of $v$, write
$\tilde{v}=\tilde{v}_{-}+\tilde{v}_{+}$ according the decomposition (1.1),
then the map

$\rho:\pi_{-}(U)\to M_{+}/(M_{+}\cap
U),\,\,\,\,v\mapsto\tilde{v}_{+}+M_{+}\cap U$ (1.6)

is well-defined. One can prove that

$\pi(r)\phi(0)=\int_{(\pi_{-}(U)\otimes V)_{\bf A}}\psi(\frac{1}{2}\langle
x,\rho x\rangle)\phi(x)dx\stackrel{{\scriptstyle\rm def}}{{=}}E(\phi,U),$
(1.7)

where $\psi$ is the additive character of ${\bf A}/F$ used in the definition
of the Weil representation, and the measure on the right hand side is the Haar
measure on $(\pi_{-}(U)\otimes V)_{\bf A}$ normalized by the condition that
the covolume of the lattice $\pi_{-}(U)\otimes V$ is $1$. So the Eisenstein
series (1.3) can be written as

${\rm E}(\phi)=\sum_{U\in Gr(M)}E(\phi,U).$

In our generalization of the Siegel-Weil formula (1.5), we assume the
symplectic space $M$ has an additional structure, which we call an snt-module.
The groups involved are no longer reductive groups (examples:
$Sp_{n}(F[t]/(t^{k}))$ and $G(F[t]/(t^{k}))$) . To state our generalization,
we define

###### Definition 1.1

By a symplectic, nilpotent $t$-module (=snt-module) $M$, we mean an
$F[t]$-module which is finite-dimensional when considered as a vector space
over $F,$ and which is equipped a symplectic form $\langle\,,\,\rangle$ such
that the following conditions are satisfied
(i) there exists a positive integer $N>0$ such that

$t^{N}\cdot\xi=0,\,\,\,{\rm for\,all}\,\,\xi\in M.$

(ii). The operator $t$ is self-dual, i.e.

$\langle t\xi,\eta\rangle=\langle\xi,t\eta\rangle$ (1.8)

Since $t^{N}=0$ on $M$, we may regard $M$ as an $F[[t]]$-module. We give a
simple example of an snt-module. Consider $F[[t]]$-module

$H_{k}=F[[t]]/(t^{k})\oplus F[[t]]/(t^{k}),$ (1.9)

with a symplectic form $\langle,\rangle$ defined by the conditions
(i). Each of the two summands is isotropic.
(ii).

$\langle(t^{i},0),(0,t^{j})\rangle=\left\\{\begin{array}[]{c}0,\;i+j\neq
k-1\\\ 1,\;i+j=k-1\end{array}\right.,$ where $i,j=0,1,....,k-1.$

We shall prove later (see Lemma 2.2) that every snt-module is a direct sum of
the above examples. For a given snt-module $M$, $g\in GL(M)$ is called an snt-
module automorphism if $g$ preserves both the $F[[t]]$-module structure and
the symplectic structure. We denote by

$Sp(M,t)$

the group of all snt-automorphisms on $M$.

For a given snt-module $M$ and a space $V$ with non-degenerate, bilinear,
symmetric form $(\,,\,)$, then the space

$M\otimes_{F}V=M\otimes_{F[[t]]}V[[t]]$ (1.10)

has a natural snt-module structure, where $V[[t]]=V\otimes_{F}F[[t]]$. And the
symplectic form is defined using the first tensor product:

$\langle x_{1}\otimes v_{1},x_{2}\otimes v_{2}\rangle=\langle
x_{1},x_{2}\rangle(v_{1},v_{2})$

and the $F[[t]]$-module structure is defined using the second tensor product.
Let $Sp_{2N}$ ($2N={\rm dim}\,M{\rm dim}\,V$) denote the symplectic group of
the symplectic space $M\otimes_{F}V$, the group $Sp(M,t)$ is a subgroup of
$Sp_{2N}$. The orthogonal group $G(F)$ acts on $M\otimes_{F}V$ preserving the
snt-module structure. But a larger group $G(F[[t]])$ acts on (1.10): for
$x\otimes v\in M\otimes_{F[[t]]}V[[t]]$, $g\in G(F[[t]])$,

$(x\otimes v)\cdot g=x\otimes(v\cdot g).$

It is easy to see that the action preserves both the symplectic structure and
the $F[[t]]$-module structure, so we also have a group morphism

$G(F[[t]])\to Sp_{2N}.$

We denote $G^{q}(F[[t]])$ the image. The subgroups $Sp(M,t)$ and
$G^{q}(F[[t]])$ in $Sp(M\otimes V,t)$ obviously commute. Our generalization of
the Siegel-Weil formula is concerned with the commuting pair
($Sp(M,t),G^{q}(F[[t]])$), which are not reductive groups in general. Let

$Gr(M,t)$

denote the set of all $F[[t]]$-stable Lagrangian subspaces of $M$, so
$Gr(M,t)\subset Gr(M)$.

We take a direct sum decomposition $M=M_{-}\oplus M_{+}$ such that $M_{-}\in
Gr(M),M_{+}\in Gr(M,t)$. As before $L^{2}((M_{-}\otimes V)_{\bf A})$ is a
representation of ${\widehat{S}p}_{2N}({\bf A})$, with the theta functional

$\theta:{\cal S}((M_{-}\otimes V)_{\bf
A})\to{\mathbb{C}},\,\,\,\phi\mapsto\theta(\phi)=\sum_{r\in M_{-}\otimes
V}\phi(r).$

For a subspace $U\in Gr(M)$, let $E(\phi,U)$ be as in (1.7), and we define

${\rm Et}(\phi)\stackrel{{\scriptstyle\rm def}}{{=}}\sum_{W\in
Gr(M,t)}E(\phi,W).$ (1.11)

And we define

${\rm It}(\phi)\stackrel{{\scriptstyle\rm
def}}{{=}}\int_{G^{q}(F[[t]])\backslash G^{q}({\bf
A}[[t]])}\theta(h\cdot\phi)dh,$ (1.12)

where $dh$ denotes the Haar measure on $G^{q}({\bf A}[[t]])$ such that the
volume of $G^{q}(F[[t]])\backslash G^{q}({\bf A}[[t]])$ is $1$.

By Lemma 2.2, $M$ is isomorphic to a direct sum of $n$ copies of $H_{k}$’s:

$M\cong H_{k_{1}}\oplus\dots\oplus H_{k_{n}}$ (1.13)

assume ${\rm dim}\,V>6n+2$, and the quadratic form $(\,)$ on $V$ is
$F$-anisotropic or ${\rm dim}V-r>\frac{1}{2}{\rm dim}\,M+1$, where $r$ is the
dimension of a maximal isotropic subspace of $V$, then we have the following
generalization of the Siegel-Weil formula (Theorem 7.3, Theorem 8.1) :

${\rm Et}(\phi)={\rm It}(\phi).$ (1.14)

The condition ${\rm dim}\,V>6n+2$ is for the convergence of ${\rm Et}(\phi)$,
and the condition that $(V,(,))$ is $F$-anisotropic or ${\rm
dim}\,V-r>\frac{1}{2}{\rm dim}\,M+1$ is for convergence of ${\rm It}(\phi)$.
This formula reduces to the classical formula (1.5) when
$k_{1}=\dots=k_{n}=1$.

In general, the $Sp(M,t)$-action on $Gr(M,t)$ is not transitive, but there are
only finitely many orbits, so the sum ${\rm Et}(\phi)=\sum E(U,\phi)$ is a sum
of Eisenstein series induced from several ”parabolic” subgroups (rather than
only one), and each corresponds to a $Sp(M,t)$-orbit in $Gr(M,t)$.

In the case that $M$ is a direct sum of $n$-copies of the snt-module $H_{k}$,
then $Sp(M,t)=Sp_{2n}(F[t]/(t^{k}))$ and $G^{q}(F[[t]])=G(F[t]/(t^{k}))$. The
formula (1.14) means that the Siegel-Weil formula holds for symplectic and
orthogonal groups over $F[t]/(t^{k})$. It is implies that the Siegel-Weil
formula holds for symplectic and orthogonal groups over $F[t]/(p(t))$ for
arbitrary polynomial $p(t)$.

We now give an explicit example of our formula. Let $F={\mathbb{Q}}$, and the
snt - module $M$ be

$M={\mathbb{Q}}[t]/(t^{2})\oplus{\mathbb{Q}}/(t^{2})$ (1.15)

with the snt-module structure given as in (1.9). And we take a positive
definite even unimodular lattice $L$ of rank $N$ with the bilinear form given
by $(,)$, and let $V={\mathbb{Q}}L$. It is well-known that $N$ is divisible by
$8$. Let $L_{1},\dots,L_{g}$ be the list of the positive definite even
unimodular lattices of rank $N$ (up to isomorphism), let $()_{j}$ denote the
the pairing of $L_{j}$, and $|{\rm Aut}_{j}|$ be the order of automorphism
group of $L_{j}$. We denote $1\oplus 0$ and $0\oplus 1$ in (1.15) by $e_{1}$
and $e_{2}$ respectively, then $e_{1},te_{1},e_{2},te_{2}$ is a
${\mathbb{Q}}$-basis of $M$. Let $M_{-}={\mathbb{Q}}e_{1}+{\mathbb{Q}}te_{1}$
and $M_{+}={\mathbb{Q}}te_{2}+{\mathbb{Q}}e_{2}$. It is clear that $M_{-}$ and
$M_{+}$ are Lagrangian subspaces of $M$ and

$M=M_{-}\oplus M_{+}.$

We have the Weil representation of $Sp_{4N}({\bf A})$ on

${\cal S}((M_{-}\otimes V)_{\bf A}).$

Take $\phi=\Pi_{v}\phi_{v}\in{\cal S}((M_{-}\otimes V)_{\bf A})$ as follows:
for a finite place $p$ of ${\mathbb{Q}}$, $\phi_{p}$ is the characteristic
function of $e_{1}\otimes L_{{\bf Z}_{p}}+te_{1}\otimes L_{{\bf Z}_{p}}$
(where ${\bf Z}_{p}$ denotes the ring of $p$-adic integers and $L_{{\bf
Z}_{p}}=L\otimes{\bf Z}_{p}$); for the real place $\infty$ of ${\mathbb{Q}}$,

$\phi_{\infty}(e_{1}\otimes v_{1}+te_{1}\otimes v_{2})=e^{\pi
i\tau_{1}(v_{1},v_{1})+\pi i\frac{-1}{\tau_{2}}(v_{2},v_{2})},$

where $\tau_{1},\tau_{2}$ are complex numbers in the upper half plane. With
the above choice of $\phi$, our new Siegel-Weil formula (1.14) becomes

$\displaystyle 1+\frac{1}{2}\sum_{a\geq 1,b\in{\bf
Z}:(a,b)=1}\,\sum_{m,n\in{\bf
Z}:(m,n)=1}(am^{2}\tau_{1}+an^{2}\tau_{2}+b)^{-\frac{N}{2}}$ (1.16)
$\displaystyle=$ $\displaystyle C\sum_{j=1}^{g}\frac{1}{|{\rm
Aut}_{j}|}\sum_{u,v\in L_{j},u,v\,{\rm colinear}}e^{\pi i\tau_{1}(u,u)+\pi
i\tau_{2}(v,v)}$

where $(m,n)=1$ ( $(a,b)=1$) means $m,n$ (resp. $a,b$) are relatively prime,
and $u,v$ colinear means the ${\mathbb{Q}}$-span of $u,v$ is at most
$1$-dimensional. And the constant $C$ is given by (1.18) below.

We compare (1.16) with the classical Siegel-Weil formula

$\frac{1}{2}\sum_{m,n\in{\bf
Z},(m,n)=1}(m\tau+n)^{-\frac{N}{2}}=C\sum_{j=1}^{g}\frac{1}{|{\rm
Aut}_{j}|}\sum_{u\in L_{j}}e^{\pi i\tau(u,u)_{j}}$ (1.17)

This formula expresses the $\frac{N}{2}$-th Eisenstein series of
$SL(2,{\mathbb{Z}})$ as a sum of Theta series. The constant term in
$q$-expansion of both sides gives the density formula

$1=C\sum_{j=1}^{g}\frac{1}{|{\rm Aut}_{j}|},$ (1.18)

which determines $C$.

If we take more general test function $\phi$, our Siegel-Weil formula (1.14)
is

$\displaystyle 1+\frac{1}{2}\sum_{a\geq 1,b\in{\bf
Z}:(a,b)=1}\,\sum_{m,n\in{\bf
Z}:(m,n)=1}(am^{2}\tau_{11}+2amn\tau_{12}+an^{2}\tau_{22}+b)^{-\frac{N}{2}}$
(1.19) $\displaystyle=$ $\displaystyle C\sum_{j=1}^{g}\frac{1}{|{\rm
Aut}_{j}|}\sum_{u,v\in L_{j},u,v\,{\rm colinear}}e^{\pi
i\left(\tau_{11}(u,u)+2\tau_{12}(u,v)+\tau_{22}(v,v)\right)},$

where $C$ is determined by (1.18).

The paper is organized as follows: In §2 we study the structure of an snt-
module $M$ (Lemma 2.2) and the structure of the corresponding group $Sp(M,t)$
of snt-automorphisms of $M$ (Cor. 2.6). In §3 we recall basic facts about the
Weil representation and study the Eisenstein series ${\rm Et}(\phi)$ for the
Weil representation associated to an snt-module. In particular, we state the
convergence theorem (Theorem 3.3) for such Eisenstein series. As we already
stated, the proof depends on the convergence theorem for Eisenstein series on
loop groups, and will be given in Part II of this paper ([3]). We also state a
consequence, Theorem 3.4, of Theorem 3.3 together with Prop. 1 and Prop. 2 of
[7].

In §4 we extend a result in Weil [7] and obtain Theorem 4.7, which identifies
certain abstract measures associated with the map $T_{W}$ (see (3.13) for the
definition of $T_{W})$ with certain gauge measures in the sense of [7]. In §5
we obtain Theorems 5.4 and 5.8. In particular, we identify the space of
$G(F[[t]])$ \- orbits in $M_{\\_}\otimes V$ with the set of pairs $(W,i)$ with
$W\in Gr(M_{\\_},t)$ (the $t$ \- Grassmannian), $i\in S_{t}^{2}(W)$ such that
$U(i)_{F}$ is non-empty ( $U(i)_{F}$ is defined in §5, just before the
statement of Theorem 5.8).

In §6 we discuss $\theta$ \- series and finally, we prove Theorems 7.3 and
8.1, our versions of the Siegel-Weil theorem for snt-modules.

##  2 The structure of symplectic, nilpotent $t$-modules.

In this section, we prove some results about the structure of symplectic,
nilpotent $t$-modules (snt-modules) defined in Section 1 (Definition 1.1).

###### Lemma 2.1

Let $M$ be an snt-module. For all $\xi\in M$, $k\in\mathbb{Z}_{>0}$, we have

$\langle\xi,t^{k}\xi\rangle=0.$

Proof.  We have for all $\xi\in M$, $k\in\mathbb{Z}_{\geq 0},$

$\langle\xi,t^{k}\cdot\xi\rangle=-\langle t^{k}\cdot\xi,\xi\rangle,$

since $\langle\cdot,\cdot\rangle$ is skew symmetric. On the other hand

$\langle\xi,t^{k}\cdot\xi\rangle=\langle t^{k}\cdot\xi,\xi\rangle$

by (1.8). Hence

$\langle\xi,t^{k}\cdot\xi\rangle=0,$

$\Box$

###### Lemma 2.2

Every snt-module $M$ is isomorphic to a direct sum

$M\cong H_{k_{1}}\oplus....\oplus H_{k_{n}}$

$k_{1}\geq k_{2}\geq....\geq k_{n}$. Where $H_{k}$ is given as (1.9).
Moreover, $n$ and the $k_{i}$ are uniquely determined by $M$.

Proof. The uniqueness is clear from the theory of elementary divisors. For
$\xi\in M$, we define the order of $\xi$ to be the smallest positive integer
$k,$ such that $t^{k}\cdot\xi=0.$ Pick $\xi\in M$ of maximal order $(N$ say).
Then the vectors

$\xi,t\cdot\xi,....,t^{N-1}\cdot\xi$

are linearly independent over $F.$ To see this, we consider $F[[t]]$-submodule
$F[[t]]\xi$. Since $F[[t]]$ is a PID, $F[[t]]\xi$ is isomorphic to
$F[[t]]/(t^{k})$. It is clear that $k=N$.

Since $\langle\,,\,\rangle$ is non-degenerate, we can find $\eta\in M$, so
that

$\langle t^{N-1}\cdot\xi,\eta\rangle=1,\,\,\,\,\langle
t^{j}\cdot\xi,\eta\rangle=0,\,\,\,j=0,1,...,N-2$

But then,

$\langle\xi,t^{N-1}\cdot\eta\rangle=\langle t^{N-1}\cdot\xi,\eta\rangle=1$

Hence

$t^{N-1}\cdot\eta\neq 0,$

and so

$\eta,t\cdot\eta,....,t^{N-1}\cdot\eta$

are linearly independent (by the above argument) and

$t^{N}\cdot\eta=0$

(since $N$ was assumed the maximal order of any element of $M$). But then if

$C_{1}=F\text{-span of }\xi,t\cdot\xi,....,t^{N-1}\cdot\xi,$
$C_{2}=F\text{-span of }\eta,t\cdot\eta,....,t^{N-1}\cdot\eta,$

we see that

$H=C_{1}\oplus C_{2}$

with symplectic structure given by the restriction of $\langle,\rangle$ on $M$
is isomorphic to the snt-module $H_{N}$, with $C_{1},$ $C_{2}$ corresponding
to the two direct summands $F[[t]]/(t^{N})$. Since $\langle,\rangle$
restricted to $H$ is non-degenerate, $M$ decomposes as a direct sum of
$snt$-modules

$M\cong H\oplus H^{\bot},$

and applying the induction hypothesis to $H^{\bot},$ we obtain the lemma.
$\Box$

For an snt-module $M$, so $M$ is in particular a symplectic space. As in
Section 1, we let $Gr(M)$ denote the set of all Lagrangian subspaces, and
$Gr(M,t)$ denote the set of Lagrangian subspaces which are stable under the
action of $F[[t]]$, so $Gr(M,t)\subset Gr(M)$. We call elements in $Gr(M,t)$
$t$-Lagrangian subspaces. We have:

###### Lemma 2.3

If $U\subset M$ is a $F[[t]]$-stable, isotropic and it is not properly
contained in any other $F[[t]]$-stable, isotropic subspaces, then $U$ is
$t$-Lagrangian, i.e. $U\in Gr(M,t)$.

Proof. Assume $U$ is not maximal isotropic. Then there exists $v\in M$,
$v\notin U,$ such that

$\langle v,u\rangle=0,\,\,\text{ all }\,u\in U.$

But then for all $u\in U,$ $i\in\mathbb{Z}_{\geq 0}$

$\langle t^{i}\cdot v,u\rangle=\langle v,t^{i}\cdot u\rangle=0,$

since $t^{i}\cdot u\in U.$

Hence, the space $Span_{F[[t]]}\\{v,U\\}$ is $F[[t]]$-stable, isotropic and
contains $M$ properly. This contradicts the maximality of $M$. $\Box$

As in Section 1, we let $Sp(M,t)$ be the subgroup of all $\sigma\in Sp(M)$
such that $\sigma$ is an $F[[t]]$-module automorphism. For a finite
dimensional $F[[t]]$-module $U$, a subset of non-zero elements
$e_{1},\dots,e_{n}$ is called a quasi-basis of $U$ if every element in $U$ can
be written as an $F[[t]]$-linear combination of $e_{i}$’s and
$a_{1}e_{1}+\dots+a_{n}e_{n}=0$ implies that all $a_{i}e_{i}=0$. For example,
for

$U=F[[t]]/(t^{k_{1}})\oplus\dots\oplus F[[t]]/(t^{k_{n}}),$

the set (n elements) $(1,0,\dots,0),(0,1,\dots,0),\dots(0,\dots,1)$ is a
quasi-basis. If $U$ has a quasi-basis consisting of $n$-elements, then the
$F$-vector space $\bar{U}\stackrel{{\scriptstyle\rm def}}{{=}}U/tU$ is
$n$-dimensional. If we have two such $F[[t]]$-modules $U_{1}$ and $U_{2}$, and
$T:U_{1}\to U_{2}$ is a morphism, then $T$ induces a $F$-linear map
$\bar{T}:{\bar{U}}_{1}\to{\bar{U}}_{2}$.

An snt-module $M$ with decomposition as Lemma 2.2 is called homogeneous if
$k_{1}=k_{2}=....=k_{n}=k$. In this case the group $Sp(M,t)$ is determined by

###### Lemma 2.4

Let $M$ be a homogeneous snt-module as in (1.13) with $k_{1}=\dots=k_{n}=k$.
Then $Sp(M,t)$ is isomorphic to $Sp_{2n}(F[[t]]/(t^{k}))$. In particular, its
reduction ${\rm mod}\,t$ defines a surjective group homomorphism

$\pi_{0}:Sp(M,t)\rightarrow Sp_{2n}(F),$

Proof. We first consider the free $F[[t]]/(t^{k})$-module
$(F[[t]]/(t^{k}))^{2n}$. It has as standard $F[[t]]/(t^{k})$-valued symplectic
form $\langle\,,\,\rangle^{\wedge}$. We define an $F$-valued symplectic form
$\langle\,,\,\rangle$ on $(F[[t]]/(t^{k}))^{2n}$ by

$\langle a,b\rangle={\rm coefficient\,\,of\,\,}\,\,t^{k-1}\,\,{\rm
in}\,\,\langle a,b\rangle^{\wedge}.$

With this $\langle\,,\,\rangle$, $(F[[t]]/(t^{k}))^{2n}$ is an snt-module,
which is clearly isomorphic to $M$ in the lemma. It follows from the
construction that $Sp_{2n}(F[[t]]/(t^{k}))$ preserves the snt-module
structure, so we have

$Sp_{2r}(F[[t]]/(t^{k}))\subset Sp(M,t).$

For the converse inclusion is also clear. $\Box$

For a homogeneous snt-module $M$ as in Lemma 2.4 , $\bar{M}=M/tM$ has a
symplectic structure defined as follows, if $\bar{a},\bar{b}\in\bar{M}$, let
$a,b\in M$ be their liftings, then

$\langle\bar{a},\bar{b}\rangle\stackrel{{\scriptstyle\rm def}}{{=}}\langle
a,t^{k-1}b\rangle.$ (2.1)

Now we turn to the case of general (possibly non-homogeneous) snt-modules $M$,
with direct sum decomposition as in Lemma 2.2. For fixed $k$, we let

$M(k)=\oplus_{k_{i}=k}H_{k_{i}},$

so

$M=M(l_{1})\oplus...\oplus M(l_{s}),\,\,\,\,\,l_{1}>l_{2}>....>l_{s},$ (2.2)

with each $M(l_{i})$ a homogeneous snt-submodule. The $M(l_{i})$’s are
mutually orthogonal with respect to $\langle,\rangle$.

Now if $\sigma\in Sp(M,t),$ then $\sigma$ (acting on the right, recall) has a
block decomposition with respect to (2.2),

$\sigma=\left[\begin{array}[]{c}\sigma_{1}^{1}....\sigma_{s}^{1}\\\
........\\\ \sigma_{1}^{s}....\sigma_{s}^{s}\end{array}\right]$ (2.3)

where $\sigma_{j}^{i}:M(l_{i})\rightarrow M(l_{j}),$ and for
$\xi=(\xi_{1},....,\xi_{s})\in M$ $(\xi_{i}\in M(l_{i}))$

$\xi\sigma=(\xi_{1},....,\xi_{s})\left[\begin{array}[]{c}\sigma_{1}^{1}....\sigma_{s}^{1}\\\
........\\\ \sigma_{1}^{s}....\sigma_{s}^{s}\end{array}\right].$

We have also the decomposition ${\bar{M}}=M/tM$ induced from the decomposition
(2.2),

${\bar{M}}={\bar{M}}(l_{1})\oplus...\oplus{\bar{M}}(l_{s}),\,\,\,\,l_{1}>l_{2}>....>l_{s},$
(2.4)

So $\bar{\sigma}:{\bar{X}}\to{\bar{X}}$ has a block decomposition:

$\bar{\sigma}=\left[\begin{array}[]{c}\bar{\sigma}_{1}^{1}....\bar{\sigma}_{s}^{1}\\\
........\\\ \bar{\sigma}_{1}^{s}....\bar{\sigma}_{s}^{s}\end{array}\right]$
(2.5)

###### Lemma 2.5

The matrix $\bar{\sigma}$ is block-upper triangular, i.e.

$\bar{\sigma}_{i}^{j}=0\,\,{\rm for}\,\,i<j$ (2.6)

and the diagonal block $\bar{\sigma}_{i}^{i}$ is in $Sp(\bar{X}(l_{i}))$
(recall $\bar{X}(l_{i})$ has the symplectic structure defined by (2.1)).

Proof. For $i<j$, $v_{j}\in M(l_{j})$, we have $t^{l_{j}}v_{j}=0$ and
$\sigma_{i}^{j}$ is $t$-linear, this implies that
$t^{i}(v_{j}\sigma_{i}^{j})=0$, then

$v_{j}\sigma_{i}^{j}\in t^{l_{i}-l_{j}}M(l_{i})$ (2.7)

This implies that $\bar{\sigma}_{i}^{j}=0$. For $a,b\in M(l_{i})$, we have

$\langle t^{l_{i}-1}a,b\rangle=\langle
t^{l_{i}-1}a\sigma,b\sigma\rangle=\sum_{j=1}^{s}\langle
t^{l_{i}-1}a\sigma_{j}^{i},b\sigma_{j}^{i}\rangle,$ (2.8)

if $j<i$, by (2.7) we have

$t^{l_{i}-1}a\sigma_{j}^{i}\in
t^{l_{i}-1}t^{l_{j}-l_{i}}M(l_{j})=t^{l_{j}-1}M(l_{j})$

and

$b\sigma_{j}^{i}\in t^{l_{j}-l_{i}}M(l_{j})$

it implies that

$\langle t^{l_{i}-1}a\sigma_{j}^{i},b\sigma_{j}^{i}\rangle=0.$

For $j>i$, then $l_{i}-1\geq l_{j}$, we have

$t^{l_{i}-1}a\sigma_{j}^{i}\in t^{l_{i}-1}M(l_{j})=0.$

So (2.8) gives that

$\langle t^{l_{i}-1}a,b\rangle=\langle
t^{l_{i}-1}a\sigma_{i}^{i},b\sigma_{i}^{i}\rangle,$

by the definition (2.1) of the symplectic form on ${\bar{M}}(l_{i})$, we prove
that $\bar{\sigma}_{i}^{i}$ is a symplectic isomorphism of ${\bar{M}}(l_{i})$.
$\Box$

###### Corollary 2.6

Let $M$ be an snt-module as in (2.2), then $Sp(M,t)$ is the semi-direct
product

$Sp(M,t)=N\ltimes H,$

where $N$ is the unipotent radical of $Sp(M,t)$ and

$H\cong\Pi_{i=1}^{s}Sp_{2r_{i}}(F),$

where $r_{i}$ is the number $H_{l_{i}}$’s in the decomposition of $M(l_{i})$.

Next we discuss the classification of $t$-Lagrangian subspaces. First for the
snt-module (1.9), let $e_{1},e_{2}$ denote $(1,0),(0,1)$ respectively. For
each $0\leq i\leq k-1$, let $L_{i}$ be the $F$-subspace with basis

$t^{i}e_{1},t^{i+1}e_{1},\dots,t^{k-1}e_{1},t^{k-i}e_{2},t^{k-i+1}e_{2},\dots,t^{k-1}e_{2}.$

It is clear that $L_{i}$ is a $t$-Lagrangian subspace. For an snt-module $M$
with decomposition as in Lemma 2.2, let $L_{i_{j}}\subset H_{k_{j}}$ be the
subspace described as above, it is clear that

$L_{i_{1}}\oplus\dots\oplus L_{i_{n}}$ (2.9)

is an $t$-Lagrangian subspace of $M$. We have

###### Proposition 2.7

Let $M$ be a snt-module with decomposition as in Lemma 2.2, then every
$t$-Lagrangian subspace can be tranformed by some $g\in Sp(M,t)$ to an
$t$-Lagrangian subspace as in (2.9).

This proposition will not be used later, we skip its proof.

##  3 The Weil representation and $t$-Eisenstein series.

In this section, we first recall some basic facts about the Weil
representation associated to a symplectic space over $F$, then we study the
Eisenstein series ${\rm Et}(\phi)$ (1.11) for the Weil representations
associated to an snt-module.

We shall fix a non-trivial additive character $\psi:{\bf A}\to S^{1}$ that is
trivial on $F$. Let $F^{2N}$ be the standard symplectic space over $F$, and
$C=C_{-}\oplus C_{+}$ be a direct sum into Lagrangian subspaces; then a two-
fold cover, denoted by $\widehat{Sp}_{2N}({\bf A})$, of the adelic group
$Sp_{2N}({\bf A})$ acts on $L^{2}(C_{-,{\bf A}})$. The subspace ${\cal
S}(C_{-,{\bf A}})$, formed by the Schwartz-Bruhat functions is invariant under
this action. We recall now the action formula. For $g\in Sp_{2N}({\bf A})$,
let

$\left[\begin{array}[]{cc}\alpha_{g}&\beta_{g}\\\
\gamma_{g}&\delta_{g}\end{array}\right]$ (3.1)

be the block decomposition of $g$ with respect to the decomposition

${\bf A}^{2N}=C_{-,{\bf A}}\oplus C_{+,{\bf A}}.$

In this paper, we always assume the action of $Sp_{2N}$ on $F^{2N}$ (as well
as other symplectic group actions on symplectic spaces) is from the right. So
$\gamma_{g}$ in (3.1) is a map from $C_{+}\to C_{-}$. Let
$\tilde{g}\in\widehat{Sp}_{2N}({\bf A})$ be a lifting of $g$. For
$\phi\in{\cal S}(C_{-,{\bf A}})$, $({\tilde{g}}\cdot\phi)(x)$ equals to

$\lambda\int_{{\rm
Im}\,\gamma_{g}}S_{g}(x+x^{*})\phi(x\alpha_{g}+x^{*}\gamma_{g})d(x^{*}\gamma_{g}),$
(3.2)

where $\lambda\in{\mathbb{C}}^{*}$ is a certain scalar depending only on
$\tilde{g}$, $d(x^{*}\gamma_{g})$ is a Haar measure on ${\rm Im}\,\gamma_{g}$
and

$S_{g}(x+x^{*})=\psi\left(\frac{1}{2}\langle
x\alpha_{g},x\beta_{g}\rangle+\frac{1}{2}\langle
x^{*}\gamma_{g},x^{*}\delta_{g}\rangle+\langle
x^{*}\gamma_{g},x\beta_{g}\rangle\right);$

it is easy to see that $\langle x^{*}\gamma_{g},x^{*}\delta_{g}\rangle$
depends only on $x^{*}\gamma_{g}$ (not on the choice of $x^{*}$), therefore
$S_{g}(x+x^{*})$ is a function of $x$ and $x^{*}\gamma_{g}$. Since $\tilde{g}$
is unitary, $\lambda$ can be determined up to a factor in $S^{1}$.

Let $P$ be the subgroup of $Sp_{2N}$ that consists of elements that maps
$C_{+}$ to itself. An element $g$ is in $P({\bf A})$ iff $\gamma_{g}=0$. Then
there is a lifting $P({\bf A})\subset\widehat{Sp}_{2N}({\bf A})$ so that for
$g\in P({\bf A})$ and $\phi\in{\cal S}(C_{-,{\bf A}})$,

$(g\cdot\phi)(x)=|det(\alpha_{g})|_{\bf
A}^{\frac{1}{2}}\,\psi(\frac{1}{2}\langle
x\alpha_{g},x\beta_{g}\rangle)\phi(x\alpha_{g}),$ (3.3)

where the factor $|det(\alpha_{g})|_{\bf A}^{\frac{1}{2}}$ guarantees the
unitarity of the operator $g$. There is also a lifting

$Sp_{2N}(F)\subset\widehat{Sp}_{2N}({\bf A})$

such that theta functional

$\theta:{\cal S}(C_{-,{\bf A}})\to{\mathbb{C}}$

given by

$\theta(\phi)=\sum_{r\in C_{-}}\phi(r)$

is invariant under $Sp_{2N}(F)$. The action of $Sp_{2N}(F)$ is given by (3.2)
with $\lambda=1$ and the Haar measure is given by the condition that the
covolume of $({\rm Im}\,\gamma_{g})(F)$ is $1$.

For a given snt-module $M$ with

$M=H_{k_{1}}\oplus\dots\oplus H_{k_{n}}$ (3.4)

with $H_{k_{i}}$ is as in (1.9), and a space $V$ with non-degenerate,
bilinear, symmetric form $(\,,\,)$. Let $G$ denote the orthogonal group of
$V$. Recall that

$M\otimes_{F}V=M\otimes_{F[[t]]}V[[t]]$ (3.5)

has a natural snt-module structure (Section 1). Let $Sp_{2N}$ (where $2N={\rm
dim}\,M{\rm dim}\,V$) denote the symplectic group of the symplectic space
$M\otimes_{F}V$. The group $Sp(M,t)$ is a subgroup of $Sp_{2N}$. The group
$G(F[[t]])$ acts on $M\otimes_{F[[t]]}V[[t]]$ in the second factor, so we have
a group morphism

$G(F[[t]])\to Sp(M\otimes V,t).$

Suppose $l={\rm max}(k_{1},\dots,k_{n})$; then the image $G^{q}(F[[t]])$ of
the above homomorphism is isomorphic to $G(F[[t]]/(t^{l}))$. We have a
commuting pair ($Sp(M,t),G(F[[t]]/(t^{l}))$) in $Sp(M\otimes_{F}V,t)$.

Suppose we have a direct sum decomposition

$M=M_{-}\oplus M_{+}$ (3.6)

such that $M_{+},M_{-}\in Gr(M,t)$. We put

$X\stackrel{{\scriptstyle\rm def}}{{=}}M_{-}\otimes V.$

The space $L^{2}(X_{\bf A})$ is a representation of metaplectic group
${\widehat{S}p}_{2N}({\bf A})$, and we have the theta functional

$\theta:{\cal S}(X_{\bf
A})\to{\mathbb{C}},\,\,\,\phi\mapsto\theta(\phi)=\sum_{r\in X}\phi(r).$ (3.7)

Recall the Eisenstein series (1.3), (1.7) for $\phi\in{\cal S}(X_{\bf A})$ is
given by

${\rm E}(\phi)=\sum_{U\in Gr(M)}E(\phi,U)=\sum_{U\in
Gr(M)}\int_{(\pi_{-}(U)\otimes V)_{\bf A}}\psi(\frac{1}{2}\langle x,\rho
x\rangle)\phi(x)dx.$

Let $\pi_{-}:M\to M_{-}$ be the projection map with respect to (3.6). It gives
a map

$Gr(M)\to Gr(M_{-}):\,\,\,\,\,U\mapsto\pi_{-}(U).$ (3.8)

We wish to describe the inverse image of a given $W\in Gr(M_{-})$. Let

$W^{\bot}=\\{x\in M_{+}\,|\,\langle W,x\rangle=0\\}.$

If $U\in Gr(M)$ satisfying $\pi_{-}(U)=W$, then it is easy to see that

$W^{\bot}=M_{+}\cap U.$

The symplectic pairing $W\times M_{+}\to F$ factors through a non-degenerate
pairing

$\langle\,\,\rangle:W\times M_{+}/W^{\bot}\to F.$

We may identify $W^{*}$ with $M_{+}/W^{\bot}$ using this pairing. Recall we
have the map

$\rho_{U}:W\to M_{+}/W^{\bot}$

as defined in (1.6). It is easy to prove that $\rho_{U}$ is self-dual. We have

###### Lemma 3.1

For a given $W\in Gr(M_{-})$, the map

$U\mapsto\rho_{U}$

is a bijection from the set of $U\in Gr(M)$ such that $\pi_{-}(U)=W$ to the
set self-dual linear maps from $W$ to $W^{*}=M_{+}/W^{\bot}$

Proof. It is clear that the map $U\mapsto\rho_{U}$ is one-to-one. If
$\rho:W\to W^{*}=M_{+}/W^{\bot}$ is self-dual, then

$U\stackrel{{\scriptstyle\rm def}}{{=}}\\{w+\rho(w)+W^{\bot}\,|\,w\in W\\}$
(3.9)

is a Lagrangian subspace of $M$, and $\rho_{U}=\rho$. So the map in the lemma
is also onto. $\Box$

Recall the $t$-Eisenstein series defined in (1.11) is a sub-series of ${\rm
E}(\phi)$ given by

${\rm Et}(\phi)=\sum_{U\in Gr(X,t)}E(\phi,U).$ (3.10)

Since $M_{-}$ and $M_{+}$ are $F[[t]]$-submodules of $M$, the projection map

$\pi_{-}:M\to M_{-}$

is an $F[[t]]$-module homomorphism. For each $U\in Gr(M,t)$, $\pi_{-}(U)$ is
an $F[[t]]$-submodule of $M_{-}$. Denote $Gr(M_{-},t)$ the set of
$F[[t]]$-submodules of $M_{-}$, so we have map

$P:Gr(M,t)\to Gr(M_{-},t):\,\,\,U\mapsto\pi_{-}(U).$ (3.11)

For a $W\in Gr(M_{-},t)$, we wish to describe the inverse image

$P_{W}\stackrel{{\scriptstyle\rm def}}{{=}}P^{-1}(W).$

For any $U\in P_{W}$, i.e. $U\in Gr(M,t)$ and $\pi_{-}(U)=W$, we have the
linear map

$\rho_{U}:W\to M_{+}/W^{\bot},$

as defined in (1.6). As before $\rho_{U}$ is self-dual. We now prove
$\rho_{U}$ is $F[t]$-linear. If $w\in W$, since $\pi(U)=W$, there is
$w^{\prime}\in W_{+}$ such that $w+w^{\prime}\in U$. By our definition of
$\rho_{U}$, $\rho_{U}(w)=w^{\prime}\,\,\,mod(W^{\bot})$. Since $U$ is
$t$-Lagrangian, $tw+tw^{\prime}\in U$, this implies
$\rho_{U}(tw)=tw^{\prime}\,\,\,mod(W^{\bot})$.

###### Lemma 3.2

For each $U\in P_{W}$,

$\rho_{U}:W\to M_{+}/W^{\bot}=M_{+}/(M_{+}\cap U)$

is $F[[t]]$-linear and self-dual. Conversely for each $\rho:W\to
M_{+}/W^{\bot}$ that is $F[[t]]$-linear and self-dual, there is a unique $U\in
P_{W}$ such that $\rho_{U}=\rho$. Therefore $U\mapsto\rho_{U}$ is a bijection
from $P_{W}$ to $F_{W}$, the space of all $\rho:W\to U_{+}/W^{\bot}$ that is
self-dual and $F[[t]]$-linear .

This lemma is an $t$-analog of Lemma 3.1. Suppose $\rho$ is $F[[t]]$-linear
and self-dual, then $U$ given as (3.9) is the unique $t$-Lagrangian subspace
such that $\rho_{U}=\rho$.

We set

${\rm Et}_{W}(\phi)=\sum_{U\in P_{W}}E(\phi,U),$

By Lemma 3.2, we have

${\rm Et}_{W}(\phi)=\sum_{\rho\in F_{W}}\int_{(W\otimes V)_{\bf
A}}\phi(x)\psi(\frac{1}{2}\langle x,\rho(x)\rangle)dx,$ (3.12)

where $dx$ denotes the Haar measure on $(W\otimes V)_{\bf A}$ such that the
covolume of $W\otimes V$ is $1$. We have

${\rm Et}(\phi)=\sum_{W\in Gr(M_{-},t)}{\rm Et}_{W}(\phi).$

For an $F[[t]]$-submodule $W\subset M_{-},$ we let

$S_{t}^{2}(W)\subset W\otimes_{F[[t]]}W$

denote the $F[[t]]$-submodule of symmetric tensors. We define

$T_{W}:W\otimes_{F}V=W\otimes_{F[[t]]}V[[t]]\rightarrow S_{t}^{2}(W)$

by

$T_{W}:\sum_{i=1}^{s}w_{i}\otimes
v_{i}\mapsto\sum_{i,j=1}^{s}(v_{i},v_{j})w_{i}\otimes w_{j},$ (3.13)

where $w_{i}\in W,$ $v_{i}\in V$, $i=1,....,s,$ and where $w_{i}\otimes w_{j}$
denotes the tensor product of $w_{i},$ $w_{j}$ in $S_{t}^{2}(W).$ Of course
$T_{W}$ can be extended to an adelic map

$T_{W}:(W\otimes_{F}V)_{{\bf A}}\rightarrow S_{t}^{2}(W)_{{\bf A}},$

and for $r\in S_{t}^{2}(W)_{\bf A},$ we set

$\mathcal{U}_{r}=\\{x\in(W\otimes_{F}V)_{{\bf A}}\,|\,T_{W}(x)=r\\}.$

We now consider ${\rm Et}_{W}(\phi)$ as defined in (3.12). In that expression
we consider $\rho\in F_{W}$. We have a pairing

$S_{t}^{2}(W)\times F_{W}\to F,\,\,\,\,\,\,(\sum w_{i}\otimes
u_{i},\rho)=\sum\langle w_{i},\rho(u_{i})\rangle,$ (3.14)

which is clearly non-degenerate. And we have

$\langle w,\rho(w)\rangle=(T_{W}(w),\rho),\,\,\,\,\,w\in(W\otimes V)_{\bf A}.$

We may then rewrite the right hand side of (3.12) as

$\sum_{\rho\in F_{W}}\int_{(W\otimes V)_{{\bf
A}}}\phi(x)\psi(\frac{1}{2}(T_{W}(x),\rho))dx.$ (3.15)

The following result is a corollary of convergence of Eisenstein series on
loop groups, it will be proved in part II.

###### Theorem 3.3

Let $M$ be an snt-module as in (3.4), where $H_{k}$ is as in (1.9). If ${\rm
dim}V>6n+2$, then the series (3.10) converges absolutely and the convergence
is uniform for $\phi$ varying over a compact subsets in ${\cal S}(X_{\bf A})$.
It follows that the series (3.12)= (3.15) converges absolutely and the
convergence is uniform for $\phi$ varying over a compact subset of ${\cal
S}((W\otimes V)_{\bf A})$.

We can apply Proposition 1 and Proposition 2 of [7]. Using Weil’s notation as
in [7]:

$X=(W\otimes V)_{\bf A},\,\,\,\,\,G=S_{t}^{2}(W)_{\bf
A},\,\,\,\,\,\Gamma=S_{t}^{2}(W),\,\,\,\,\,f=T_{W}.$ (3.16)

and $G^{*}=(F_{W})_{\bf A}$, $\Gamma_{*}=F_{W}$, where we regard
$G^{*}=(F_{W})_{\bf A}$ as Pontryagin dual of $G$ by the pairing

$S_{t}^{2}(W)_{\bf A}\times(F_{W})_{\bf A}\to
S^{1},\,\,\,\,\,\,\\{a,\rho\\}=\psi(\frac{1}{2}(a,\rho)).$

We have

$F_{\phi}^{*}(g^{*})=\int_{(W\otimes V)_{{\bf
A}}}\phi(x)\psi(\frac{1}{2}\\{T_{W}(x),g^{*}\\})dx=\int_{X}\phi(x)\\{f(x),g^{*}\\}dx.$

Theorem 3.3 implies that the condition of Proposition 2 in [7] is satisfied;
that is

$\sum_{r^{*}\in\Gamma_{*}}|F_{\Phi}^{*}(g^{*}+\gamma^{*})|$

converges and the convergence is uniform as $(\phi,g^{*})$ varies over a
compact subset of ${\cal S}(X)\times G^{*}$. By using of Proposition 1 and
Proposition 2 [7], we obtain

###### Theorem 3.4

Suppose ${\rm dim}V>6n+2$. To every $r\in S_{t}^{2}(W)_{\bf A},$ there
corresponds a unique positive measure $\mu_{r}$ on $(W\otimes V)_{\bf A}$
whose support is contained in $\mathcal{U}_{r},$ so that for every function
$\phi$ on $(W\otimes V)_{\bf A}$ which is continuous with compact support, the
function $F_{\phi}(r)=\int\phi d\mu_{r}$ is continuous and satisfies

$\int F_{\phi}(r)dr=\int\phi(x)dx$

where $dr,$ $dx$ are fixed Haar measures on $S_{t}^{2}(W)_{\bf A},$
$(W\otimes_{F}V)_{\bf A},$ respectively. Moreover, the $\mu_{r}$’s are
tempered measures and for $\phi\in{\cal S}((W\otimes V)_{\bf A}),$ $F_{\phi}$
is continuous, is an element of $L^{1}(S_{t}^{2}(W)_{\bf A})$, and is the
Fourier transform of the function $F_{\phi}^{\ast}(\cdot)$ on
$S_{t}^{2}(W^{\ast})_{\bf A}$ given by

$F_{\phi}^{\ast}({\rho})=\int_{(W\otimes V)_{\bf
A}}\phi(x)\psi(\frac{1}{2}(T_{W}(x),{\rho}))dx.$

Finally,

${\rm Et}_{W}(\phi)=\sum_{r\in S_{t}^{2}(W)}\int_{(W\otimes V)_{\bf
A}}{\phi}d\mu_{r},$ (3.17)

the series on the right being absolutely convergent.

Since the convergence of the right hand side of (3.17) is uniform as $\phi$
varies on a compact subset of ${\cal S}((W\otimes V)_{\bf A})$, ${\rm Et}_{W}$
is a tempered measure on $(W\otimes V)_{\bf A}$. The formula (3.17) can be
restated as:

###### Corollary 3.5

We have the identity of the tempered distributions:

${\rm Et}_{W}=\sum_{r\in S_{t}^{2}(W)}\mu_{r}.$ (3.18)

##  4 An extension of Weil’s abstract lemma

In this section we study the measure $d\mu_{r}$ in Theorem 3.4. We begin with
a statement of Proposition 1 in [7] (page 6)

###### Lemma 4.1

Let $X$ and $G$ be two locally compact, abelian groups with fixed Haar
measures $dx$, $dg$ respectively. Let

$f:X\rightarrow G$

be a continuous map such that

  1. (A)

For any $\Phi\in\mathcal{S}(X),$ the function $F_{\Phi}^{\ast}$ on $G^{\ast}$
defined by

$F_{\Phi}^{\ast}(g^{\ast})=\int_{X}\Phi(x)\\{f(x),g^{\ast}\\}dx,$ (4.1)

(where $dx$ $\\{\,,\,\\}$ denotes the pairing between $G$ and $G^{\ast})$ is
integrable on $G^{\ast}$ and the integral $\int|F_{\Phi}^{\ast}|dg^{\ast}$
(where $(dg^{\ast}$ is Haar measure on $G^{\ast})$ dual to $dg$) converges
uniformly on every compact subset of $\mathcal{S}(X).$

Then one can find a uniquely determined family of positive measures
$\\{\mu_{g}\\}_{g\in G},$ on $X,$ where support($\mu_{g})\subseteq
f^{-1}(\\{g\\}),$ and so that for every continuous function with compact
support $\Phi$ on $X,$ the function $F_{\Phi}$ on $G$ defined by

$F_{\Phi}(g)=\int\Phi d\mu_{g},$ (4.2)

is continuous and satisfies

$\int F_{\Phi}dg=\int\Phi dx.$ (4.3)

Moreover, the measures $\mu_{g}$ are tempered measures and for
$\Phi\in\mathcal{S}(X),$ $F_{\Phi}$ is continuous, belongs to $L^{1}(G),$
satisfies (4.3), and is the Fourier transform of $F_{\Phi}^{\ast}.$

We call $(X,G,f)$ as in Lemma 4.1, an admissible triple.

Let $M,M_{-},M_{+},V$ be as in Section 3. And as Section 3, $W$ denotes a
$F[[t]]$-submodule of $M_{-}$. For a fixed place $v$ of $F$, we set

$X_{v}=(W\otimes V)_{F_{v}},\,\,\,\,\,\,\,\,G_{v}=S_{t}^{2}(W)_{F_{v}},$

and let $T_{v}:X_{v}\to G_{v}$ be the $F_{v}$-linear extension of $T_{W}$
defined in (3.13). The dual group $G_{v}^{*}$ of $G_{v}$ is identified with
$(F_{W})_{F_{v}}$.

###### Lemma 4.2

If ${\rm dim}\,V>6n+2$, then above triple $(X_{v},G_{v},T_{v})$ is an
admissible triple, equivalently,

$F_{\Phi}^{\ast}(g^{\ast})=\int_{X}\Phi(x)\\{f(x),g^{\ast}\\}dx,$

satisfies condition (A) in Lemma 4.1.

This lemma is an analog of Proposition 5 in [7] (page 45). We expect that the
condition ${\rm dim}\,V>6n+2$ can be replaced by the weaker condition ${\rm
dim}\,V>6n+1$. For our purpose, the condition in the lemma is enough.

Proof. For simplicity, we write $X,G,T$ for $X_{v},G_{v},T_{v}$. Let $X_{\bf
A}=(W\otimes V)_{{\bf A}}$, $G_{\bf A}=S_{t}^{2}(W)_{{\bf A}}$, and $T_{\bf
A}:X_{\bf A}\to G_{\bf A}$ be $T_{W}\otimes{\bf A}$ then we have

$X_{\bf A}=X\times X^{c},\,\,\,\,\,\,\,G_{\bf A}=G\times G^{c}$

where $X^{c}$ is the restricted product of $(W\otimes V)_{F_{w}}$’s for $w\neq
v$, $G^{c}$ is the restricted product of $S_{t}^{2}(W)_{F_{w}}$’s for $w\neq
v$. And $T_{\bf A}=T\times T_{c}$, where $T_{c}:X^{c}\to G^{c}$ is defined
similarly as $T$. Let $C$ be a compact subset of ${\cal S}(X)$. We choose a
function $\phi_{0}\in{\cal S}(X^{c})$ such that

$\int_{X^{c}}\phi_{0}(x_{c})dx_{c}\neq 0.$

Then $F^{*}_{\phi_{0}}(g_{c}^{*})$ given by

$F^{*}_{\phi_{0}}(g_{c}^{*})=\int_{X^{c}}\phi_{0}(x_{c})\\{T_{c}(x_{c}),g_{c}^{*}\\}dx_{c},$

satisfies that $F^{*}_{\phi_{0}}(0)\neq 0$. Since
$F^{*}_{\phi_{0}}(g_{c}^{*})$ is continuous, we have

$\int_{G^{c}}|F_{\phi_{0}}(g_{c}^{*})|dg_{c}^{*}=M\neq 0.$

For each function $\phi(x)\in{\cal S}(X)$ , $\phi(x)\phi_{0}(x_{c})$ is in
${\cal S}(X_{\bf A})$. Since $C$ is a compact subset of ${\cal S}(X)$,
$C\phi_{0}$ is a compact subset of ${\cal S}(X_{\bf A})$. By Theorem 3.3, and
the Proposition 2 in [7], we know that $X_{\bf A},G_{\bf A},T_{\bf A}$ is an
admissible triple. We consider the function

$\int_{X}\int_{X^{c}}\phi(x)\phi_{0}(x_{c})\\{T(x),g^{*}\\}\\{T_{c}(x_{c}),g_{c}^{*}\\}dxdx_{c}=F_{\phi}(g^{*})F_{\phi_{0}}(g_{c}^{*}).$

Since $C\phi_{0}$ is compact, the integral

$\int_{G^{*}\times
G_{c}^{*}}|F_{\phi}(g^{*})F_{\phi_{0}}(g_{c}^{*})|dg^{*}dg_{c}^{*}$

converges uniformly as $\phi$ varies over $C$. By the Fubuni theorem, we have

$\displaystyle\int_{G^{*}\times
G_{c}^{*}}|F_{\phi}(g^{*})F_{\phi_{0}}(g_{c}^{*})|dg^{*}dg_{c}^{*}$
$\displaystyle=\int_{G^{*}}|F_{\phi}(g^{*})|dg^{*}\int_{G_{c}^{*}}|F_{\phi_{0}}(g_{c}^{*})|d\bar{g}_{c}^{*}$
$\displaystyle=M\int_{G^{*}}|F_{\phi}(g^{*})|dg^{*}.$

This implies that

$\int_{G^{*}}|F_{\phi}(g^{*})|dg^{*}$

converges uniformly as $\phi$ varies on $C$. $\Box$.

Since $W$ is a finite dimensional over $F$ and $t^{N}W=0$ for $N$ large, $W$
is isomorphic to

$F[t]/(t^{k_{1}})e_{1}\oplus\dots\oplus F[t]/(t^{k_{m}})e_{m}$

as a $F[[t]]$-module, where $e_{1},\dots,e_{m}$ is a quasi-basis of $W$. Let
$\bar{W}$ denote the quotient $W/tW$, we have

$\bar{W}\cong F\bar{e}_{1}\oplus\dots\oplus F\bar{e}_{m},$

where $\bar{e}_{i}$ is the projection of $e_{i}$. Let $\bar{G}_{v}$ denote
$S^{2}(\bar{W}_{F_{v}})$, where $S^{2}(\bar{W}_{F_{v}})$ is the subspace of
the symmetric tensors in $\bar{W}_{F_{v}}\otimes\bar{W}_{F_{v}}$. For
simplicity, we shall write $G,X,\bar{G}$ for $G_{v},X_{v},\bar{G}_{v}$. We
have $\bar{T}$ given by

$\bar{T}:\bar{X}\stackrel{{\scriptstyle\rm def}}{{=}}(\bar{W}\otimes
V)_{v}\to\bar{G}\stackrel{{\scriptstyle\rm
def}}{{=}}S^{2}(W_{v}),\,\,\,\,\,\,\,\sum_{i}u_{i}\otimes
v_{i}\mapsto\sum_{i,j}(v_{i},v_{j})u_{i}\otimes u_{j}.$

The condition ${\rm dim}V>6n+2$ implies in particular ${\rm dim}V>6n+2\geq
6m+2$; this implies that the condition for Proposition 5 in [7] (page 45) is
satisfied, so $(\bar{X},\bar{G},\bar{T})$ is an admissible triple. The
canonical map $W\to\bar{W}$ induces surjective linear maps

$\pi_{X}:X\to\bar{X},\,\,\,\,\,\,\,\pi_{G}:G\to\bar{G}.$

We have the commutative diagram

$\begin{array}[]{ccc}X&\overset{\pi_{X}}{\longrightarrow}&\bar{X}\\\
\downarrow T&&\downarrow\bar{T}\\\
G&\overset{\pi_{G}}{\longrightarrow}&\bar{G}\end{array}$ (4.4)

Let $f:X\to\bar{G}$ denote $\pi_{G}\circ T=\bar{T}\circ\pi_{X}$.

###### Lemma 4.3

For $x\in X$, the following conditions are equivalent
(1). $T$ is submersive at $x$.
(2). $f$ is submersive at $x$.
(3). $\bar{T}$ is submersive at $\pi_{X}(x)$.

Proof. Since $\pi_{X}$ is linear and surjective, it is submmersive at every
point. It follows that (2) and (3) are equivalent. Since $\pi_{G}$ is linear
and surjective, it is submmersive at every point, it follows that (1) implies
(3). The fact that (3) implies (1) follows directly from Lemma 5.7 in Section
5.

###### Lemma 4.4

The map $f:X\to\bar{G}$ satisfies the condition (A) in Lemma 4.1, so
$(X,\bar{G},f)$ is an admissible triple.

Proof. We use the following diagram to prove the lemma:

$\begin{array}[]{ccc}X&\overset{\pi_{X}}{\longrightarrow}&\bar{X}\\\
&f\searrow&\downarrow\bar{T}\\\ &&\bar{G}\end{array}$

Let $K$ denote the kernel of $\pi_{X}$. We have a map from ${\cal
S}(X)\to{\cal S}(\bar{X})$ given by

$\Phi\mapsto\bar{\Phi}(\bar{x})=\int_{K}\Phi(k+\bar{x})dk,$ (4.5)

where $k$ denotes the Haar measure on $K$. It is clear that this map is
continuous. Consider

$F_{\Phi}^{\ast}({\bar{g}}^{\ast})=\int_{X}\Phi(x)<f(x),{\bar{g}}^{*}>dx$

In the right hand side, we integrate over $K$ first, and notice that

$<f(x),{\bar{g}}^{*}>=<\bar{T}(\bar{x}),{\bar{g}}^{*}>$

where $\bar{x}=\pi_{X}(x)$, we get

$F_{\Phi}^{\ast}({\bar{g}}^{\ast})=\int_{\bar{X}}\bar{\Phi}(\bar{x})<\bar{T}(\bar{x}),{\bar{g}}^{\ast}>d\bar{x},$

since $(\bar{X},\bar{G},\bar{T})$ is an admissible triple, the right hand side
is in $L^{1}(\bar{G})$. And if $\Phi$ runs through a compact subset of ${\cal
S}(X)$, then $\bar{\Phi}$ which is related to $\Phi$ by (4.5) runs through a
corresponding compact subset of ${\cal S}(\bar{X})$, so the integral

$\int|F_{\Phi}^{\ast}({\bar{g}}^{\ast})|d\bar{g}^{*}$

converges uniformly. This proves the lemma. $\Box$

By Lemma 4.1, we have a family of measures
$\\{\mu_{\bar{g}}\\}_{\bar{g}\in\bar{G}}$ on $X$, with
${\rm support}(\mu_{\bar{g}})\subset f^{-1}(\bar{g})$, such that for every
$\Phi\in C_{c}(X)$, $\int\Phi d\mu_{\bar{g}}$ is continuous function of
$\bar{g}$ and

$\int_{\bar{G}}\left(\int\Phi d\mu_{\bar{g}}\right)d\bar{g}=\int_{X}\Phi dx.$
(4.6)

On the other hand, apply Lemma 4.1 to the admissible triple
$(\bar{G},\bar{X},\bar{T})$, we have a family of measures
$\\{\mu_{\bar{g}}^{\bar{T}}\\}_{\bar{g}\in\bar{G}}$ on $\bar{X}$, with ${\rm
support}(\mu_{\bar{g}}^{\bar{T}})\subset\bar{T}^{-1}(\bar{g})$, and

$\int_{\bar{G}}\left(\int\bar{\Phi}d\mu_{\bar{g}}^{\bar{T}}\right)d\bar{g}=\int_{\bar{X}}\bar{\Phi}d\bar{x}.$
(4.7)

Suppose that $\Phi$ and $\bar{\Phi}$ are related by (4.5) and the Haar
measures $dg,d\bar{g},dk$ are compatible so that the right hand sides of (4.6)
and (4.7) are equal. We then have

$\int_{\bar{G}}\left(\int\Phi
d\mu_{\bar{g}}\right)d\bar{g}=\int_{\bar{G}}\left(\int\bar{\Phi}d\mu_{\bar{g}}^{\bar{T}}\right)d\bar{g}$
(4.8)

We claim that the truth of (4.8) for all $\Phi\in C_{c}(X)$ implies that

$\int\Phi d\mu_{\bar{g}}=\int\bar{\Phi}d\mu_{\bar{g}}^{\bar{T}}.$ (4.9)

To prove this, take arbitrary $h(\bar{g})\in C_{c}(\bar{G})$, let
$f^{*}h=h\circ f$, replace $\Phi$ in (4.8) by $f^{*}h\Phi$. We get

$\int_{\bar{G}}h(\bar{g})\left(\int\Phi
d\mu_{\bar{g}}\right)d\bar{g}=\int_{\bar{G}}h(\bar{g})\left(\int\bar{\Phi}d\mu_{\bar{g}}^{\bar{T}}\right)d\bar{g}.$

The above is true for all $h(\bar{g})\in C_{c}(\bar{G})$, and $\int\Phi
d\mu_{\bar{g}}$, $\int\bar{\Phi}d\mu_{\bar{g}}^{\bar{T}}$ are continuous
functions of $\bar{g}$, so we have (4.9). We rewrite (4.9) as

$\int\Phi
d\mu_{\bar{g}}=\int(\int_{K}\Phi(k+\bar{x})dk)d\mu_{\bar{g}}^{\bar{T}}$ (4.10)

Recall Lemma 17 [7] (page 52), the support of $\mu_{\bar{g}}^{\bar{T}}$ is on
the $\bar{T}^{-1}(\bar{g})_{\rm re}$ (the regular points (= submerssive
points) in $\bar{T}^{-1}(\bar{g})$). By (4.10), the support of $\mu_{\bar{g}}$
is in $\pi_{X}^{-1}\bar{T}^{-1}(\bar{g})_{\rm re}$, which is precisely the set
of the regular points in $f^{-1}(\bar{g})$ by Lemma 4.3. We have proved

###### Lemma 4.5

The measure $\mu_{\bar{g}}$ is supported on $f^{-1}(\bar{g})_{\rm re}$, the
subset of regular points of $f^{-1}(\bar{g})$.

We consider the diagram

$\begin{array}[]{ccc}X&&\\\ \downarrow T&\searrow f&\\\
G&\overset{\pi_{G}}{\longrightarrow}&\bar{G}\end{array}$

For the admissible triple $(X,G,T)$, Lemma 4.1 implies that we have a family
of measures $\mu_{g}$ ($g\in G$) supported on $T^{-1}(g)$ such that for
$\Phi\in C_{c}(X)$, $F_{\Phi}(g)\stackrel{{\scriptstyle\rm def}}{{=}}\int\Phi
d\mu_{g}\in C(G)$ , we have

$\int_{G}F_{\Phi}(g)dg=\int_{X}\Phi(x)dx$ (4.11)

We take a subspace of $G$ that maps isomorphically onto $\bar{G}$, we denote
this space by $\bar{G}$, so we have the identification $G=K\times\bar{G}$,
where $K$ is the kernal of $\pi_{G}$. Since $\Phi$ has compact support,
$F_{\Phi}$ has compact support, and it is continuous, so we have

$\bar{g}\to\int_{K}F_{\Phi}(\bar{g}+k)dk$

is in $C(\bar{G})$. The left hand side of (4.11) can be written as

$\int_{\bar{G}}\int_{K}F_{\Phi}(\bar{g}+k)dkd\bar{g},$

so we have

$\int_{\bar{G}}\int_{K}F_{\Phi}(\bar{g}+k)dkd\bar{g}=\int_{X}\Phi(x)dx$ (4.12)

On the other hand, use the triple $(X,\bar{G},f)$, we have by (4.6

$\int_{\bar{G}}(\int\Phi d\mu_{\bar{g}})d\bar{g}=\int_{X}\Phi(x)dx$ (4.13)

Comparing (4.12) and (4.13), we get

$\int_{\bar{G}}\int_{K}F_{\Phi}(\bar{g}+k)dkd\bar{g}=\int_{\bar{G}}(\int\Phi
d\mu_{\bar{g}})d\bar{g}.$ (4.14)

Take an arbitrary $h\in C_{c}(\bar{G})$, let $f^{*}h=h\circ f\in C(X)$, and
replacing $\Phi$ in (4.14) by $f^{*}h\Phi$, we get

$\int_{\bar{G}}h(\bar{g})\int_{K}F_{\Phi}(\bar{g}+k)dkd\bar{g}=\int_{\bar{G}}h(\bar{g})(\int\Phi
d\mu_{\bar{g}})d\bar{g}.$

This is true for arbitrary $h\in C_{c}(\bar{G})$, and both
$\int_{K}F_{\Phi}(\bar{g}+k)dk$ and $\int\Phi d\mu_{\bar{g}}$ are continuous
functions, so we have

$\int_{K}F_{\Phi}(\bar{g}+k)dk=\int\Phi d\mu_{\bar{g}}.$

But the measure $\mu_{\bar{g}}$ is supported on $f^{-1}(\bar{g})_{\rm re}$, so
it is a gauge measure (see section 5 of [7] for the definition of ”gauge”
measure),

$f^{-1}(\bar{g})=T^{-1}(\pi_{G}^{-1}(\bar{g}))=T_{X}^{-1}(g+K)$

Note that

$f^{-1}(\bar{g})_{\rm re}=\cup_{k\in K}T^{-1}(g+k)_{\rm re}$

For each given $\bar{g}$, $T^{-1}(\bar{g}+k)_{\rm re}$ is non-singular
subvariety of $X$, we have a gauge form $d\delta_{k}$ on it, we have

$\int\Phi d\mu_{\bar{g}}=\int_{K}\int_{T^{-1}(\bar{g}+k)}\Phi d\delta_{k}dk$

we obtain

$\int_{K}F_{\Phi}(\bar{g}+k)dk=\int_{K}\int{T^{-1}(\bar{g}+k)}\Phi(x)d\delta_{k}dk.$
(4.15)

The above holds for arbitrary $\Phi\in C_{c}(X)$, use the same method we used
to deduce (4.9) from (4.8), we deduce from (4.15) that

$F_{\Phi}(\bar{g}+k)=\int{T^{-1}(\bar{g}+k)_{\rm re}}\Phi d\delta_{k}$

So we have proved

###### Lemma 4.6

The measure $\mu_{g,v}$ for the triple $(X_{v},G_{v},T_{v})$ is supported on
$T_{v}^{-1}(g)_{\rm re}$ and is the gauge measure.

Lets recall the meaning of ”gauge” measure ( [7], section 5). In the situation
as Lemma 4.6. We first take an invariant top form $\eta$ on $G$ and an
invariant top form $\omega$ on $X$. Let $X^{\prime}$ be the open set of $X$
that consists of all the points where $T$ is submersive. Near each point $x\in
X^{\prime}$, there is a form $\theta_{x}$ such that $\theta\wedge
T^{*}\eta=\omega$. For each $y\in G$, the local forms $\theta$, restrict to
$T^{-1}(y)_{\rm re}=T^{-1}(y)\cap X^{\prime}$, give a top form $\theta_{y}$ on
$T^{-1}(y)_{\rm re}$, which defines a measure which is equal to $\mu$ in Lemma
4.6.

Now we consider the global situation. Let $X,G,T_{W}$ as in (3.16). For each
$r\in S^{2}_{t}(W)$, we consider the inverse image $T_{W}^{-1}(r)$. Notice
that $T$ is submersive at a generic point, the space $X^{\prime}$ formed by
the points at which $f$ is submersive is $F$-open in $X$. We take a top form
$\eta$ over $G$ and a top form $\omega$ on $X$, we assume that the Tamagawa
measures on $G({\bf A})$ ($X_{\bf A}$ resp.) with respect to $\eta$ (resp
$\omega$ ) are the Haar measure normalized by the condition that the covolume
of $G(F)$ ($X(F)$) is $1$. The space $X^{\prime}$ can be covered by $F$-open
subsets $U_{\lambda}$ such that, there is a form $\theta_{\lambda}$ rational
over $F$ satisfying $\theta\wedge f^{*}=\omega$. For each $i\in G(F)$, then
the $\theta$’s restrict on $f^{-1}(i)\cap X^{\prime}$ to get a top form
$\theta_{i}$ on $f^{-1}(i)\cap X^{\prime}$. By Lemma 4.6, $d\mu_{i,v}$ is
given by $|\theta_{i}|_{v}$. Using a similar argument as in [7], Section 42,
we can prove that $1$ is a system of convergence factor of $|\theta_{i}|_{\bf
A}$. And we have

###### Theorem 4.7

For each $r\in S_{t}^{2}(W)$, the measure $\mu_{r}$ in (3.17) is supported in
$T_{W}^{-1}(r)_{\rm re}$ and it is the same as $|\theta_{r}|_{\bf A}$, the
measure define by the form $\theta_{r}$.

##  5 Classification of orbits of orthogonal groups

As in Section 3, we denote $M$ an snt-module with decomposition $M=M_{-}\oplus
M_{+}$ into $t$-Lagrangian subspaces and $V$ a finite dimensional vector space
over $F$ with a non-degenerate, bilinear symmetric form $(,)$. The orthogonal
group $G(F[[t]])$ acts on $M\otimes V$, leaving the subspace $M_{-}\otimes V$
invariant. The purpose of this section to give a complete set of invariants of
$G(F[[t]])$-orbits in $M_{-}\otimes V$ (see Theorem 5.4 below).

###### Definition 5.1

Let $W$ be a finitely generated $F[[t]]$-module. A submodule $L\subset W$ is
called a primitive submodule if one of the following equivalent conditions is
satisfied:
(1). there is a complement $F[[t]]$-submodule $L^{\prime}$, i.e. $W=L\oplus
L^{\prime}$.
(2). the natural map $L/tL\to W/tW$ induced form the embedding
$L\hookrightarrow M$ is injective.

Examples. (1) Let $W=F[[t]]/(t^{k_{1}})\oplus\dots\oplus F[[t]]/(t^{k_{m}})$,
For any $l\leq m$, $L=F[[t]]/(t^{k_{1}})\oplus\dots\oplus F[[t]]/(t^{k_{l}})$
is a primitive submodule of $M$.
(2). Let $F[[t]]^{m}$ be a free $F[[t]]$-module of rank $m$, then $F[[t]]^{l}$
(consists of elements with last $(m-l)$-components $0$ is a primitive
submodule. (3). $\\{0\\}$ is a primitive submodule for any $M$.

Let $W$ be a finitely generated $F[[t]]$-module, $e_{1},\dots,e_{m}$ is called
a quasi-basis of $W$ if every $e_{i}\neq 0$, every element $x\in W$ can be
written as a $F[[t]]$-linear combination of $e_{1},\dots,e_{m}$, and
$a_{1}e_{1}+\dots+a_{m}e_{m}=0$ ($a_{i}\in F[[t]]$) implies that all
$a_{i}e_{i}=0$. If $W$ is a finite dimensional $F[[t]]$-module, the ”quasi-
basis” defined above is the same notion as defined in Section 2. If $W$ is a
free $F[[t]]$-module, then a quasi-basis of $W$ is the same as a basis of $W$.
In the example (1) above, there is a quasi-basis of $m$ elements. It is clear
that $e_{1},\dots,e_{m}$ is a quasi-basis of $W$ if and only if the images of
$e_{1},\dots,e_{m}$ in $W/tW$ form an $F$-basis of vector space $W/tW$.
Therefore any two quasi-bases have the same number of elements. The
cardinality of a quasi-basis is called the rank of $W$.

Every

$x=\sum_{i}u_{i}\otimes v_{i}\in M_{-}\otimes V=M_{-}\otimes_{F[[t]]}V[[t]]$

gives rise to an $F[[t]]$-linear map

$f_{x}:V[[t]]\to M_{-},\,\,\,\,\,\,f_{x}(v)=\sum_{i}(v_{i},v)u_{i}.$ (5.1)

We denote by ${\rm Im}\,f_{x}$ the image of $f_{x}$, which is an
$F[[t]]$-submodule of $M_{-}$. Let $e_{1},\dots,e_{m}$ be a quasi-basis of
${\rm Im}\,f_{x}$, then

${\rm Im}\,f_{x}\cong F[[t]]/(t^{k_{1}})e_{1}\oplus\dots\oplus
F[[t]]/(t^{k_{m}})e_{m},$ (5.2)

where $k_{i}$ is the smallest positive integer such that $t^{k_{i}}e_{i}=0$.

###### Lemma 5.2

Let $e_{1},\dots,e_{m}$ be a quasi-basis of ${\rm Im}\,f_{x}$, and suppose
$v_{1},\dots,v_{m}\in V[[t]]$ satisfy $f_{x}(v_{i})=e_{i}$ ($i=1,\dots,m)$,
then ${\rm Span}_{F[[t]]}\\{v_{1},\dots,v_{m}\\}$ is a primitive submodule of
$V[[t]]$ and $v_{1},\dots,v_{m}$ is a basis of ${\rm
Span}_{F[[t]]}\\{v_{1},\dots,v_{m}\\}$.

Proof. Set $L={\rm Span}_{F[[t]]}\\{v_{1},\dots,v_{m}\\}$, then
$f_{x}|_{L}:L\to{\rm Im}\,f_{x}$ induces a linear isomorphism:

$\bar{f_{x}}:L/tL\to{\rm Im}\,f_{x}/t({\rm Im}\,f_{x}).$

This implies in particular, ${\rm dim}\,L/tL=m$, so the map $L/tL\to
V[[t]]/tV[[t]]$ induced from $L\subset V[[t]]$ is injective, so $L$ is
primitive submodule of $V[[t]]$. The other conclusions are clear. $\Box$

The bilinear form $(,)$ on $V$ can be extended to a $F[[t]]$-valued bilinear
form on $V[[t]]=V\otimes_{F}F[[t]]$. This $F[[t]]$-valued bilinear form on
$V[[t]]$ is non-degenerate. It is easy to prove

###### Lemma 5.3

Let $e_{1},\dots,e_{m}$ be a quasi-basis of ${\rm Im}\,f_{x}$, then there are
elements $w_{1},\dots,w_{m}\in V[[t]]$ such that
(1) ${\rm Span}_{F[[t]]}\\{w_{1},\dots,w_{m}\\}$ is a primitive submodule of
$V[[t]]$ and $w_{1},\dots,w_{m}$ is basis of ${\rm
Span}_{F[[t]]}\\{w_{1},\dots,w_{m}\\}$.
(2)

$x=e_{1}\otimes w_{1}+\dots+e_{m}\otimes w_{m}.$

Proof. Choose $v_{i}\in V[[t]]$ ( $i=1,\dots,m$) such that
$f_{x}(v_{i})=e_{i}$. By Lemma 5.2, ${\rm
Span}_{F[[t]]}\\{v_{1},\dots,v_{m}\\}$ is a primitive submodule of $V[[t]]$
and $v_{1},\dots,v_{m}$ is a basis. It is clear that

$V[[t]]={\rm Span}_{F[[t]]}\\{v_{1},\dots,v_{m}\\}\oplus ker(f_{x}).$

Let $v_{m+1},\dots,v_{N}$ be a basis of $ker(f_{x})$. Then $v_{1},\dots,v_{N}$
is basis of $V[[t]]$. Now let $w_{1},\dots,w_{N}$ be the dual basis if
$v_{1},\dots,v_{N}$, i.e., $(v_{i},w_{j})=\delta_{ij}$. This is clear that
$v=\sum_{i=1}^{m}e_{i}\otimes w_{i}$. $\Box$

From Lemma 5.3, we see that $x\in{\rm Im}\,f_{x}\otimes_{F[[t]]}V[[t]]$. We
define a map

$T:{\rm Im}\,f_{x}\otimes_{F[[t]]}V[[t]]\to S_{t}^{2}({\rm
Im}\,f_{x}),\,\,\,\,\,\,\sum_{i}^{N}u_{i}\otimes
v_{i}\mapsto\sum_{i,j=1}^{N}(v_{i},v_{j})u_{i}\otimes u_{j}.$

Where $S_{t}^{2}({\rm Im}\,f_{x})$ denote the subspace of symmetric tensors in
${\rm Im}\,f_{x}\otimes_{F[[t]]}{\rm Im}\,f_{x}$. We remark that though ${\rm
Im}\,f_{x}$ is a submodule of $M_{-}$, but the natural map $S_{t}^{2}({\rm
Im}\,f_{x})\to S_{t}^{2}(M_{-})$ is in general not an embedding.

###### Theorem 5.4

Two elements $x,y\in M_{-}\otimes V$ are in the same $G(F[[t]])$-orbit iff
${\rm Im}\,f_{x}={\rm Im}\,f_{y}$ and $T(x)=T(y)$ .

We need some preparations for proving the theorem.

We recall a special case of Witt’s theorem (see [4] ):

###### Theorem 5.5

If $L_{1},$ $L_{2}\subseteq V[[t]]$ are two primitive submodules of $V[[t]]$
and if $\sigma:L_{1}\rightarrow L_{2}$ is an isometry, then $\sigma$ can be
extended to an isometry in $G(F[[t]])$.

It is clear that if $x,y$ are in the same $G(F[[t]])$-orbit, then ${\rm
Im}\,f_{x}={\rm Im}\,f_{y}$ and $T(x)=T(y).$ Conversely, if ${\rm
Im}\,f_{x}={\rm Im}\,f_{y}\stackrel{{\scriptstyle\rm def}}{{=}}W$ and
$T(x)=T(y)$. Let $e_{1},....,e_{m}$ be a quasi-basis for $W$, and $W$ be as
(5.2). We may assume that $k_{1}\geq k_{2}\geq\dots\geq k_{m}$. Then
$S_{t}^{2}(W)$ has a quasi-basis $e_{ij}\stackrel{{\scriptstyle\rm
def}}{{=}}e_{i}\otimes e_{j}+e_{j}\otimes e_{i}$ ($1\leq i\leq j\leq m$), and

$S_{t}^{2}(W)=\sum_{1\leq i\leq j\leq m}F[[t]]/(t^{k_{j}})e_{ij}.$

By lemma 5.3, we may write

$x=e_{1}\otimes a_{1}+\dots+e_{m}\otimes a_{m},\,\,\,\,\,\,y=e_{1}\otimes
b_{1}+\dots+e_{m}\otimes b_{m}$

with $\\{a_{1},\dots,a_{m}\\}$ and $\\{b_{1},\dots,b_{m}\\}$ satisfy condition
(1) in Lemma 5.3. Then $T(x)=T(y)$ implies that

$(a_{i},a_{j})=(b_{i},b_{j})\,\,\,\,\,{\rm mod}\,\,t^{{\rm
min}(k_{i},k_{j})}.$ (5.3)

###### Lemma 5.6

Let $L_{1}$ and $L_{2}$ be two primitive submodules of $V[[t]]$ with bases
$a_{1},\dots,a_{m}$ and $b_{1},\dots,b_{m}$. Let $1\geq k_{1}\geq\dots\geq
k_{m}$. If (5.3) holds, then the set $b_{1},\dots,b_{m}$ can be altered to
another set ${\tilde{b}}_{1},\dots,{\tilde{b}}_{m}$ such that

${\tilde{b}}_{i}=b_{i}\,\,\,\,\,\,{\rm mod}\,t^{k_{i}},\,\,\,\,{\rm
for}\,1\leq i\leq m,$ (5.4)

and

$({\tilde{b}}_{i},{\tilde{b}}_{i})=(a_{i},a_{j})\,\,\,\,\,\,{\rm for}\,1\leq
i,j\leq m,$ (5.5)

and the $F[[t]]$-span of ${\tilde{b}}_{1},\dots,{\tilde{b}}_{m}$ is a
primitive submodule of $V[[t]]$ with ${\tilde{b}}_{1},\dots,{\tilde{b}}_{m}$
as a basis.

Suppose the truth of Lemma 5.6, then Theorem 5.4 can be proved as follows. The
equation (5.4) implies that

$y=\sum e_{i}\otimes b_{i}=\sum e_{i}\otimes\tilde{b}_{i}.$

The equation (5.5) implies that the map $\sigma:L_{1}\rightarrow L_{2}$ give
by $a_{i}\mapsto{\tilde{b}}_{i}$ is an isometry, by Theorem 5.5, $\sigma$ can
be extended to $g\in G(F[[t]])$. then

$y\cdot g=(\sum e_{i}\otimes\tilde{b}_{i})\cdot g=\sum
e_{i}\otimes\sigma(\tilde{b}_{i})=\sum e_{i}\otimes{a}_{i}=x.$

It remains to prove Lemma 5.6.

Proof of Lemma 5.6. We use induction on $m$. For case $m=1$, we first take
$c\in V[[t]]$ such that $(b_{1},c)=1$, we want to find
${\tilde{b}}_{1}=b_{1}+t^{k_{1}}h(t)c$ where
$h(t)=h_{0}+h_{1}t+h_{2}t^{2}+\dots\in F[[t]]$ such that We have

$(a_{1},a_{1})=({\tilde{b}}_{1},{\tilde{b}}_{1})$

which is equivalent to

$(a_{1},a_{1})-(b_{1},b_{1})=2t^{k_{1}}h(t)+t^{2k_{1}}h(t)^{2}(c,c).$ (5.6)

Since $(a_{1},a_{1})=(b_{1},b_{1})\,\,{\rm mod}\,t^{k_{1}}$, we see that 5.6
holds mod $t^{k_{1}}$ for arbitrary $h(t)$. Compare the coefficient of
$t^{k_{1}}$, we solve for $h_{0}$, after $h_{0}$, we compare coefficient of
$t^{k_{1}+1}$, we solve $h_{1}$. It is clear that the similar process can be
continued to solve all $h_{i}$. Assume the Lemma holds for $m-1$, so we can
find ${\tilde{b}}_{1},\dots,{\tilde{b}}_{m-1}$ such that

${\tilde{b}}_{i}\cong b_{i}\,\,\,\,\,\,{\rm mod}\,t^{k_{i}},\,\,\,\,{\rm
for}\,1\leq i\leq m-1,$ (5.7)

and

$({\tilde{b}}_{i},{\tilde{b}}_{i})=(a_{i},a_{j})\,\,\,\,\,\,{\rm for}\,1\leq
i,j\leq m-1.$ (5.8)

We may assume $b_{i}={\tilde{b}}_{i}$ for $i=1,\dots,m-1$. Since $(\,)$ on
$V[[t]]$ is non-degenerate, we can find $c_{1},c_{2},\dots,c_{m}\in V[[t]]$
such that

$(b_{i},c_{j})=\delta_{i,j}.$ (5.9)

We want to find $h_{1}(t),\dots,h_{m}(t)\in F[[t]]$ such that

${\tilde{b}}_{m}=b_{m}+t^{k_{m}}\left(h_{1}(t)c_{1}+\dots+h_{m}(t)c_{m}\right)$

satisfies the following $m$ equations

$({\tilde{b}}_{i},{\tilde{b}}_{m})=(a_{i},a_{m}),\,\,\,\,\,\,\,i=1,\dots,m$
(5.10)

Let $h_{i}(t)=\sum_{s=0}^{\infty}h_{i,s}t^{s}$. The equations (5.10) already
hold mod $t^{k_{m}}$. Compare the coefficient of $t^{k_{m}}$ of (5.10), we get
a linear system with $m$-variables $h_{1,0},\dots,h_{m,0}$ and $m$ equations,
this system has non-zero determinant, thanks to (5.9), we can solve for
$h_{1,0},\dots,h_{m,0}$. Then we compare coefficient of $t^{k_{m}+1}$ of
(5.10), we get a linear system with $m$ equations and $m$ variables
$h_{1,1},\dots,h_{m,1}$, and again because of (5.9), the system has a
solution. It is clear that this process can be continued to solve for all
$h_{i,s}$’s. $\Box$

###### Lemma 5.7

Let $v$ be a place of $F$, $W$ be a $F[[t]]$-submodule of $M_{-}$, let
$W_{v}=W\otimes F_{v},V_{v}=V\otimes F_{v}$. The map $T:W_{v}\otimes V_{v}\to
S_{t}^{2}(W_{v})$ given by

$T(\sum_{i}u_{i}\otimes v_{i})=\sum_{i,j}(v_{i},v_{j})u_{i}\otimes u_{j}$

is submersive at $x_{0}\in W_{v}\otimes V_{v}$ iff ${\rm Im}\,f_{x_{0}}=W_{v}$

Proof. For simplicity, we denote $W_{v},V_{v},F_{v}$ by $W,V,F$ respectively.
Let $a_{1},\dots,a_{m}$ be a quasi-basis of the $F[[t]]$-module ${\rm
Im}\,f_{x_{0}}$. By Lemma 5.3, $x_{0}$ can be written as

$x_{0}=\sum_{i}a_{i}\otimes b_{i}=a_{1}\otimes b_{1}+\dots+a_{m}\otimes b_{m}$

where $b_{1},\dots,b_{m}$ is a basis of $Span\\{b_{1},\dots,b_{m}\\}$ and
$Span\\{b_{1},\dots,b_{m}\\}$ is a primitive $F[[t]]$-submodule of $V[[t]]$.
We first find a formula for the tangent map

$dT_{x_{0}}:T_{x_{0}}=W\otimes_{F[[t]]}V[[t]]\to T_{y_{0}}=S^{2}_{t}(W)$

where $y_{0}=T(x_{0})$. Take a line $x(\epsilon)=\sum_{i}a_{i}\otimes
b_{i}+\epsilon\sum_{j}{u_{j}\otimes v_{j}}$ passing through $x_{0}$ in the
direction $\sum_{j}{u_{j}\otimes v_{j}}$, Then

$T(x(\epsilon))=\sum(b_{i},b_{j})a_{i}\otimes
a_{j}+\epsilon\sum(b_{i},v_{j})(a_{i}\otimes u_{j}+u_{j}\otimes
a_{i})+\epsilon^{2}\sum(v_{i},v_{j})u_{i}\otimes u_{j}.$

So we have

$dT_{x_{0}}(\sum_{j}{u_{j}\otimes v_{j}})=\sum(b_{i},v_{j})(a_{i}\otimes
u_{j}+u_{j}\otimes a_{i}).$ (5.11)

From this formula, we see that $dT_{x_{0}}$ is $F[[t]]$-linear. If ${\rm
Im}\,f_{x_{0}}\not=W$, then ${\rm Im}\,dT_{x_{0}}\subset{\rm
Im}\,f_{x^{0}}\otimes W+{\rm Im}\,f_{x^{0}}\otimes W$, $dT_{x_{0}}$ is not
surjective, i.e. $T$ is not submersive at $x_{0}$. If ${\rm Im}\,f_{x^{0}}=W$,
since $(\,)$ on $V[[t]]$ is non-degenerate, for each $1\leq k\leq m$, we can
find $v\in V[[t]]$ such that $(b_{i},v)=\delta_{i,k}$, then
$dT_{x_{0}}(a_{l}\otimes v)=a_{l}\otimes a_{k}+a_{k}\otimes a_{l}$. So all
$a_{l}\otimes a_{k}+a_{k}\otimes a_{l}$ are in ${\rm Im}\,f_{x_{0}}$. So
$dT_{x_{0}}$ is surjective and $T$ is submersive at $x_{0}$. $\Box$

For an $F[[t]]$-submodule $W$ of $M_{-}$, $T_{W}:W\otimes V\to S_{t}^{2}(W)$
as in Section 3. For $i\in S_{t}^{2}(W)$, We denote by $U(i)$ the variety of
elements in $T_{W}^{-1}(i)$ where $T_{W}$ is submerssive. Of course the set of
$F$-points $U(i)_{F}$ may be empty. By Lemma 5.7, Theorem 5.4 can be
reformulated as

###### Theorem 5.8

The $G(F[[t]])$-orbits in $M_{-}\otimes V$ are in one-to-one correspondence
with the set of pairs $(W,i)$ with $W\in Gr(M_{-},t)$, $i\in S_{t}^{2}(W)$
such that $U(i)_{F}$ is not empty. The correspondence is the following, for
$x\in M_{-}\otimes V$, its orbit corresponds to the pair $(W,i)$, where
$W={\rm Im}\,f_{x}$, and $i=T_{W}(x)$.

##  6 Theta series.

We continue with the snt-module $M$ with decomposition $M=M_{-}\oplus M_{+}$
and $V$ a finite dimensional vector space with a non-degenerate, bilinear
symmetric form $(\,,\,)$ as in Section 3. For each $\phi\in{\cal S}(X_{\bf
A})={\cal S}((M_{-}\otimes V)_{\bf A})$, the theta functional $\theta(\phi)$
is defined by (3.7). Recall $G^{q}({\bf A}[[t]])$ acts on ${\cal S}(X_{\bf
A})$, the action formula is given as follows. An element $g\in G^{q}({\bf
A}[[t]])$ has the block decomposition

$\left[\begin{array}[]{cc}\alpha_{g}&\beta_{g}\\\
\gamma_{g}&\delta_{g}\end{array}\right]$

with respect the decomposition

$(M\otimes V)_{\bf A}=(M_{-}\otimes V)_{\bf A}\oplus(M_{+}\otimes V)_{\bf A}.$

Since $G^{q}({\bf A}[[t]])$ preserves $(M_{\pm}\otimes V)_{\bf A}$, so we have
$\beta_{g}=0$ and $\gamma_{g}=0$ . Then the action of $g$ on ${\cal
S}((M_{-}\otimes V)_{\bf A})$ is given by

$(g\cdot\phi)(x)=\phi(x\alpha_{g})=\phi(xg)$ (6.1)

If $g\in G^{q}(F[[t]])$, it is clear that

$\theta(g\cdot\phi)=\theta(\phi).$

So $\theta(g\cdot\phi)$ is a continuous function on $G^{q}(F[[t]])\backslash
G^{q}({\bf A}[[t]])$. Let $dg$ be the Haar measure on $G^{q}({\bf A}[[t]])$
such that the volume of $G^{q}(F[[t]])\backslash G^{q}({\bf A}[[t]])$ is $1$.

###### Lemma 6.1

If $(V,(\,,\,))$ is anisotropic over $F$ or ${\rm dim}V-r>\frac{1}{2}{\rm
dim}\,M+1$, where $r$ is the dimension of a maximal isotropic subspace of $V$,
then the integral

${\rm It}(\phi)\stackrel{{\scriptstyle\rm
def}}{{=}}\int_{G^{q}(F[[t]])\backslash G^{q}({\bf
A}[[t]])}\theta(g\cdot\phi)dg$ (6.2)

converges

Proof. If $(V,(\,,\,))$ is anisotropic over $F$, then $G^{q}(F[[t]])\backslash
G^{q}({\bf A}[[t]])$ is compact, so (6.2) converges. Let $U$ be the unipotent
radical of $G^{q}$, then $G^{q}=G\ltimes U$. Let $D$ be a compact fundamental
domain of $U(F)\backslash U({\bf A})$. Then

$(\ref{4.5})=\int_{G(F)\backslash G^{q}(A)}\int_{a\in D}\sum_{r\in
M_{-}\otimes V}\phi(rag)dadg=\int_{G(F)\backslash
G^{q}(A)}\theta(g\cdot\bar{\phi})dg,$ (6.3)

where $\bar{\phi}(x)=\int_{D}\phi(xa)da$. Since $D$ is compact,
$\bar{\phi}\in{\cal S}((M_{-}\otimes V)_{\bf A})$. By the convergence
criterion in Proposition 8 [7], the right hand side of (6.3) converges under
the condition ${\rm dim}V-r>\frac{1}{2}{\rm dim}\,M+1$. $\Box$

We can write the integral (6.2) as a sum of orbital integrals. Let ${\cal O}$
be a set of representatives of $G(F[[t]])$-orbit in $M_{-}\otimes V$. For each
$\xi\in{\cal O}$, let $G_{\xi}$ denote its isotropy subgroup in
$G^{q}(F[[t]])$. Then the integral (6.2) can be written as

$\int_{G^{q}(F[[t]])\backslash G^{q}({\bf A}[[t]])}\sum_{\xi\in{\cal
O}}\sum_{\tau\in G_{\xi}\backslash G^{q}(F[[t]])}\phi(\xi\tau g)dg,$ (6.4)

which can be further written as

$\sum_{\xi\in\mathcal{O}}vol(G_{\xi}\backslash G_{\xi,{\bf
A}})\int_{G_{\xi,{\bf A}}\backslash G^{q}({\bf A}[[t]])}\phi(\xi g)dg$ (6.5)

and we have thereby expressed ${\rm It}(\phi)$ as

${\rm It}(\phi)=\sum_{\xi\in\mathcal{O}}vol(G_{\xi}\backslash G_{\xi,{\bf
A}})\int_{G_{\xi,{\bf A}}\backslash G^{q}({\bf A}[[t]])}\phi(\xi g)dg.$ (6.6)

##  7 Siegel-Weil formula

We assume in this section $M$ and $V$ satisfies the conditions that ${\rm
dim}\,V>6n+2$, where $n$ is the number of $H_{k}$’s in the decomposition of
$M$ as in (3.4) and $V$ satisfies the conditions in Lemma 6.1. By Theorem 3.3,
the $t$-Eisenstein series $\phi\in{\cal S}((M_{-}\otimes V)_{\bf
A})\mapsto{\rm Et}(\phi)$ is a tempered distribution on $(M_{-}\otimes V)_{\bf
A}$. And for each $W\in Gr(M_{-},t)$, we have a tempered distribution
$\phi\mapsto{\rm Et}_{W}(\phi)$, given by (3.12). We have

${\rm Et}=\sum_{W\in Gr(M_{-},t)}{\rm Et}_{W}.$

By Theorem 4.7,

${\rm Et}_{W}=\sum_{i\in{\rm St}^{2}(W)}\mu_{i}.$

We denote $\mu_{i}$ by ${\rm Et}_{W,i}$. Therefore we have

${\rm Et}=\sum_{W\in Gr(M_{-},t)}\sum_{i\in{\rm St}^{2}(W)}\mu_{W,i}.$ (7.1)

Moreover the measure ${\rm Et}_{W,i}=\mu_{i}$ is the gauge measure as
described in Theorem 4.7, which implies in particular $\mu_{i}$ is $0$ if
$U(i)_{\bf A}$ in empty.

On the other hand,

$\phi\in{\cal S}((M_{-}\otimes V)_{\bf A})\mapsto{\rm It}(\phi)$

given in (6.2) is a tempered distribution and it has a decomposition given by
(6.6). By Theorem 5.8, each orbit corresponds uniquely to a pair $(W,i)$ where
$W\in Gr(U_{-},t)$ and $i\in{\rm St}^{2}(W)$. So we may write

${\rm It}=\sum_{W,i}{\rm I}_{W,i}$

where

${\rm I}_{W,i}(\phi)=vol(G_{\xi}\backslash G_{\xi,{\bf A}})\int_{G_{\xi,{\bf
A}}\backslash G^{q}({\bf A}[[t]])}\phi(\xi g)dg.$

where $(W,i)$ corresponds to the orbit containing $\xi$, i.e., ${\rm
Im}\,f_{\xi}=W$ and $T_{W}(\xi)=i$. If $U(i)$ is empty, $(W,i)$ doesn’t
corresponds to any orbit, in this case we define

${\rm I}_{W,i}\stackrel{{\scriptstyle\rm def}}{{=}}0.$

We shall prove that ${\rm Et}={\rm It}$, and actually we shall prove more:
${\rm Et}_{W,i}={\rm It}_{W,i}$ for any pair $(W,i)$ . We use the induction on
${\rm dim}\,M$. The case ${\dim}\,M=2$ is the classical result in [7].

Our proof is entirely parallel to that of [7]. To start with, we introduce
some notations. Let $\pi:\widehat{Sp}_{2N}({\bf A})\to Sp_{2N}({\bf A})$
denote the double cover (recall $2N={\rm dim}\,M{\rm dim}\,V$). Let
$\widehat{Sp}(M,t)_{\bf A}$ denote $\pi^{-1}({Sp}(M,t)_{\bf A})$. We let

$M(k)=\oplus_{k_{i}=k}H_{k_{i}},$

so

$M=M(l_{1})\oplus\dots\oplus M(l_{s}),\,\,\,\,\,l_{1}>\dots>l_{s}.$

Recall Corollary 2.6, $Sp(M,t)=N\ltimes H$, where $N$ is the unipotent
radical, and

$H=\Pi_{i=1}^{s}Sp_{2r_{i}}(F),$

where $r_{i}$ is the number $H_{k_{i}}$’s in the decomposition of $M_{l_{i}}$.
Since $M$ has decomposition (3.4), $M_{-}$ has decomposition

$M_{-}=F[t]/(t^{k_{1}})\oplus\dots\oplus F[t]/(t^{k_{n}}).$ (7.2)

Let $T$ be a maximal torus of $H$, we may take $T$ such that $T$ preserves
$M_{\pm}$ and preserves each component in (7.2). Then $T={\rm G}_{\rm m}^{n}$,
where the $i$-th $G_{m}$ acts on the $i$-th component in (7.2). We have
$T_{\bf A}=I_{F}^{n}$, where $I_{F}$ denotes the idele group of $F$. Let
$T_{\bf A}^{\prime}$ be the subset of $T_{\bf A}$ formed by

$T_{\bf A}^{\prime}=\\{(t_{1},\dots,t_{n})\,|\,|t_{1}|_{\bf
A}\geq\dots\geq|t_{r_{1}}|_{\bf A}\geq 1,\dots,|t_{n-r_{s}+1}1|_{\bf
A}\geq\dots\geq|t_{n}|_{\bf A}\geq 1\\}.$

Since $N$ is unipotent, the space $N(F)\backslash N({\bf A})$ is compact. By
the reduction theory for the semi-simple group $H$, there exists a compact
$C\subset Sp(M,t)_{\bf A}$ such that

$Sp(M,t)_{\bf A}=Sp(M,t)_{F}T_{\bf A}^{\prime}C.$ (7.3)

Since $T$ preserves $M_{+}$, we may regard $T_{\bf A}$ as a subgroup of
$\widehat{Sp}(M,t)_{\bf A}$, the decomposition (7.3) implies that

${\widehat{S}p}(M,t)_{\bf A}=Sp(M,t)_{F}T_{\bf A}^{\prime}C.$ (7.4)

for some compact subset $C\subset{\widehat{S}p}(M,t)_{\bf A}$.

As in [7], for each $\tau\in{\mathbb{R}}_{>0}$, we let $a_{\tau}\in I_{F}$
denote the idele such that $(a_{\tau})_{v}=\tau$ for each infinite place $v$
and $(a_{\tau})_{v}=1$ for each finite place. We let $\Theta(T)$ denote the
set of all $(a_{\tau_{1}},\dots,a_{\tau_{n}})$, and set
$\Theta(T)^{\prime}=\Theta(T)\cap T_{\bf A}^{\prime}$.

###### Lemma 7.1

If $\hat{E}$ is a positive tempered measure on $X_{\bf A}=(M_{-}\otimes
V)_{\bf A}$, and is a sum of positive measures $\hat{\mu}_{i}$ supported on
$U(i)_{\bf A}$ ($i\in S_{t}^{2}(M_{-})$ ), and is $T_{F}$-invariant, and there
is a place $v$ of $F$ and a subgroup $G_{v}^{\prime}$ of $G^{q}(F_{v}[[t]])$
that acts transitively on $U(i)_{v}$ such that $\hat{E}$ is invariant under
$G_{v}^{\prime}$. Then the function $S\mapsto\hat{E}(S\phi)$ is bounded on
$T^{\prime}_{\bf A}$, uniformly for $\phi$ in a compact subset in ${\cal
S}(X_{\bf A})$.

This lemma is a generalization of Lemma 23 in [7]. Our proof below closely
follows that of [7].

Proof. Let $e_{1},\dots,e_{n}$ be a quasi-basis of $M_{-}$:

$M_{-}=F[t]/(t^{k_{1}})e_{1}\oplus\dots\oplus F[t]/(t^{k_{n}})e_{n}.$

Then $e_{i}\otimes e_{j}+e_{j}\otimes e_{i}$ is a quasi-basis of ${\rm
St}^{2}(M_{-})$. For each $\alpha\in\\{0,1,\dots,n\\}$, let
$S_{t}^{2}(M_{-})^{(\alpha)}$ be the set that consists of elements

$\sum_{i,j>\alpha}k_{ij}(e_{i}\otimes e_{j}+e_{j}\otimes e_{i})$

such that

$k_{\alpha+1,j}(e_{\alpha+1}\otimes e_{j}+e_{j}\otimes e_{\alpha+1})\neq 0$

for at least one $j\geq{\alpha+1}$. We set

$S_{t}^{2}(M_{-})^{(n)}=\\{0\\}$

by convention. It is clear that $S_{t}^{2}(M_{-})$ is a disjoint union of
$S_{t}^{2}(M_{-})^{(\alpha)}$. Let $\hat{E}_{\alpha}$ be the sum of
$\hat{\mu}_{i}$ for $i\in S_{t}^{2}(M_{-})^{(\alpha)}$. It is clear that

$\hat{E}=\hat{E}_{0}+\hat{E}_{1}+\dots+\hat{E}_{n}$

and $\hat{E}_{\alpha}$ satisfies all the conditions in the lemma. it is enough
to prove the result for each $\hat{E}_{\alpha}$.

Now we fix $0\leq\alpha\leq n$. Let $q$ be the constant which is $1$ if $v$ is
infinite and is equal to the cardinality of the residue field if $v$ is a
finite place. As in [7], there is a compact subset $C\subset I_{F}$ (where
$I_{F}$ is the idele group of $F$), such that every $t\in I_{F}$ with
$1\leq|t|\leq S$ (where $S$ is a fixed constant ) can be written as $rc$ with
$r\in F$, $c\in C$. We denote $C^{n}$ the subset of $T_{\bf A}$ formed by
elements $(c_{1},\dots,c_{n})$ with all $c_{i}\in C$, and let

$\Theta^{\prime}_{\alpha}=\\{(a_{\tau_{1}},\dots,a_{\tau_{n}})\,|\,\tau_{1}=\dots=\tau_{\alpha+1}\geq\dots\geq\tau_{n}\geq
1\\}.$

We will apply Lemma 6 of [7] to the space $X_{F}$. We consider $X_{F}$ as a
product space $\Pi_{i=1}^{n-\alpha}X_{F}^{(i)}$, where $X_{F}^{(1)}$ is the
$F[[t]]$-submodule generated by $e_{1}\otimes V,\dots,e_{\alpha+1}\otimes V$,
and for $n-\alpha\geq i\geq 2$, $X_{F}^{(i)}$ is $F[[t]]e_{i+\alpha}$,
$Y_{F}\subset M_{-}\otimes_{F[[t]]}M_{-}$ is the submodule over $F[[t]]$
spanned by $e_{i}\otimes e_{j}$ with $j\leq\alpha+1$ . And $p:X_{F}\to Y_{F}$
is given by

$p(\sum_{i=1}^{j=1}e_{i}\otimes
v_{i})=\sum_{i=1}^{n}\sum_{j=1}^{\alpha+1}(v_{i},v_{j})e_{i}\otimes e_{j}.$

Apply Lemma 6 of [7], there is $\phi_{0}\in S(X_{\bf A})$ such that

$|((\theta c)\cdot\phi)(x)|\leq\phi_{0}(x)$ (7.5)

for all $x\in X_{\bf A}$ with $T(x)\in S_{t}(M_{-})^{(\alpha)}$, all
$\theta\in\Theta^{\prime}_{\alpha}$, $c\in C^{n}$, and $\phi\in C_{0}$.

It can be proved ([7] page 71) that each element $t=(t_{1},\dots,t_{n})\in
T^{\prime}_{\bf A}$ can be written as

$t=ry\theta c$

where $r\in T_{F}$, $y=(y_{1},\dots,y_{n})\in T_{v}$ with all $|y_{i}|_{v}\geq
1$ and $y_{\alpha+1}=\dots=y_{n}=1$, $\theta\in\Theta^{\prime}_{\alpha}$ and
$c\in C^{n}$. Since $\hat{E}_{\alpha}$ is invariant under $T_{F}$, we have

$|\hat{E}_{\alpha}(t\cdot\phi)|\leq\hat{E}_{\alpha}(y\cdot\phi_{0}).$

We may assume that $\phi_{0}=\phi_{v}\phi^{\prime}$ where $\phi_{v}\in{\cal
S}(X_{v})$, $\phi^{\prime}\in{\cal S}(\Pi^{\prime}_{w\neq v}X_{w})$. By Lemma
22 [7], we have

$\hat{E}_{\alpha}(y\cdot\phi_{0})=\sum_{i\in
St^{2}(M_{-})^{(\alpha)}}c_{i}(\phi^{\prime})\int_{U(i)_{v}}y\cdot\phi_{v}|\theta_{i}|_{v}.$

From this, we obtain that

$\hat{E}_{\alpha}(y\cdot\phi_{0})=\Pi_{i=1}^{n}|y_{i}|_{v}^{k_{1}+\dots+k_{i-1}+(n-m/2-i+2)k_{i}}\hat{E}_{\alpha}(\phi_{0})$

since $k_{i}\leq k_{j}$ for $i\leq j$, and $m>6n+2$, we see that the exponent
of $|y_{i}|_{v}$ is $\leq 0$, and since $|y_{i}|_{v}\geq 1$, so have

$\hat{E}_{\alpha}(y\cdot\phi_{0})\leq\hat{E}_{\alpha}(\phi_{0}).$

This proves the lemma. $\Box$

The following theorem is a generalization of Theorem 4 in [7].

###### Theorem 7.2

Suppose ${\rm dim}\,V>6n+2$. If there is a place $v$ of $F$ such that
$U(0)_{v}$ is not empty, and a subgroup $G_{v}^{\prime}$ of
$G^{q}(F_{v}[[t]])$ acts transitively on $U(i)_{v}$ for every $i\in
S_{t}^{2}(M_{-})$. And if $E^{\prime}$ is a positive tempered measure on
$X_{\bf A}$ invariant under $Sp(M,t)$ and $G_{v}^{\prime}$ and
$E^{\prime}-{\rm Et}$ is supported on the union of $U(i)_{\bf A}$ for $i\in
S_{t}^{2}(M_{-})$ . Then $E^{\prime}={\rm Et}$.

Sketch of Proof.  Using Lemma 7.1 and 7.4, it is easy to see that the function
$\widehat{Sp}(M,t)_{\bf A}\to{\mathbb{C}}$ given by $S\mapsto(E^{\prime}-{\rm
Et})(S\phi)$ is bounded on $\widehat{Sp}(M,t)_{\bf A}$ uniformly for $\phi$ in
every compact subset of ${\cal S}(X_{\bf A})$. The remainder of the proof is
similar to that of Theorem 4 in [7]. $\Box$

Now we can prove the main theorem of this work:

###### Theorem 7.3

Suppose ${\rm dim}\,V>6n+2$ and $V$ is anisotropic or ${\rm
dim}V-r>\frac{1}{2}{\rm dim}\,M+1$, where $r$ is the dimension of a maximal
isotropic subspace of $V$, then

${\rm Et}={\rm It}.$

The condition on $V$ in the Theorem is for the convergence of ${\rm It}$ (see
6.1). The proof uses the induction on ${\rm dim}\,M$. The induction assumption
implies that $E^{\prime}={\rm It}$ satisfies the conditions of Theorem 7.2,
therefore ${\rm Et}={\rm It}$.

##  8 Corollaries of Siegel-Weil formula

In this section we prove a slightly more general form of the Siegel-Weil
formula (Theorem 8.1) for snt-modules that will be used in part II. Let $M,V$
be as in Section 3. Let

$M=M_{-}\oplus M_{+}$ (8.1)

be a direct sum such that $M_{+}\in Gr(M,t)$, $M_{-}\in Gr(M)$ but not
necessarily in $Gr(M,t)$. Let $X=M_{-}\otimes V$. The space $L^{2}(X_{\bf A})$
is a representation of the metaplectic group ${\widehat{S}p}_{2N}({\bf A})$
($2N={\rm dim}M{\rm dim}V$) with the usual theta functional

$\theta:{\cal S}(X_{\bf
A})\to{\mathbb{C}},\,\,\,\phi\mapsto\theta(\phi)=\sum_{r\in X}\phi(r).$

Recall the Eisenstein series (1.3), (1.7) for $\phi\in{\cal S}(X_{\bf A})$ is
given by

${\rm E}(\phi)=\sum_{U\in Gr(M)}E(\phi,U)=\sum_{U\in
Gr(M)}\int_{(\pi_{-}(U)\otimes V)_{\bf A}}\psi(\frac{1}{2}\langle x,\rho
x\rangle)\phi(x)dx.$

The $t$-Eisenstein series defined in (1.11) is a subseries given by

${\rm Et}(\phi)=\sum_{U\in Gr(X,t)}E(\phi,U).$ (8.2)

Though $M_{-}$ is not an $F[[t]]$-submodule of $M$, it has $F[[t]]$-module
structure via the isomorphism $M_{-}=M/M_{+}$. In the case that $M_{-}$ is an
$F[[t]]$-submodule, the two $F[[t]]$-module structures on $U_{-}$ clearly
coincide. The projection map

$\pi_{-}:M\to M/M_{+}=M_{-}$

is an $F[[t]]$-module homomorphism. For each $U\in Gr(M,t)$, $\pi_{-}(U)$ is
an $F[[t]]$-submodule of $M_{-}$. Denote $Gr(M_{-},t)$ the set of
$F[[t]]$-submodules of $M_{-}$, so we have a map

$P:Gr(M,t)\to Gr(M_{-},t):\,\,\,U\mapsto\pi_{-}(U).$ (8.3)

For $W\in Gr(M_{-},t)$, we set

${\rm Et}_{W}(\phi)=\sum_{U\in Gr(M,t):\pi_{-}(U)=W}E(\phi,U).$

We recall the action formula of $G^{q}({\bf A}[[t]])$ on ${\cal
S}((M_{-}\otimes V)_{\bf A})$. An element $g\in G^{q}({\bf A}[[t]])$ has the
block decomposition

$\left[\begin{array}[]{cc}\alpha_{g}&\beta_{g}\\\
\gamma_{g}&\delta_{g}\end{array}\right]$

with respect the decomposition

$X_{\bf A}=(M_{-}\otimes V)_{\bf A}\oplus(M_{+}\otimes V)_{\bf A}.$

Since $G^{q}({\bf A}[[t]])$ preserves $(M_{+}\otimes V)_{\bf A}$, so we have
$\gamma_{g}=0$. Since $M_{-}$ is not an $F[[t]]$-submodule in general,
$\beta_{g}$ may not be $0$. Then the action of $g$ on ${\cal S}((X\otimes
V)_{\bf A})$ is given by

$(g\cdot\phi)(x)=\psi(\frac{1}{2}\langle
x\alpha_{g},x\beta_{g}\rangle)\phi(x\alpha_{g})$ (8.4)

If $g\in G^{q}(F[[t]])$ and $\xi\in V$, then

$\frac{1}{2}\langle\xi\alpha_{g},\xi\beta_{g}\rangle\in F,$

and so $\psi(\frac{1}{2}\langle\xi\alpha_{g},\xi\beta_{g}\rangle)=1$ and

$(g\cdot\phi)(\xi)=\phi(\xi\alpha_{g}).$

Therefore, we have

$\theta(g\cdot\phi)=\theta(\phi),\,\,\,\,\,{\rm for}\,\,g\in G^{q}(F[[t]]).$
(8.5)

The function $\theta(g\cdot\phi)$ (as a function on $G^{q}({\bf A}[[t]])$ is
actually a function on $G^{q}(F[[t]])\backslash G^{q}({\bf A}[[t]])$. Assume
$(V,(\,,\,))$ satisfies the conditions in Lemma 6.1, then we can form the
convergent integral

${\rm It}(\phi)\stackrel{{\scriptstyle\rm
def}}{{=}}\int_{G^{q}(F[[t]])\backslash G^{q}({\bf
A}[[t]])}\theta(g\cdot\phi)dg,$ (8.6)

the convergence can be proved in the same way as Lemma 6.1.

We wish to write the integral (8.6) as a sum of orbital integrals. We
introduce the set

$\Omega=S^{1}\times(M_{-}\otimes V)_{\bf A},$

on which $G^{q}({\bf A}[[t]])$ acts as

$(s,x)\cdot g=(s\psi(\frac{1}{2}\langle
x\alpha_{g},x\beta_{g}\rangle),x\alpha_{g}).$ (8.7)

One can then check directly that (8.7) does define a group action; i.e., that

$((s,x)g_{1})g_{2}=(s,x)g_{1}g_{2},\;\,\,\,\,g_{1},g_{2}\in G^{q}({\bf
A}[[t]]).$

On the other hand, if

$\phi\in{\cal S}((U_{-}\otimes V)_{\bf A}),$

we can extend $\phi$ to $\Omega$ by

$\varphi(s,x)=s\varphi(x),\;\,\,\,\,x\in(U_{-}\otimes V)_{{\bf
A}},\,\,\,\,s\in S^{1}.$

And note that we can rewrite the integral (8.6) as

$\int_{G^{q}(F[[t]])\backslash G^{q}({\bf A}[[t]])}\left(\sum_{\xi\in
U_{-}\otimes V}\phi((1,\xi)\cdot g)\right)dg.$ (8.8)

We note that the subset

$(1,U_{-}\otimes V)\subset\Omega$

is invariant under $G^{q}(F[[t]]).$ We let $\mathcal{O}\subset(1,U_{-}\otimes
V)$ be a family of orbit representatives for the action of $G^{q}(F[[t]])$. Of
course $\mathcal{O}$ can be identified with a subset of $U_{-}\otimes_{F}V$
$((1,\xi)\mapsto\xi,$ $\xi\in U_{-}\otimes V),$ and as such, it is a family of
orbit representatives for the action

$\xi\longmapsto\xi\alpha_{g},\;\,\,\,\,g\in G^{q}(F[[t]])$

of $G^{q}(F[[t]])$ on $U_{-}\otimes V$. We may rewrite (8.8) as

$\int_{G^{q}(F[[t]])\backslash G^{q}({\bf
A}[[t]])}\sum_{\xi\in\mathcal{O}}\sum_{\tau\in G_{\xi}\backslash
G^{q}(F[[t]])}\varphi((1,\xi)\tau g)dg.$ (8.9)

Using the fact that $G_{\xi,{\bf A}}$ is unimodular, we can prove (8.9) is
equal to

${\rm It}(\phi)=\sum_{\xi\in\mathcal{O}}vol(G_{\xi}\backslash G_{\xi,{\bf
A}})\int_{G_{\xi,{\bf A}}\backslash G^{q}({\bf
A}[[t]])}\phi((1,\xi)g_{2})dg_{2},$ (8.10)

by a similar argument as in [7] (page 16). We have thereby expressed the ${\rm
It}(\phi)$ as a sum of orbital integrals. For each orbit ${\cal O}$, we have a
corresponding $W\in Gr(M_{-},t)$ (see Theorem 5.8). Let ${\rm It}_{W}(\phi)$
be the subseries of (8.10) that is over all $\xi$ corresponding to $W$. Then
we have

${\rm It}(\phi)=\sum_{W\in Gr(M_{-},t)}{\rm It}_{W}(\phi).$

###### Theorem 8.1

Suppose ${\rm dim}\,V>6n+2$ and $(V,(\,,\,))$ satisfies the condition in Lemma
6.1. Let $M=M_{-}\oplus M_{+}$ be a decomposition such that $M_{-}\in Gr(M)$
and $M_{+}\in Gr(M,t)$. Put $X=M_{-}\otimes V$. We can define tempered
distributions ${\rm Et}$ and ${\rm It}$ on $X_{\bf A}$ as in (8.2) and (8.6).
Then

${\rm Et}={\rm It}.$ (8.11)

And for each $W\in Gr(M_{-},t)$, we have

${\rm Et}_{W}={\rm It}_{W}.$ (8.12)

Notice that we didn’t assume $M_{-}\in Gr(M,t)$, but this theorem can be
reduced to Theorem 7.3. Let

$M=\bar{M}_{-}\oplus\bar{M}_{+}$

be a decomposition of $t$-Lagrangian subspaces. Both $L^{2}((M_{-}\otimes
V)_{\bf A})$ and $L^{2}((\bar{M}_{-}\otimes V)_{\bf A})$ are models of the
Weil representation of $\widehat{Sp}_{2N}({\bf A})$. We shall define an
intertwining operator

$T:L^{2}((M_{-}\otimes V)_{\bf A})\to L^{2}((\bar{M}_{-}\otimes V)_{\bf A})$

such that $T$ sends ${\cal S}((M_{-}\otimes V)_{\bf A})$ to ${\cal
S}((\bar{M}_{-}\otimes V)_{\bf A})$, and

${\rm Et}(\phi)={\rm Et}(T\phi),\,\,\,\,\,\,\,{\rm It}(\phi)={\rm It}(T\phi).$

By Theorem 7.3, ${\rm Et}(T\phi)={\rm It}(T\phi)$, so we have , ${\rm
Et}(\phi)={\rm It}(\phi)$. The more detailed proof follows.

Proof. Recall for the symplectic space $M\otimes V$, we have the associated
Heisenberg group

$H=(M\otimes V)_{\bf A}\times S^{1}$

with the product given by

$(a_{1},s_{1})(a_{2},s_{2})=(a_{1}+a_{2},s_{1}s_{2}\psi(\frac{1}{2}\langle
a_{1},a_{2}\rangle).$

Since $M_{+}$ is a Lagrangian subspace, $A\stackrel{{\scriptstyle\rm
def}}{{=}}(M_{+}\otimes V)_{\bf A}\times S^{1}$ is a maximal abelian subgroup
of $H$. And

$\chi:(M_{+}\otimes V)_{\bf A}\times S^{1}\to S^{1},\,\,\,\,\,(a,s)\mapsto s$

is a $1$-dimensional representation. The induced representation

$Ind_{A}^{H}\chi$

consists of functions $f$ on $H$ such that $f(ax)=\chi(a)f(x)$ for all $a\in
A$ and the restriction $f$ on $(M_{-}\otimes V)_{\bf A}$ is in
$L^{2}((M_{-}\otimes V)_{\bf A})$. Let $M=\bar{M}_{-}\oplus\bar{M}_{+}$ be a
direct sum of $t$-Lagrangian subspaces. Similarly, for the maximal abelian
subgroup $(\bar{M}_{-}\otimes V)_{\bf A}\times S^{1}$, and the character

$\bar{\chi}:(\bar{M}_{-}\otimes V)_{\bf A}\times S^{1}\to
S^{1},\,\,\,\,\,(\bar{a},s)\mapsto s$

we have the induced representation

$Ind_{\bar{A}}^{H}\bar{\chi}.$

The representation space $Ind_{A}^{H}\chi$ ( $Ind_{\bar{A}}^{H}\bar{\chi}$,
resp.) can be identified with $L^{2}((M_{-}\otimes V)_{\bf A})$ (
$L^{2}((\bar{M}_{-}\otimes V)_{\bf A})$, resp.) by the restricting a function
on $H$ to $(\bar{M}_{-}\otimes V)_{\bf A}$ ($(\bar{M}_{-}\otimes V)_{\bf A}$,
reps.). The space of smooth vectors are ${\cal S}(M_{-}\otimes V)_{\bf A})$
and ${\cal S}((\bar{M}_{-}\otimes V)_{\bf A})$, respectively. Let $h\in Sp(M)$
be an element such that

$M_{-}h=\bar{M}_{-},\,\,\,\,\,\,\,\,M_{+}h=\bar{M}_{+}.$

We define

$T:{\cal S}(M_{-}\otimes V)_{\bf A})\to{\cal S}((\bar{M}_{-}\otimes V)_{\bf
A})$

by

$(Tf)(x)=(h\cdot f)(xh^{-1}).$

It is easy to check that $T$ is an isomorphism of $H$-representations. There
are two actions of $\widehat{{Sp}_{2N}}({\bf A})$ on ${\cal
S}((\bar{M}_{-}\otimes V)_{\bf A})$ (or on $L^{2}({\bar{M}}_{-}\otimes V)_{\bf
A})$): the Weil representation action which we denote by $\pi(g)$, and
$\pi^{\prime}(g)=T\pi(g)T^{-1}$ (where $\pi(g)$ denote the Weil representation
action on ${\cal S}({M}_{-}\otimes V)_{\bf A})$. They both satisfy that, for
every $\alpha\in H:$

$\pi(g^{-1})\pi(\alpha)\pi(g)=\pi(\alpha\cdot g)$
$\pi^{\prime}(g^{-1})\pi(\alpha)\pi^{\prime}(g)=\pi(\alpha\cdot g)$

So $\pi^{\prime}(g)=c_{g}\pi(g)$ for some scalar $c_{g}$. It is clear that
$c_{g_{1}g_{2}}=c_{g_{1}}c_{g_{2}}$. Since $Sp_{2N}(F)$ is a perfect group, we
have $c_{g}=1$ for all $g\in Sp_{2N}(F)$. This implies that $c_{g}=1$ for all
$g\in\widehat{{Sp}_{2N}}({\bf A})$. It then follows that $T$ is an isomorphism
of the Weil representations. We define as usual the theta function functionals

$\theta:{\cal S}((M_{-}\otimes V)_{\bf
A})\to{\mathbb{C}},\,\,\,\,\,\,\,\bar{\theta}:{\cal S}((\bar{M}_{-}\otimes
V)_{\bf A})\to{\mathbb{C}}.$

We have

$\bar{\theta}(Tf)=\sum_{r\in\bar{M}_{-}}(Tf)(r)=\sum_{r\in\bar{M}_{-}}(h\cdot
f)(rh^{-1})=\sum_{r\in M_{-}}(h\cdot f)(r)=\theta(h\cdot f)=\theta(f).$

And

$\displaystyle{\rm It}(Tf)$ $\displaystyle=$
$\displaystyle\int_{G^{q}(F[[t]]\backslash G^{q}({\bf A}[[t]]}\theta(g\cdot
Tf)dg$ $\displaystyle=$ $\displaystyle\int_{G^{q}(F[[t]]\backslash G^{q}({\bf
A}[[t]]}\bar{\theta}(T(g\cdot f))dg$ $\displaystyle=$
$\displaystyle\int_{G^{q}(F[[t]]\backslash G^{q}({\bf A}[[t]]}\theta((g\cdot
f))dg$ $\displaystyle=$ $\displaystyle{\rm It}(f).$

Let $P\subset Sp(M)$ be the parabolic subgroup that consists of all $g\in
Sp(M)$ such that $M_{+}g=M_{+}$, similarly, let $\bar{P}\subset Sp(M)$ be the
parabolic subgroup that consists of all $g\in Sp(M)$ such that
$\bar{M}_{+}g=\bar{M}_{+}$. And let $S$ denote the set of all $g\in
P\backslash Sp(M)$ such that $M_{+}g$ is a $t$-Lagrangian subspace. Similarly
let $\bar{S}$ denote the set of all $g\in\bar{P}\backslash Sp(M)$ such that
$\bar{M}_{+}g$ is a $t$-Lagrangian subspace. Since $M_{+}h=\bar{M}_{+}$, the
map $L_{h}:\bar{S}\to S,g\mapsto hg$ is a bijection.

$\displaystyle{\rm Et}(Tf)$ $\displaystyle=$
$\displaystyle\sum_{g\in\bar{S}}(g\cdot Tf)(0)$ $\displaystyle=$
$\displaystyle\sum_{g\in\bar{S}}(Tgf)(0)$ $\displaystyle=$
$\displaystyle\sum_{g\in\bar{S}}(hgf)(0)$ $\displaystyle=$
$\displaystyle\sum_{g\in S}(gf)(0)$ $\displaystyle=$ $\displaystyle{\rm
Et}(f).$

This proves ${\rm Et}(\phi)={\rm It}(\phi)$. To prove ${\rm Et}_{W}(\phi)={\rm
It}_{W}(\phi)$ for every $W\in Gr(M_{-},t)$, we first note for every $W\in
Gr(M_{-},t)$, $W+M_{+}$ is an $F[[t]]$-submodule of $M$. Let

$W^{\bot}=\\{v\in M_{+}\,|\,\langle v,W\rangle=0\\}.$

It is easy to see that $W^{\bot}$ is the radical of the restriction of
$\langle\,,\,\rangle$ on $W+M_{+}$. Therefore

$M_{W}\stackrel{{\scriptstyle\rm def}}{{=}}(W+M_{+})/W^{\bot}$

is an snt-module. And

$M_{W}=W\oplus M_{+}/W^{\bot}$

is a decomposition into Lagrangian subspaces, and $M_{+}/W^{\bot}\in
Gr(M_{W},t)$. We consider ${\cal S}((W\otimes V)_{\bf A})$, and apply (8.11)
to the pair $(Sp(M_{W},t),G)$, we have

$\sum_{W^{\prime}\in Gr(M_{-},t):W^{\prime}\subset W}{\rm
Et}_{W^{\prime}}(\phi)=\sum_{W^{\prime}\in Gr(M_{-},t):W^{\prime}\subset
W}{\rm It}_{W^{\prime}}(\phi).$

The above equality holds for all $W\in Gr(M_{-},t)$, which implies ${\rm
Et}_{W}(\phi)={\rm It}_{W}(\phi)$. $\Box$

## References

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Dept of Math, Yale University, New Haven, CT 06520-8283.
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Dept of Math, Hong Kong University of Science and Tecgnology<EMAIL_ADDRESS>