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\title {Hecke Operators for the Symplectic Group\\ and Buildings} 

\author {Thomas R. Shemanske}

\date{\today}
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\evenhead{Thomas R. Shemanske}     %Running Heads on Even Pages
\oddhead{Hecke Operators for the Symplectic Group and Buildings}       %Running Heads on Odd Pages

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%\dedication{This is dedicated to the one I love}

\subjclass {Primary 11F60; Secondary 11F46 20E42}

\keywords{Hecke Operators, Siegel modular forms, Symplectic Group, Bruhat--Tits Building}

\begin{abstract}
  We demonstrate that (isomorphic images of) the generators of the
  Hecke algebra for the symplectic group have generating series which
  are equal to highly structured rational functions.  For two of the
  $(n+1)$ generators of the algebra, the denominators of those
  rational functions correspond to the standard and spinor zeta
  functions; in the remaining cases, they represent new zeta
  functions.  We also give two representations of the classical Hecke
  algebra.  The first of these is achieved by defining Hecke operators
  which act on the vertices of the Bruhat-Tits building for
  $Sp_n(\Qp)$, while the second is related to but distinct from
  Satake's isomorphism between the local Hecke algebra and a ring of
  polynomials invariant under a certain Weyl group.
\end{abstract}


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\section{Introduction}\label{sec:introduction}
Hecke theory for automorphic forms on the symplectic group is still
very much in its infancy.  Simplisticly, the major stumbling block
is that unlike the elliptic modular case, there is no obvious connection
between the known invariants of the Hecke algebra (Satake $p$-parameters) and
the Fourier coefficients of a Hecke eigenform, although there has been some
interesting work done:  using a partial knowledge of Satake parameters
to infer complete knowledge (\cite{Veenstra-Satake}), or correlations
between Fourier coefficients and Hecke eigenvalues in degree 2
(\cite{Hafner-Walling}).  Still we are very far away from a
satisfactory general theory.  



In this paper, we have two major goals. The first is to demonstrate
that (isomorphic images of) the generators of the Hecke algebra for
the symplectic group have generating series which are equal to highly
structured rational functions.  For two of the $(n+1)$ generators of
the algebra, the denominators of those rational functions correspond
to the standard and spinor zeta functions; in the remaining cases,
they represent new zeta functions.  The second goal is to give two
representations of the Hecke algebra.  The first of these is achieved
by defining Hecke operators which act on the vertices of the
Bruhat-Tits building for $Sp_n(\Qp)$, while the second is related to
but distinct from Satake's isomorphism between the local Hecke algebra
and a ring of polynomials invariant under a certain Weyl group.  As a
byproduct of the representation on the vertices of the building, we
derive a number of what might be called ``arithmetic'' results about
the building for $Sp_n$.


It is well-known (see e.g., Cartier \cite{Cartier-Corvalis}, Theorem
4.1) that the Satake map shows that the $p$-part of the Hecke algebra
associated to the symplectic group is isomorphic to a polynomial ring
invariant under a certain Weyl group.  In \cite{AZbook}, Andrianov and
Zhuravlev refer to this isomorphism as the spherical map, and give a
rather complicated description of it in terms of right cosets of the
double cosets which generate the Hecke algebra.  In this paper we
define a slightly different isomorphism with several advantages: the
isomorphism is defined naturally in terms of the double cosets which
generate the Hecke algebra (not in terms of right cosets), and derives
its shape from a representation of the local Hecke algebra on the
vertices of the Bruhat-Tits building for $Sp_n(\Qp)$. This
correspondence makes role of the Weyl group completely explicit.  Then
we exploit the isomorphism of the Hecke algebra with the polynomial
ring by using that setting to show the images of the standard
generators of the Hecke algebra have generating series which produce
structured rational functions.  Finally, we show that our
representations are related by establishing the validity of a
commutative diagram.

The key to understanding both representations is to understand the
structure of the Bruhat-Tits building for $Sp_n(\Qp)$, in particular
how the vertices of an apartment are associated with symplectic
elementary divisors relative to a fixed a lattice.  It is of course
here that the Weyl group plays a natural role.  Moreover, it is
through the natural connection between right cosets of a double coset
and (sub)lattices of a given lattice with prescribed elementary
divisors that we begin to develop our representations.  In section 2,
we describe the connection between lattices and right cosets.  In
section 3, we look carefully at the building for $Sp_n$, give
labellings of an apartment in terms of symplectic divisors, and
discuss how special vertices are arithmetically distinguished.  We
then begin to consider polynomials invariant under the Weyl group and
a natural correspondence between vertices and monomials which forms
the basis of our representations.  Finally, we define our ``Hecke''
operators in the polynomial algebra.  In the final section, we define
the two representations and prove that they are indeed ring
homomorphisms, and are related via a commutative diagram.  At the end,
we make a brief comparison of the Satake isomorphism and our map.

\section{Preliminaries}\label{sec:preliminaries}

\subsection{The Classical Hecke Algebra}\label{sec:class-hecke-algebra}
\noindent Much of this material can be found in Chapter 3 of
\cite{AZbook}; we state it here to set the notation.  Let $\G = \Gn =
Sp_n(\Z) \subset SL_{2n}(\Z)$, and let $G = GSp_n^+(\Q)\subset
GL_{2n}(\Q)$ be the group of symplectic similitudes with scalar factor
$r(M) \in \Q_+^\times$:
\begin{align*}
  GSp_n^+(\Q)&= \{ M = \left(\begin{smallmatrix} A&B\\C&D
    \end{smallmatrix}\right) \in M_{2n}(\Q)
  \mid A^tC = C^tA, B^tD = D^tB, A^tD - C^tB = r(M)I_{2n}\}\\
  &= \{ M = \left(\begin{smallmatrix} A&B\\C&D
    \end{smallmatrix}\right) \in M_{2n}(\Q) \mid AB^t = B^tA, CD^t =
  DC^t, AD^t - BC^t = r(M)I_{2n}\}.
\end{align*}

Let $H$ denote the rational Hecke algebra associated to the pair $\G$
and $G$.  That is, as a vector space, $H$ is generated by all double
cosets $\G \xi \G$ ($\xi \in G$), and we turn $H$ into an algebra by
defining the multiplication law as follows: Given $\xi_1, \xi_2 \in
G$, define
\begin{equation}
  \G \xi_1 \G \cdot \G \xi_2 \G = \sum_\xi
  c(\xi) \G \xi \G,
\end{equation}
where the sum is over all double cosets $\G \xi \G \subseteq \G \xi_1
\G \xi_2 \G$, and the $c(\xi)$ are nonnegative integers (see
\cite{Shimurabook}).  There is an alternate characterization of the
Hecke algebra which will be convenient as well.  Let $L(\G,G)$ be the
rational vector space with basis consisting of right cosets $\G \xi$
for $\xi \in G$.  The Hecke algebra can be thought of as
those elements of $L(\G,G)$ which are right invariant under the action
of $\G$.  Thus we can and will think of a double coset $\G \xi \G = \cup \G \xi_\nu$ as
the disjoint union of right cosets and as the sum of those right
cosets $\sum \G \xi_\nu \in L(\G,G)$.





The global Hecke algebra, $H$, is generated by local Hecke algebras,
$H_p$, one for each prime $p$, obtained as above by replacing $G$ by
$G \cap GL_{2n}(\Z[p^{-1}])$ in the above construction. $H_p$ is
generated by double cosets $\G \xi \G$ with $\xi$ of the form
$\diag(p^{a_1}, \dots, p^{a_n}, p^{b_1}, \dots , p^{b_n})$ where $a_1
\le \cdots \le a_n \le b_n \le \cdots \le b_1$ are integers with
$p^{a_i + b_i} = r(\xi)$ for all $i$. It is occasionally useful to
consider the ``integral'' Hecke algebra $\underline H_p$ generated by
all $\xi$ as above with $\xi = \diag(p^{a_1}, \dots, p^{a_n}, p^{b_1},
\dots , p^{b_n})\in M_{2n}(\Z)$.


In this paper, we want to work with a $p$-adic version of the local
Hecke algebra.  This local Hecke algebra is isomorphic to a subalgebra
of the Hecke algebra associated to the Hecke pair $\Gp = Sp_n(\Zp)$
and $G_p = GSp_n(\Qp)$.  To be explicit we set $\H_p$ to be the Hecke
algebra generated as a $\Q$-vector space by double cosets $\Gp \xi
\Gp$ with $\xi = \diag(p^{a_1}, \dots, p^{a_n}, p^{b_1}, \dots ,
p^{b_n})$ as above, and let $\underline \H_p$ denote the integral
subring as above.

The key to observing the above isomorphism is the following
observation:

\begin{lemma}\label{lem:local-global}
  Let $\xi \in GSp_n^+(\Q) \cap GL_{2n}(\Z[p^{-1}])$.  If $\G \xi \G$
  is the disjoint union $\cup \G\alpha_i$, then $\Gp \xi \Gp$ is the
  disjoint union $\cup \Gp\alpha_i$.
\end{lemma}


\begin{proof}
  Without loss of generality, we may assume $\xi \in M_{2n}(\Z)$ since
  the general case follows by multiplying by a power of $p$.  First,
  it is clear that $\cup \Gp\alpha_i \subseteq \Gp \xi \Gp$, and that
  the $\alpha_i$ have integer entries.  Next, we see that the cosets
  $\Gp\alpha_i$ are disjoint since if not then $\alpha_i \alpha_j^{-1}
  \in \Gp \cap GL_{2n}(\Z[p^{-1}]) \subseteq \G$.  To show that the
  union is all of $\Gp \xi \Gp$, we need only show that $\xi\Gp
  \subseteq \Gp\xi\G$, for then any element $\tilde\gamma_1 \xi
  \tilde\gamma_2 = \tilde\gamma_3 \xi \gamma$ ($\tilde\gamma_i \in
  \Gp$, $\gamma \in \G$), so $\xi\gamma \in \G\alpha_i$ for some $i$,
  hence $\tilde\gamma_1 \xi \tilde\gamma_2 \in \Gp \alpha_i$.  To see
  $\xi\Gp \subseteq \Gp\xi\G$ is really just a density argument: Let
  $q = r(\xi)$ be the similitude factor associated to $\xi$.  Recall
  (see \cite{AZbook}) that there is a natural surjective homomorphism
  $Sp_n(R) \to Sp_n(R/qR)$ with $R= \Z$ or $\Zp$.  Denote by $\G_p(q)$
  the kernel of the homomorphism $Sp_n(\Zp) \to Sp_n(\Zp/q\Zp)$.  From
  Chapter 2, \S 3.3 of \cite{AZbook}, we have that $\G_p(q) \subset
  \G_p \cap \xi^{-1}\G_p\xi$.  Using the fact that $Sp_n(\Zp/q\Zp)
  \cong Sp_n(\Z/q\Z)$ (and that $Sp_n(\Z) \to Sp_n(\Z/q\Z)$ is
  surjective), we may write $\G_p = \cup \G_p(q) \delta_i$ with the
  $\delta_i \in \G$.  Let $\tilde\gamma \in \G_p$, and write
  $\tilde\gamma = \tilde\gamma_0 \delta_k$ for some $\tilde\gamma_0\in
  \G_p(q)$ and some $k$. Then
\[ \xi\tilde\gamma = \xi\tilde\gamma_0\delta_k =
  \xi\tilde\gamma_0\delta_k (\delta_k^{-1}\xi^{-1})\xi\delta_k=
(\xi\tilde\gamma_0\xi^{-1})\xi\delta_k \in \G_p\xi\G
\]
since $\xi\tilde\gamma_0\xi^{-1} \in \xi\G_p(q)\xi^{-1} \subset \xi(\G_p
\cap \xi^{-1}\G_p\xi) \xi^{-1} \subset \G_p$.

\end{proof}


The integral Hecke algebra $\underline\H_p$ is generated by the
$(n+1)$ Hecke operators
\begin{equation*}
  T(p) = \Gp
  \begin{pmatrix}
    I_n&0\\0&pI_n
  \end{pmatrix}\Gp 
\end{equation*}
and for $k = 1,\dots,n$,
\begin{equation*}
  T_k^n(p^2) = T_k(p^2) =
\Gp
\begin{pmatrix}
  I_{n-k}&0&0&0\\0&pI_k&0&0\\0&0&p^2I_{n-k}&0\\0&0&0&pI_k
\end{pmatrix}
\Gp,
\end{equation*}
while the Hecke algebra $\H_p$ is generated by the $(n+1)$ elements
above together with the element $T_n(p^2)^{-1} = (pI_{2n})^{-1}$.

The Satake isomorphism shows that the local Hecke algebra is
isomorphic to a polynomial ring invariant under a Weyl group:
  \begin{align*}
    \underline\H_p &\cong \Q[x_0, \dots, x_n]^{W_n}\\
    \H_p &\cong \Q[x_0^{\pm1}, \dots, x_n^{\pm1}]^{W_n} \cong
    \Q[x_0,\dots, x_n]^{W_n}[(x_0^2x_1\cdots x_n)^{-1}]
  \end{align*}
  where $W_n$ is the group of $\Q$-automorphisms of the rational
  function field $\Q(x_0, \ldots, x_n)$ generated by all permutations
  of the variables $x_1, \ldots, x_n$ and by the automorphisms
  $\tau_1, \ldots, \tau_n$ which are given by:
\begin{equation*}
  \tau_i(x_0) = x_0x_i, \qquad \tau_i(x_i) = x_i^{-1}, \qquad
  \tau_i(x_j) = x_j \quad (0 < j \ne i).
\end{equation*}

\noindent $W_n$ is a signed permutation group, in particular, $W_n = \langle
\tau_i\rangle \rtimes S_n \cong (\Z/2\Z)^n \rtimes S_n \cong C_n$
where $C_n$ is Coxeter group associated to the spherical building for
$Sp_n(\Q_p)$.

\subsection{Symplectic lattices and elementary divisors}
\label{sec:sympl-latt-elem}

Let $E$ be the field $\Q$ or $\Qp$, $\O$ its ring of integers, and
$(V, \la*,*\ra)$ a $2n$-dimensional symplectic space over $E$.  Fix a
symplectic basis $\B = \{u_1, \ldots, u_n, v_1,\ldots,v_n\}$ of $V$
satisfying $\langle u_i ,v_j \rangle = \delta_{ij}$ (Kronecker delta),
$\langle u_i,u_j \rangle = \langle v_i ,v_j \rangle=~0$.  Let $\Gamma
= Sp_n(\O)$, and let $G = GSp_n(E)$ be the group of symplectic
similitudes with scalar factor $r(M) \in E^\times$.
%\begin{align*}
% GSp_n(E)&= 
% \{ M = \left(\begin{smallmatrix} A&B\\C&D \end{smallmatrix}\right)
% \in M_{2n}(E)
% \mid A^tC = C^tA, B^tD = D^tB, A^tD - C^tB = r(M)I_n\}\\
% &= \{ M = \left(\begin{smallmatrix} A&B\\C&D \end{smallmatrix}\right)
% \in M_{2n}(E)
% \mid AB^t = B^tA, CD^t = DC^t, AD^t - BC^t = r(M)I_n\}\\
% \end{align*}
Note that $\Gamma$ is a normal subgroup of $GSp_n(E)$, being the
kernel of the determinant map to $E^\times$.


\dr{
\begin{nndefn}
  A linear operator $T : V \to V$ is called a similitude if there
  exists a scalar $r \in E^\times$ such that $\langle T(u),T(v)\rangle
  = r \cdot\langle u,v\rangle$ for all $u,v \in V$.  The map is
  symplectic when $r=1$.
\end{nndefn}

\begin{nnprop}
  Let $T : V \to V$ be a similitude on $V$.  The the matrix of $T$
  with respect to $\mathcal B$ is an element of $GSp_n^+(\Q)$.
\end{nnprop}

\begin{proof}
  We know that $\langle T(u_i),T(v_j)\rangle = r \cdot\langle
  u_i,v_j\rangle = r\delta_{ij}$. Put
  \begin{align*}
    T(u_i) &= \sum_{k=1}^n a_{ki}u_k + \sum_{k=1}^n c_{ki} v_k\\
    T(v_j) &= \sum_{l = 1}^n b_{lj} u_l + \sum_{l=1}^nd_{lj}v_l
  \end{align*}
  
  Using that $\langle * ,* \rangle$, is bilinear and alternating and
  the basis symplectic, we obtain
  \begin{align*}
    \langle T(u_i),T(v_j)\rangle &= \sum_{k,l=1}^n a_{ki}d_{lj}\langle
    u_k, v_l\rangle
    + \sum_{k,l=1}^n c_{ki}b_{lj}\langle v_k, u_l\rangle\\
    &= \sum_{k=1}^n (a_{ki}d_{kj} - c_{ki}b_{kj}) = \sum_{k=1}^n
    (a^t_{ik}d_{kj} - c^t_{ik}b_{kj}).
  \end{align*}
  In other words, $A^t D - C^t B = rI_n$.  The other two conditions
  defining an element of $G$ result from expanding $\langle
  T(u_i),T(u_j)\rangle = \langle T(v_i),T(v_j)\rangle = 0$.
\end{proof}

}

\begin{notation}
  We shall have need of the notion of symplectic divisors (elementary
  divisors with respect to the symplectic group).  To that end, let
  $S$ denote a set of representatives of $E^\times/\O^\times$.  For $E
  = \Q$, we let $S = \Q_+^\times$, the positive rationals, while for
  $E = \Qp$, we let $S = \{p^\nu \mid \nu \in \Z\}$.  We will denote
  by $GSp_n^S(E) = \{ M \in GSp_n(E) \mid r(M) \in S\}$, so when $E =
  \Q$, $GSp_n^S(E)$ is the classical $GSp_n^+(\Q)$.  It is worth
  noting that $Sp_n(E) \subset GSp_n^S(E)$.
\end{notation}



With obvious modification to the proof, the following is Lemma 3.6 of
\cite{AZbook}.

\begin{lemma}\label{lem:symplectic-divisors}
  Let $\xi \in GSp_n^S(E)$, then every double coset $\G \xi \G$ has a
  unique representative of the form $sd(\xi) = \diag(d_1,
  \dots,d_n,e_1,\dots,e_n)$ where $d_i, e_i \in S$ and $d_i \mid
  d_{i+1}$, $d_n \mid e_n$, $e_{i+1} \mid e_i$, and $d_ie_i = r(\xi)$
  
\end{lemma}

We call a lattice symplectic if it has an $\O$-basis which is a
symplectic basis for $V$ with respect to the alternating bilinear form
on $V$.

\begin{prop}\label{prop:characterize-Gamma}
  Let $\L$ be a symplectic lattice.  Then $\Gamma = \{A \in GSp_n^S(E)
  \mid \L A = \L\,\}$, where the action of $A$ on $\L$ is to be
  considered as the matrix of a linear transformation with
  respect to a fixed basis of $\L$.
\end{prop}

\begin{proof}
  Let $\widetilde\Gamma = \{A \in GSp_n^S(E) \mid \L A = \L\,\}.$ It
  is clear that $\Gamma \subset \widetilde\Gamma$, so we need only
  establish the other inclusion.  Let $A \in \widetilde\Gamma$.  Since
  $\widetilde\Gamma$ is a group, and $\Gamma \subset
  \widetilde\Gamma$, any element of the double coset $\Gamma A \Gamma$
  is also an element of $\widetilde\Gamma$.  By
  Lemma~\ref{lem:symplectic-divisors}, we have that $sd(A) = \diag(d_1,
  \ldots, d_n; e_1, \ldots, e_n) \in \widetilde\Gamma$, where the
  $d_i, e_i \in S$ satisfy $e_1\O \subset \cdots \subset e_n\O \subset
  d_n\O \subset \cdots \subset d_1\O$ and $e_i d_i = r(A)$.  It is
  trivial to see that the elementary divisors of $\L sd(A)$ in $\L$
  are $\{\L: \L sd(A)\} = \{d_1, \ldots, d_n, e_1, \ldots e_n\}$.  On
  the other hand, $\L A = \L$, so $\L sd(A) = \L$ and we must have
  $d_i = e_i = 1$ for all $i$.  Thus $\Gamma A \Gamma = \Gamma$, so $A
  \in \Gamma$.
\end{proof}


Fix a symplectic lattice $\L$ and put ${\mathcal R} = {\mathcal R}_\L
= \{ \L A \mid A \in GSp_n^S(E)\}.$



\begin{lemma}
  Let $\M$ and $\N$ be lattices in $\mathcal R$.  Then there exists a
  symplectic basis $\{u_1, \ldots, u_n, v_1,\ldots,v_n\}$ of $V$, and
  elements $d_i, e_i \in S$ satisfying $e_1\O \subset \cdots \subset
  e_n\O \subset d_n\O \subset \cdots \subset d_1\O$ and $e_i d_i = r
  \in S$ such that $\M = \bigoplus\limits_{i=1}^n\O u_i \oplus
  \bigoplus\limits_{i=1}^n \O v_i$ and $\N=\bigoplus\limits_{i=1}^n\O
  d_i u_i \oplus \bigoplus\limits_{i=1}^n \O e_i v_i$.
\end{lemma}

\begin{remark}
  \begin{enumerate}
    
  \item Note that in the general linear case, $GSp_n^S$ would be
    replaced by $GL_{2n}^+$, and $\mathcal R$ would be the set of all
    lattices of full rank in $V$.
  
  \item The ideals $d_i\O$ and $e_j\O$ are called the symplectic
    divisors of $\N$ in $\M$, and coincide with the standard
    elementary divisors $\{\M : \N\}$ since $\Gamma \subset
    SL_{2n}(\O)$.  That is, if we choose two lattices from $\mathcal
    R$ and consider their elementary divisors in the traditional
    sense, they are in fact symplectic elementary divisors with the
    additional properties stated above.  In particular, if $\M$ and
    $\N$ are as in the lemma, we will write $\{\M : \N\} = \{d_1,
    \dots, d_n,e_1, \dots,e_n\}$ to mean there exist bases of $\M$ and
    $\N$ as in the lemma.

  \end{enumerate}
\end{remark}



\begin{proof}
  Since $\M$ and $\N$ are in $\mathcal R$, there exists an $A \in
  GSp_n^S(E)$ with $\N = \M A$.  Assume that $\Gamma$ is identified
  with the stabilizer of $\M$.  By
  Lemma~\ref{lem:symplectic-divisors}, $sd(A) = \diag(d_1, \ldots,
  d_n,e_1,\ldots, e_n) = \gamma_1 A \gamma_2$ for some $\gamma_i \in
  \Gamma$, where $sd(A)$ is the ``symplectic divisor'' matrix of $A$.
  Finally, it is clear that since $\M\gamma_i = \M$, that $\{\M:\N\} =
  \{\M\gamma_1:\M\gamma_1 A\} = \{\M \gamma_1\gamma_2: \M\gamma_1
  A\gamma_2\} = \{\M : \M sd(A)\} = \{d_1, \ldots, d_n,e_1,\ldots,
  e_n\}$, from which the lemma follows.
\end{proof}




\begin{lemma}\label{lemma:equalcosets}
  For $A$ and $B$ in $GSP_n^S(E)$, $\Gamma A = \Gamma B$ if and only
  if $\L A = \L B$.
\end{lemma}


\begin{proof}
  $\Gamma A = \Gamma B$ if and only if $AB^{-1} \in \Gamma$, which by
  Proposition~\ref{prop:characterize-Gamma} is true if and only if $\L
  = \L A B^{-1}$.
\end{proof}

\begin{lemma}\label{lem:equal-elem-divisors}
  Let $\M$ and $\N$ be lattices in $\mathcal R$. The elementary
  divisors of $\M$ and $\N$ in $\L$ satisfy $\{\L : \M \} = \{\L : \N
  \}$ if and only if there exists an $A \in \Gamma$ such that $\M A =
  \N$.
\end{lemma}


\begin{proof}The result is clear if there exists an $A \in \Gamma$
  such that $\M A = \N$.  To prove the converse, we note that by
  definition of the symplectic elementary divisors, there exist
  elements $d_i$, $e_i \in S$ satisfying $e_1\O \subset \cdots \subset
  e_n\O \subset d_n\O \subset \cdots \subset d_1\O$ and $e_i d_i = r
  \in S$ and symplectic $\O$-bases
  \begin{equation*}
    \{u^{(j)}_1, \ldots, u^{(j)}_n; v^{(j)}_1, \ldots, v^{(j)}_n\}\qquad (j=1,2)
  \end{equation*}
  of $\L$ such that
  \begin{eqnarray*}
    \L &= \bigoplus\limits_{i=1}^n\O u_i^{(1)}
            \oplus \bigoplus\limits_{i=1}^n \O v_i^{(1)}\qquad
    \M = \bigoplus\limits_{i=1}^n\O d_i u_i^{(1)} 
            \oplus \bigoplus\limits_{i=1}^n \O e_i v_i^{(1)}\\
    \L &= \bigoplus\limits_{i=1}^n\O u_i^{(2)} 
            \oplus \bigoplus\limits_{i=1}^n \O v_i^{(2)}\qquad
    \N = \bigoplus\limits_{i=1}^n\O d_i u_i^{(2)} 
            \oplus \bigoplus\limits_{i=1}^n \O e_i v_i^{(2)}
  \end{eqnarray*}
  Let $A$ be the matrix of the linear transformation (with respect to either
  basis) taking $u_i^{(1)} \mapsto u_i^{(2)}$, and $v_i^{(1)} \mapsto
  v_i^{(2)}$.  Clearly $A \in Sp_n(E) \subset GSp_n^S(E)$ as it maps
  one symplectic basis to another.  Since $\L A = \L$, $A \in \Gamma$
  by Proposition~\ref{prop:characterize-Gamma} above. Since $A$
  obviously maps $\M$ to $\N$, the proof is complete.
\end{proof}


\begin{prop}\label{prop:one-to-one-corresp}
  Let $A \in GSp_n^S(E)$, and
\[\Gamma A \Gamma = \Gamma sd(A)
\Gamma = \Gamma \diag(d_1, \ldots, d_n,e_1,\ldots, e_n)\Gamma.
\]
Then $\Gamma \xi \mapsto \L\xi$ gives a one-to-one correspondence
between the cosets $\Gamma \xi$ in $\Gamma A \Gamma$ and lattices $M
\in \mathcal R$ with $\{\L:\M\} = \{d_1, \ldots, d_n, e_1, \ldots
e_n\}$.
\end{prop}


\begin{proof}
  We may assume that $A = \diag(d_1, \ldots, d_n,e_1,\ldots, e_n)$.
  If $\Gamma \xi = \Gamma A \delta$ with $\delta \in \Gamma$, then $\L
  \xi \in \mathcal R$ and we have $\{\L: \L\xi\} = \{\L:\L A\delta\} =
  \{\L:\L A\} = \{d_1, \ldots, d_n, e_1, \ldots e_n\}$.  Conversely,
  if $\M \in \mathcal R$ and $\{\L:\M\} = \{d_1,\ldots, d_n, e_1,
  \ldots e_n\}$, then by Lemma~\ref{lem:equal-elem-divisors} there
  exists an element $B\in \Gamma$ such that $\M = \L A B$. Clearly
  $\Gamma A B \subset \Gamma A \Gamma.$ The correspondence is
  one-to-one since by Lemma~\ref{lemma:equalcosets}, $\Gamma \xi =
  \Gamma \zeta$ if and only if $\L \xi = \L \zeta$.
\end{proof}



\section{The Bruhat-Tits Building for $Sp_n(\Qp)$}
\label{sec:bruhat-tits-building}

\subsection{Lattices}

The Bruhat-Tits building for $Sp_n(\Qp)$ is an $n$-dimensional
simplicial complex whose vertices are homothety classes of lattices in
a fixed symplectic vector space $V$.  One defines an incidence
relation on the vertices, and the resulting flag complex is the
building.  Generally our interest will be in an apartment in the
building for which we need a careful understanding of how the vertices
are indexed by classes of lattices, and later how the special vertices
are labeled by monomials in $(n+1)$ variables.  Some of the basic
material can be found in Chapter 20 of \cite{Ronan}; we supplement
where germane.  Most of this can be done over any field with a discrete
valuation, so we temporarily adopt this more general setting.

Let $K$ be a field with a discrete valuation, $\O$ the valuation ring,
$\pi$ a uniformizing parameter, and $k = \O/\pi O$ the residue field.
Let $(V, \la *,*\ra)$ be a symplectic (non-degenerate alternating)
space of dimension $2n$.

\begin{definition}
  An $\O$-lattice $\Lambda$ is primitive if $\la \Lambda, \Lambda \ra
  \subseteq \O$ and $\la *,*\ra$ induces a non-degenerate form on the
  alternating space $\Lambda/\pi\Lambda$ over $k$.
\end{definition}

\dr{
  \begin{example}
    Let $\{e_1, f_1, \dots, e_n, f_n\}$ be $n$ hyperbolic pairs ($\la
    e_i, f_i\ra = 1$, $\la e_i, e_i \ra = \la f_i,f_i\ra = 0$), so
    that $V = \perp (Ke_i + Kf_i)$.  Then $\Lambda = \oplus \O e_i
    \oplus \O f_i$ is a primitive lattice.
\end{example}
}


We describe an apartment system for the building as follows.  A
\textit{frame} is an unordered $n$-tuple $\{\lambda_1^1,
\lambda_1^2\}, \dots, \{\lambda_n^1, \lambda_n^2\}$ of pairs of lines
$\{\lambda_i^1, \lambda_i^2\}$ so that $V = \sum_1^n (\lambda_i^1
+\lambda_i^2)$, $(\lambda_i^1 + \lambda_i^2)$ is orthogonal to
$(\lambda_j^1 + \lambda_j^2)$ for $i\ne j$, and each $(\lambda_i^1 +
\lambda_i^2)$ is a hyperbolic plane.  We say that the frame determines
the apartment $\Sigma$.  Vertices in $\Sigma$ are homothety classes of
lattices, denoted~$[\Lambda]$.  A vertex $[\Lambda]$ lies in $\Sigma$
(determined by the above frame), if there are free $\O$-modules $M_i^j
\subset \lambda_i^j$ so that $\Lambda = \oplus_{i,j} M_i^j$ for some
(and hence every) representative $\Lambda$ of the homothety class.
More concretely, vertices of the building are homothety classes of
lattices $[\Lambda]$ which possess a representative $\Lambda$ such
that
\begin{align*}
  &\textrm{There exists a lattice }\Lambda_0\textrm{ with
  }\pi^{-1}\Lambda_0\textrm{ primitive,}\hspace{3in}\\
  &\Lambda_0 \subseteq \Lambda \subseteq \pi^{-1}\Lambda_0, \textrm{
    and}\\
  &\la\Lambda, \Lambda\ra \subseteq \pi\O
\end{align*}
or equivalently, $\Lambda/\Lambda_0$ is a totally isotropic
$k$-subspace of the non-degenerate alternating space
$\pi^{-1}\Lambda_0/\Lambda_0$.

The maximal simplicies (chambers) are unordered $(n+1)$-tuples
$[\Lambda_0], [\Lambda_1] \dots, [\Lambda_n]$ of homothety classes of
lattices with representatives $\Lambda_i$ satisfying:
\begin{align*}
  &\pi^{-1}\Lambda_0 \textrm{ is primitive, }\\
  &\Lambda_0 \subseteq \Lambda_i \subseteq \pi^{-1}\Lambda_0, \textrm{
    and}\\
  &\Lambda_1/\Lambda_0 \subset \Lambda_2/\Lambda_0 \subset \cdots
  \subset \Lambda_n/\Lambda_0 \textrm{ is a maximal isotropic flags of
  }k\textrm{-subspaces in }\pi^{-1}\Lambda_0/\Lambda_0.
\end{align*}



With respect to a fixed symplectic basis $\{e_1, f_1, \dots, e_n,
f_n\}$ ($\la e_i, f_i\ra = 1$, $\la e_i, e_i \ra = \la f_i,f_i\ra =
0$), let $\Lambda$ be the $\O$-lattice $\Lambda = \O \pi^{a_1} e_1
\oplus \cdots \O \pi^{a_n} e_n \oplus \O \pi^{b_1} f_1 \oplus \cdots
\oplus \O\pi^{b_n} f_n$.  With the basis fixed, we often denote this
lattice as $(\pi^{a_1},\dots \pi^{a_n}; \pi^{b_1},\dots, \pi^{b_n})$.

We note that $\la \Lambda, \Lambda \ra \subseteq \O$ iff
$\la\pi^{a_i}e_i,\pi^{b_i}f_i\ra = \pi^{a_i+b_i} \in \O$ which is true
iff $a_i + b_i \ge 0$.  Given $a_i + b_i \ge 0$, the induced
alternating form on $\Lambda/\pi\Lambda$ is non-degenerate over $k =
\O/\pi\O$ iff $a_i + b_i = 0$ for all $i$.
% for if $a_i + b_i = c > 0$ then the vector $v =
%\pi^{a_i}e_i$ is orthogonal to all vectors in $\Lambda/\pi\Lambda$.





\begin{example}
  Let $\{e_1, \dots, e_n, f_1, \dots, f_n\}$ be a symplectic basis for
  $V$ (with $\la e_i,f_i \ra = 1$), and put $\lambda_i^1 = Ke_i$ and
  $\lambda_i^2 = Kf_i$.  The frame $ \{\lambda_i^1, \lambda_i^2\}$
  defines an apartment $\Sigma$. Let $\Lambda_0 = \pi(\oplus \O e_i
  \oplus \O f_i)$.  Then $\pi^{-1}\Lambda_0$ is primitive.  Denote by
  $[\pi^{a_1}, \dots, \pi^{a_n};\pi^{b_1}, \dots, \pi^{b_n}]$ the
  class of the lattice $\O\pi^{a_1}e_1 \oplus \cdots \oplus
  \O\pi^{a_n}e_n\oplus \O\pi^{b_1}f_1 \oplus \cdots \oplus
  \O\pi^{b_n}f_n$.  Then the following flags determine (fundamental)
  chambers in $\Sigma$.
  
  $[\Lambda_0] = [\pi, \dots, \pi; \pi, \dots, \pi] \subset
  [\Lambda_1] = [1, \pi, \dots, \pi; \pi, \dots, \pi] \subset
  [\Lambda_2] = [1,1, \pi, \dots, \pi; \pi, \dots, \pi] \subset \dots
  \subset [\Lambda_n] = [1,1, \dots, 1; \pi, \dots, \pi]$.
  
  $[\Lambda_0] = [\pi, \dots, \pi; \pi, \dots, \pi] \subset
  [\Lambda_1] = [\pi, \dots, \pi; 1, \pi, \dots, \pi] \subset
  [\Lambda_2] = [\pi, \dots, \pi; 1,1,\pi \dots, \pi] \subset \dots
  \subset [\Lambda_n] = [\pi, \dots, \pi;1,1, \dots, 1]$.
\end{example}



Now to define the building, we start with the set of vertices $S$
which is the set of homothety classes of lattices defined above.  We
define an incidence relation of $S$ as follows:

Let $s, s' \in S$. We say $s \sim s'$ if there are lattices $\Lambda_s
\in s$ and $\Lambda_{s'}\in s'$ and a lattice $\Lambda_0$ such that
$\pi^{-1}\Lambda_0$ is primitive, $\Lambda_0 \subseteq \Lambda_s
\subseteq \pi^{-1}\Lambda_0$, $\pi\Lambda_0 \subseteq \Lambda_{s'}
\subseteq \pi^{-1}\Lambda_0$, and either $\Lambda_s \subset
\Lambda_{s'}$ or $\Lambda_{s'} \subset \Lambda_s$. The associated flag
complex yields the building.


\begin{remark}
  This definition is somewhat subtle.  For example, consider the case
  $n=2$.  Then it would appear if $\Lambda_0 = \pi (\O e_1 \oplus \O
  e_2 \oplus \O f_1 \oplus \O f_2)$ that any lattices $\Lambda_s$ and
  $\Lambda_{s'}$ satisfying $\pi \Lambda_0 \subset \Lambda_{s'}
  \subset \Lambda_s \subset \pi^{-1}\Lambda_0$ would give rise to
  incident vertices $s$ and $s'$.  One must be careful to remember the
  definition of a vertex, in this case that there must exist an
  integer $\nu$ and a lattice $\Lambda_1$ with $\pi^{-1}\Lambda_1$
  primitive and $\Lambda_1 \subset \pi^\nu\Lambda_{s'} \subset
  \pi^{-1}\Lambda_1$ which eliminates some of the possibilities.
\end{remark}

\begin{example}
  Consider the case $n=2$, with $\Lambda_0 = \pi (\O e_1 \oplus \O e_2
  \oplus \O f_1 \oplus \O f_2)$, and let $[\Lambda_0]$ denote any
  special vertex in the apartment $\Sigma$ (a vertex with the most
  hyperplanes through it)
  
  We restrict our attention to the apartment $\Sigma$ and hence to the
  frame determined by the lines spanned by the $e_i$ and $f_i$.  As
  above, any vertex in $\Sigma$ is the class of a lattice of the form
  $\O\pi^{a_1}e_1 \oplus \cdots \oplus \O\pi^{a_n}e_n\oplus
  \O\pi^{b_1}f_1 \oplus \cdots \oplus \O\pi^{b_n}f_n$, denoted
  $[\pi^{a_1}, \dots, \pi^{a_n};\pi^{b_1}, \dots, \pi^{b_n}]$.
  
  For $n=2$, we need to provide pairs of lattices $\Lambda_1$ and
  $\Lambda_2$ with $\Lambda_1/\Lambda_0 \subset \Lambda_2/\Lambda_0$ a
  maximal isotropic flag in $\pi^{-1}\Lambda_0/\Lambda_0$, so we only
  list the pairs of lattices $\Lambda_1$ and $\Lambda_2$.
  
  Since $Sp_n$ is of type $C_n$, the Weyl group is isomorphic to
  $(\Z/2\Z)^n \rtimes S_n$ (the signed permutation group) and has
  order $2^n n!$, so for $n=2$ we expect 8 chambers containing the
  given vertex = $[\Lambda_0] = [1,1,1,1] = [\pi,\pi,\pi,\pi]$.  The
  other pairs of vertices defining the chambers are:

\begin{enumerate}
\item $[1,\pi,\pi,\pi] \subset [1,1,\pi,\pi]$
\item $[\pi,1,\pi,\pi] \subset [1,1,\pi,\pi]$
\item $[\pi,\pi,1,\pi] \subset [\pi,\pi,1,1]$
\item $[\pi,\pi,\pi,1] \subset [\pi,\pi,1,1]$
\item $[1,\pi,\pi,\pi] \subset [1,\pi,\pi,1]$
\item $[\pi,\pi,\pi,1] \subset [1,\pi,\pi,1]$
\item $[\pi,1,\pi,\pi] \subset [\pi,1,1,\pi]$
\item $[\pi,\pi,1,\pi] \subset [\pi,1,1,\pi]$

\end{enumerate}
\end{example}



To see how the rest of the apartment is laid out, one must understand
the action of the reflections which generate the Weyl group on the
lattices.  In this context (in contrast to considering the residue of
a vertex), we must contend with the affine Weyl group associated to
the Bruhat-Tits building for $Sp_n(K)$.  The affine Weyl group is of
type $\widetilde C_n$ which has Coxeter diagram:

\[\xymatrix{{\bullet}_1\ar@{=}[r]&
  {\bullet}_2\ar@{-}[r]& {\bullet}_3\ar@{.}[r]&
  {\bullet}_{n-1}\ar@{-}[r]& {\bullet}_n\ar@{=}[r]&{\bullet}_{n+1}}
\]
with $(n+1)$ vertices, and the two endpoints being ``special''
vertices in the sense of \cite{Ronan}.  The Coxeter diagram for $C_n$
is the same with the last special vertex (and associated ``edge'')
deleted.  Associated to each vertex $i$ is a reflection $s_i$, and the
reflections satisfy the standard rules $s_i^2 = 1$, $s_is_j$ has order
$m_{ij}$ indicated by the Coxeter diagram ($m_{12}= m_{n(n+1)} = 4$,
$m_{i(i+1)} = 3, i \ne 1,n$, and $m_{ij} = 2$ otherwise).


Acting on the symplectic basis $\{e_1, \dots,e_n,f_1,\dots,f_n\}$,
define the reflections (any basis vector not specified is fixed):

\begin{itemize}
\item $s_1$: Interchange $e_1$ and $f_1$
\item $s_j$ ($2\le j \le n$): Interchange $e_{j-1}\leftrightarrow e_j$
  and $f_{j-1}\leftrightarrow f_j$
\item $s_{n+1}$: $e_n \mapsto \pi^{-1}f_n$, $f_n \mapsto \pi e_n$
\end{itemize}

That is, acting on a vertex $[\pi^{a_1}, \dots, \pi^{a_n};
\pi^{b_1},\dots,\pi^{b_n}]$,
\begin{itemize}
\item $s_1$ takes $[\pi^{a_1}, \dots, \pi^{a_n};
  \pi^{b_1},\dots,\pi^{b_n}]$ to $[\pi^{b_1},\pi^{a_2}, \dots,
  \pi^{a_n}; \pi^{a_1},\pi^{b_2}\dots,\pi^{b_n}]$
  
\item $s_j$ ($2\le j \le n$): takes $[\pi^{a_1}, \dots, \pi^{a_n};
  \pi^{b_1},\dots,\pi^{b_n}]$ to \newline
  $[\pi^{a_1},\dots,\pi^{a_{j-1}},\pi^{a_j},\dots,\pi^{a_n};
  \pi^{b_1},\dots,\pi^{b_{j-1}},\pi^{b_j},\dots,\pi^{b_n}]$
  
\item $s_{n+1}$: takes $[\pi^{a_1}, \dots, \pi^{a_n};
  \pi^{b_1},\dots,\pi^{b_n}]$ to $[\pi^{a_1}, \dots,
  \pi^{a_{n-1}},\pi^{b_n +1}; \pi^{b_1},\dots,\pi^{b_{n-1}},\pi^{a_n
    -1}]$
\end{itemize}


Now let's proceed to label the apartment $\Sigma$.  Label a special
vertex $v_0 = [\pi,\dots,\pi; \pi,\dots,\pi]$, and pick a fundamental
chamber containing it.  Label each of the codimension 1 faces in the
chamber containing the fixed vertex $s_1, \dots, s_n$.  Label the
remaining codimension 1 face $s_{n+1}$.

Each chamber in the building contains two special vertices.  In the
case of the fundamental chamber, one of them is the vertex $v_0$
(fixed by the reflections $s_1, \dots, s_n$), and the other is fixed
by $s_2, \dots, s_{n+1}$.  From above, we see that the vertex
$[\pi^{a_1}, \dots, \pi^{a_n}; \pi^{b_1},\dots,\pi^{b_n}]$ is fixed by
$s_j$ ($2 \le j \le n$) iff $a_{j-1}=a_j$ and $b_{j-1} = b_j$. The
vertex is fixed by $s_{n+1}$ iff $a_n = b_n + 1$, thus $a_i = b_i + 1$
for all $i$, and so the vertex $[\pi^{a_1}, \dots, \pi^{a_n};
\pi^{b_1},\dots,\pi^{b_n}] = [\pi,\dots,\pi; 1,\dots,1]$ is the other
special vertex in the fundamental chamber. Now using the reflections,
we may label the apartment.


\begin{example}
\label{example:lattices}
For $Sp_2(\Qp)$ we have the following (partial) labeling of an
apartment by classes of lattices:

\tiny
  \begin{center}
    
    $\xymatrix{
%%%%%  row one
      [p^2,1;1,p^2] \ar@{-}[d]\ar@{--}[dr]\ar@{-}[r] &
      [p^2,1;1;p]\ar@{.}[d]\ar@{-}[r] & [p^2,p;1,p]
      \ar@{-}[d]\ar@{--}[dl]\ar@{--}[dr]\ar@{-}[r] &
      [p^2,p;1,1]\ar@{.}[d]\ar@{-}[r]& [p^2,p^2;1,1]
      \ar@{--}[dl]\ar@{--}[dr]\ar@{-}[r] \ar@{-}[d]&
      \bullet \ar@{.}[d] \\
%%%%%  row two
      \bullet \ar@{-}[d]\ar@{.}[r] & [p,1;1,p]
      \ar@{--}[dl]\ar@{--}[dr]\ar@{.}[r]\ar@{.}[d] &
      [p,p;1,p]\ar@{-}[d] \ar@{.}[r] & [p,p;1,1]
      \ar@{--}[dl]\ar@{--}[dr]\ar@{.}[r]\ar@{.}[d] & [p,p^2;1,1]
      \ar@{.}[r] \ar@{-}[d]&
      [p^2,p^3;p,1] \ar@{.}[d] \ar@{--}[dl] \\
%%%%%  row three
      [p,1;p,p^2]\ar@{--}[dr]\ar@{-}[r]\ar@{-}[d] & [p,1;p,p]
      \ar@{-}[r]\ar@{.}[d]&
      {\bf[p,p;p,p]}\ar@{--}[dl]\ar@{--}[dr]\ar@{-}[r]\ar@{-}[d] &
      [p,p;p,1]\ar@{-}[r]\ar@{.}[d]&
      [p,p^2;p,1]\ar@{--}[dl]\ar@{--}[dr]\ar@{-}[r]\ar@{-}[d]&
      [p^2,p^3;p^2,1]\ar@{.}[d]\\
%%%%%  row four
      \bullet \ar@{-}[d]\ar@{.}[r] & [1,1;p,p] \ar@{.}[d]\ar@{.}[r]
      \ar@{--}[dl]\ar@{--}[dr] & [1,p;p,p] \ar@{-}[d]\ar@{.}[r] &
      [1,p;p,1]\ar@{.}[d]\ar@{.}[r]\ar@{--}[dr] \ar@{--}[dl]&
      [1,p^2;p,1] \ar@{.}[r] \ar@{-}[d]&
      [p,p^3;p^2,1]\ar@{.}[d]\ar@{--}[dl] \\
%%%%%  row five
      \bullet\ar@{-}[r] & [1,1;p^2,p]\ar@{-}[r] & [1,p;p^2,p]
      \ar@{-}[r] & [1,p;p^2,1] \ar@{-}[r] & [1,p^2;p^2,1] \ar@{-}[r]&
      \bullet \\
    }$
\end{center}

\end{example}

Let $\Lambda = \O \pi^{a_1} e_1 \oplus \cdots \oplus \O \pi^{a_n} e_n
\oplus \O \pi^{b_1} f_1 \oplus \cdots \oplus \O\pi^{b_n} f_n$.  The
dual lattice $\Lambda^\sharp$ is defined to be $\{ v \in V \mid \la v,
\Lambda\ra \subseteq \O\}$.  It too is a lattice, and it it easily
seen from the bilinearity of the alternating form that $\Lambda^\sharp
= \O \pi^{-b_1} e_1 \oplus \cdots \oplus \O \pi^{-b_n} e_n \oplus \O
\pi^{-a_1} f_1 \oplus \cdots \oplus \O\pi^{-a_n} f_n$.  It is also
clear that $(\pi^\nu \Lambda)^\sharp = \pi^{-\nu} \Lambda^\sharp$, so
$[\Lambda^\sharp]$ depends only on $[\Lambda]$, and in particular
$[\Lambda] = [\Lambda^\sharp]$ iff $\pi^\mu \Lambda^\sharp = \Lambda$
for some $\mu\in \Z$.

\begin{prop}
  Let $\Lambda = \O \pi^{a_1} e_1 \oplus \cdots \oplus \O \pi^{a_n}
  e_n \oplus \O \pi^{b_1} f_1 \oplus \cdots \oplus \O\pi^{b_n} f_n$.
  Then $[\Lambda] = [\Lambda^\sharp]$ iff there exists an integer
  $\mu$, so that for all $i$, $a_i + b_i = \mu$. In this case we call
  the vertex self-dual.
\end{prop}

\begin{proof}
  Using our explicit characterization of the dual lattice, $[\Lambda]
  = [\Lambda^\sharp]$ iff there exists an integer $\mu$ so that
  $\pi^\mu \Lambda^\sharp = \Lambda$, that is iff $\pi^\mu \pi^{-b_i}
  = \pi^{a_i}$ and $\pi^\mu \pi^{-a_i} = \pi^{b_i}$ which is iff $\mu
  = a_i + b_i$ for all $i$.
\end{proof}

\begin{prop}
  If $\Lambda = \O \pi^{a_1} e_1 \oplus \cdots \oplus \O \pi^{a_n} e_n
  \oplus \O \pi^{b_1} f_1 \oplus \cdots \oplus \O\pi^{b_n} f_n$, and
  the vertex $[\Lambda]$ is self-dual, then it image under the Weyl
  group is again a self-dual lattice.
\end{prop}


\begin{proof}
  We need only check this for the generators of the Weyl group, the
  $s_i$, and all of these are obvious from the definitions above.
\end{proof}

\begin{remark}
  Actually we can say a little more about the action.  The two special
  vertices in our fundamental chamber ($[\pi, \dots, \pi; \pi, \dots,
  \pi]$ and $[\pi, \dots, \pi;1,\dots,1]$) have the value of $\mu$
  equal (modulo 2) to 0 and 1 respectively.  This value of $\mu$ is
  preserved by the action of the Weyl group which acts transitively on
  the chambers in the apartment.
  
  It is now also clear that the Weyl group acts transitively on the
  special vertices in the apartment (in a ``type''-preserving way).
\end{remark}

\subsection{Symmetric polynomials and Hecke Operators}
Our goal is to define $(n+1)$ families of Hecke operators (analogous
to the $T_k(p^2)$, $T(p)$) whose generating series produce highly
structured rational functions, and hence which are arithmetically
interesting.  In two of the $n+1$ cases, these rational functions
correspond to the spinor and standard zeta functions. In the other
cases, they are new.  In particular, in no case except for $T(p)$ have
any generating series for Hecke operators been expressed as rational
functions.

We will make the
definitions not in the Hecke algebra (defined by double cosets), but
in its representation space, the ring of $W_n$-invariant polynomials.
Doing so will produce a correspondence between the local Hecke
algebra slightly different from the Satake correspondence, but who
generators are more arithmetically distinguished.

We first give a labeling of the special vertices in an apartment of
the Bruhat-Tits building for $Sp_n(\Qp)$ by monomials in
$\Q[x_0^{\pm1}, \dots, x_n^{\pm1}]$ which corresponds in a natural way
to symplectic divisors of lattices.  We recall that the Satake
isomorphism gives us that the local rational Hecke algebra is
isomorphic to the ring of polynomials $\Q[x_0^{\pm1}, \dots,
x_n^{\pm1}]$ which are invariant under the Weyl group $W_n$.
Actually, our labelling will be in $\Q[x_0^{\pm1}, \dots,
x_n^{\pm1}]^{W_n}$ modulo the relation $x_0^2x_1\cdots x_n = 1$.
Since $\H_p \cong \Q[x_0^{\pm1}, \dots, x_n^{\pm1}]^{W_n} \cong
\Q[x_0,\dots, x_n]^{W_n}[(x_0^2x_1\cdots x_n)^{-1}]$ (see
\cite{AZbook}), reducing by the relation $x_0^2x_1\cdots x_n = 1$
produces a subring of $\Q[x_0, x_1, \dots, x_n]^{W_n} \cong \underline
\H_p$, the integral local Hecke algebra. Specifically, we are defining
a representation of the local Hecke algebra in the polynomial ring in
which double cosets $\G p^\nu I_{2n} \G$ act trivially.  This is
essentially the case when Hecke operators act on automorphic forms.
One should also recall that it is only the invertibility of $\G p
I_{2n}\G$ which distinguishes $\H_p$ for $\underline\H_p$.


Fix a (fundamental) apartment $\Sigma$ in the building by means of a frame and symplectic
basis $\{e_1, \dots, e_n,f_1,\dots,f_n\}$ as in the previous section.
Let $[\Lambda_0]$ be the class of the lattice $\Lambda_0 = \Zp e_1
\oplus \cdots \oplus \Zp e_n \oplus \Zp f_1 \oplus \cdots \oplus \Zp
f_n$, labeling a fixed special vertex in the apartment $\Sigma$.  From
the previous section, we saw that a typical vertex $[\Lambda]$ in
$\Sigma$ is special iff the vertex is self-dual, that is $\Lambda =
\Zp p^{a_1} e_1 \oplus \cdots \oplus \Zp p^{a_n}e_n \oplus \Zp p^{b_1}
f_1 \oplus \cdots \oplus \Zp p^{b_n}f_n$ for which there is an integer
$\mu$ with $\mu = a_i + b_i$ for all $i$.  With this notation, we now
have a one-to-one correspondence between the classes of lattices
(labeling special vertices), and monomials in $\Q[x_0^{\pm1}, \dots, x_n^{\pm1}]$:

\begin{equation*}
  [p^{a_1}, \dots,p^{a_n}; p^{b_1}, \dots, p^{b_n}]
  \longleftrightarrow x_0^\mu x_1^{a_1} \cdots x_n^{a_n}
\end{equation*}

\noindent modulo the relation $x_0^2x_1\cdots x_n = 1$ which corresponds to the
class $[p, \dots, p; p, \dots, p] = [\Lambda_0]$.  That is, if
$\Lambda$ is replaced by $p^c\Lambda$, $x_0^\mu x_1^{a_1} \cdots
x_n^{a_n}$ is replaced by $(x_0^2x_1\cdots x_n)^c x_0^\mu x_1^{a_1}
\cdots x_n^{a_n}$, so that classes of lattices correspond to classes
of monomials.  To keep the notation from getting too involved, we will
simply write $ x_0^\mu x_1^{a_1} \cdots x_n^{a_n}$ rather than something
like $[x_0^\mu x_1^{a_1} \cdots x_n^{a_n}]$.  This avoids obvious
confusion in statements like $\Q[x_0^{\pm1}, \dots, x_n^{\pm1}]^{W_n}
= \Q[x_0, \dots, x_n]^{W_n}[(x_0^2x_1\cdots x_n)^{-1}]$.

\begin{remark}\label{remark:monomials}
  We also note one more important property of this labelling.  Any
  special vertex $v = [p^{a_1}, \dots,p^{a_n}; p^{b_1}, \dots, p^{b_n}]$ in
  the apartment can be viewed as an element of $GSp_n^+(\Q)$ via $v
  \leftrightarrow \diag(p^{a_1}, \dots,p^{a_n}, p^{b_1}, \dots,
  p^{b_n})$. As such, there is a natural multiplication of the
  vertices.  If (with $v$ as above), we denote $x_0^\mu x_1^{a_1}
  \cdots x_n^{a_n}$ by $x^v$, then we have the simple rule $x^v x^{v'}
  = x^{vv'}$.  We will have need of this observation in the next
  section.
\end{remark}

We take a moment to foreshadow a bit.  While the Satake isomorphism
provides a correspondence between the local Hecke algebra and the
polynomial ring, the correspondence does not appear at all obvious.
On the other hand, with the given notation, there is an obvious
correspondence (though not completely correct as yet) with the local
Hecke algebra: Given, $[p^{a_1}, \dots,p^{a_n}; p^{b_1}, \dots,
p^{b_n}] \longleftrightarrow x_0^\mu x_1^{a_1} \cdots x_n^{a_n}$ with
$\mu = a_i+b_i$, we immediately note that $\diag(p^{a_1},
\dots,p^{a_n}; p^{b_1}, \dots, p^{b_n}) \in GSp_n^+(\Q)$, so that
$\Gamma_n \diag(p^{a_1}, \dots,p^{a_n}; p^{b_1}, \dots,
p^{b_n})\Gamma_n$ is in the local Hecke algebra $\Hp$.  Thus there is
a clear connection between the Hecke operator 
$\Gamma_n \diag(p^{a_1}, \dots,p^{a_n}; p^{b_1}, \dots,
p^{b_n})\Gamma_n$ and the monomial $x_0^\mu x_1^{a_1} \cdots
x_n^{a_n}$ which we will turn into a valid representation.


Before proceeding to develop the representation, we provide a labeling of a piece of
the apartment $\Sigma$ for $Sp_2$, corresponding to our previous
labelling by classes of lattices:

\begin{example} A partial labeling of the special vertices in an apartment for $Sp_2(\Qp)$ by monomials

\label{example:monomials}

\begin{center}
  $\xymatrix{
%%%%%  row one
    x_1x_2^{-1} \ar@{-}[d]\ar@{--}[dr]\ar@{-}[r] & \bullet
    \ar@{.}[d]\ar@{-}[r] & x_1
    \ar@{-}[d]\ar@{--}[dl]\ar@{--}[dr]\ar@{-}[r] & \bullet
    \ar@{.}[d]\ar@{-}[r]& x_1x_2 \ar@{--}[dl]\ar@{--}[dr]\ar@{-}[r]
    \ar@{-}[d]&
    \bullet \ar@{.}[d] \\
%%%%%  row two
    \bullet \ar@{-}[d]\ar@{.}[r] & x_0x_1
    \ar@{--}[dl]\ar@{--}[dr]\ar@{.}[r]\ar@{.}[d] & \bullet \ar@{-}[d]
    \ar@{.}[r] & x_0x_1x_2
    \ar@{--}[dl]\ar@{--}[dr]\ar@{.}[r]\ar@{.}[d] & \bullet \ar@{.}[r]
    \ar@{-}[d]&
    x_0x_1x_2^2 \ar@{.}[d] \ar@{--}[dl] \\
%%%%%  row three
    x_2^{-1}\ar@{--}[dr]\ar@{-}[r]\ar@{-}[d] & \bullet
    \ar@{-}[r]\ar@{.}[d]& 1
    \ar@{--}[dl]\ar@{--}[dr]\ar@{-}[r]\ar@{-}[d] &
    \bullet\ar@{-}[r]\ar@{.}[d]& x_2
    \ar@{--}[dl]\ar@{--}[dr]\ar@{-}[r]\ar@{-}[d]&
    \bullet\ar@{.}[d]\\
%%%%%  row four
    \bullet \ar@{-}[d]\ar@{.}[r] & x_0 \ar@{.}[d]\ar@{.}[r]
    \ar@{--}[dl]\ar@{--}[dr] & \bullet \ar@{-}[d]\ar@{.}[r] & x_0x_2
    \ar@{.}[d]\ar@{.}[r]\ar@{--}[dr] \ar@{--}[dl]& \bullet \ar@{.}[r]
    \ar@{-}[d]&
    x_0x_2^2 \ar@{.}[d]\ar@{--}[dl] \\
%%%%%  row five
    x_0^2 \ar@{-}[r] & \bullet \ar@{-}[r] & x_1^{-1} \ar@{-}[r] &
    \bullet \ar@{-}[r] & x_1^{-1}x_2 \ar@{-}[r]&
    \bullet \\
  }$
\end{center}
\end{example}
\bigbreak
\begin{remark}
  We note that if we think of each monomial as acting on the vertex 1
  as a motion (e.g. $x_2$ represents motion two vertices ``to the
  right''), then multiplication by monomials is consistent with this
  action (e.g., $x_1x_2$ is two vertices to the right of $x_1$).
\end{remark}
 

The motivation behind the definition of the new Hecke operators comes
from several sources.  On the one hand, Hecke operators, thought of as
double cosets, have an action on automorphic forms defined in terms of
a ``sum'' of their right cosets, which by
Proposition~\ref{prop:one-to-one-corresp} corresponds to a sum of
(classes of) lattices with prescribed symplectic divisors.

On the other hand, Hecke operators (via the Satake isomorphism)
correspond to $W_n$-invariant polynomials which correspond to sums of
vertices in the building.  In fact, we are really thinking only of the
underlying graph and not worrying about the full simplicial structure
the building embodies.  That said, natural operators on graphs are
adjacency operators (a sum of certain neighboring vertices), so
combining these ideas, we are led in the following direction: A Hecke
operator should be a sum of monomials (corresponding to lattices)
which are invariant under $W_n$.





To define our Hecke operators in the context of this polynomial ring
we need a definition and simple lemma: For a nonnegative integer
$\ell$, define $\displaystyle h^r(\ell) = \sum_{\substack{\sum j_k =
    \ell\\j_k \ge 0}}z_1^{j_1}z_2^{j_2} \cdots z_r^{j_r}$.  Note that
$h^r(\ell)$ is a symmetric polynomial in the $r$ variables $z_1,
\dots, z_r$, and in particular, $h^r(0) = 1$ and $h^r(1) = z_1 +
\cdots + z_r$.

\begin{prop}
\label{prop:basiceuler}
The generating series associated to the $h^r(\ell)$ satisfies
\[
\sum_{\ell \ge 0} h^r(\ell) u^\ell = \left[(1 - uz_1) \cdots
  (1-uz_r)\right]^{-1}
\]
\end{prop}

\begin{proof}
  This is essentially obvious:

\begin{align*}
  \left[(1 - uz_1) \cdots (1-uz_r)\right]^{-1} & = \left(\sum_{a_1 \ge
      0}(uz_1)^{a_1}\right) \cdots
  \left(\sum_{a_r \ge 0}(uz_r)^{a_r}\right)\\
  &= \sum_{\ell \ge 0} u^\ell\cdot \big[\sum_{\substack{\sum a_i =
      \ell\\a_i \ge 0}} z_1^{a_1} \cdots z_r^{a_r}\big]
\end{align*}
It is clear from the definitions above that the coefficient of
$u^\ell$ in the given expression is $h^r(\ell)$.
\end{proof}


Next we need to use the above polynomial to create a $W_n$-invariant
polynomial.  The simplest examples are simply to fix a monomial and to
sum its images under the action of $W_n$.  To that end, we compute a
few simple orbits.

\begin{lemma}
  \label{lem:orbits}
  Under the action of $W_n$, we obtain the following orbits:
  \begin{enumerate}
  \item $\Orbit_{W_n}(x_0) = \{ x_0 x_1^{\varepsilon_1}\cdots
    x_n^{\varepsilon_n} \mid \varepsilon_i = 0,1 \}$.
  \item $\Orbit_{W_n}(x_1\cdots x_k) = \{x_{i_1}^{\delta_{i_1}} \cdots
    x_{i_k}^{\delta_{i_k}} \mid 1\le i_1\le \cdots \le i_k\le n,
    \delta_{i_j} = \pm 1\}$.
  \end{enumerate}
  
  In particular, the orbits have size $2^n$ and $2^k\binom{n}{k}$ respectively.
\end{lemma}

\begin{proof}
  Recall that $W_n$ is the group of $\Q$-automorphisms of the rational
  function field $\Q(x_0, \ldots, x_n)$ generated by all permutations
  of the variables $x_1, \ldots, x_n$ and by the automorphisms
  $\tau_1, \ldots, \tau_n$ which are given by:
\begin{equation*}
  \tau_i(x_0) = x_0x_i, \qquad \tau_i(x_i) = x_i^{-1}, \qquad
  \tau_i(x_j) = x_j \quad (0 < j \ne i).
\end{equation*}

\noindent In particular, $W_n = \langle
\tau_i\rangle \rtimes S_n \cong (\Z/2\Z)^n \rtimes S_n$.  Since
$\tau_{k_s}\tau_{k_{s-1}}\cdots\tau_{k_1}(x_0) = x_0x_{k_1}\cdots
x_{k_s}$ for distinct $k_j \ge 1$, it is clear that $\Orbit_{W_n}(x_0)
\supseteq \{ x_0 x_1^{\varepsilon_1}\cdots x_n^{\varepsilon_n} \mid
\varepsilon_i = 0,1 \}$, and so the orbit has cardinality at least
$2^n$.  On the other hand, all of $S_n$ is contained in the stabilizer
of $x_0$, so the size of the orbit is $[W_n: \Stab(x_0)] \le [W_n :
S_n] = 2^n$, which gives the first result.

For the second, it is easy to see directly: $S_n$ can take $x_1\cdots
x_k$ to any monomial $x_{i_1}\cdots x_{i_k}$ with $1\le i_1\le \cdots
\le i_k\le n$. Applying $\tau_{i_j}$ takes $x_{i_j}$ to $x_{i_j}^{-1}$
fixing all other indices.  Since these generate the group $W_n$, the
orbit and its size are clear.
\end{proof}

With these orbits determined, the following definitions
become less mysterious.  We start with $h^r(\ell)$ where $r$ is the size
of one of the above orbits and substitute for the variables $z_i$ the
elements in the orbit.  Thus we define the families of Hecke
operators:

\begin{equation*}
  t_0^n(p^\ell) = h^{2^n}(\ell)\Big\vert_
  {\substack{z_i \mapsto \sigma_i(x_0)\\
      \sigma_i \in W_n/\Stab(x_0)}}.
\end{equation*}
and for $1 \le k \le n$,
\begin{equation*}
  t_k^n(p^\ell) = h^{2^k\binom{n}{k}}(\ell)\Big\vert_
  {\substack{z_i \mapsto \sigma_i(x_1\cdots x_k)\\
      \sigma_i \in W_n/\Stab(x_1\cdots x_k)}}.
\end{equation*}

In particular,
\begin{equation*}
  t_0^n(p) = \sum_{\varepsilon_i = 0,1}x_0x_1^{\varepsilon_1}\cdots 
  x_n^{\varepsilon_n}.\ \ (2^n \textrm{ summands})
\end{equation*}
and
\begin{equation*}
  t_k^n(p) = \sum_
  {\substack{1\le i_1 < \cdots < i_k \le n\\
      \delta_{i_j} = \pm 1}} x_{i_1}^{\delta_{i_1}} \cdots 
  x_{i_k}^{\delta_{i_k}}.\ \ (2^k \binom{n}{k} \textrm{ summands})
\end{equation*}

Here once again, we are suppressing the fact that we are looking at
classes of monomials, and that really we have defined (for example)
\begin{equation*}
  t_0^n(p) = \sum_{\varepsilon_i = 0,1} [x_0x_1^{\varepsilon_1}\cdots 
  x_n^{\varepsilon_n}].
\end{equation*}
Since $x_0^2x_1\cdots x_n$ is fixed by every element of $W_n$, these
operators are well-defined.

\begin{remark}
  We want to make clear the naturalness of these operators as well as
  their significance.  As alluded to above, $t_j(p)$ represents an
  adjacency operator. To see this observe, consider the case $n=2$ and
  the apartments labeled in diagrams in Examples
  \ref{example:lattices} and \ref{example:monomials}. 

\begin{align*}
  t_0(p) &= x_0 + x_0x_1 + x_0x_2 + x_0x_1x_2 = [1,1;p,p] + [p,1;1,p] +
  [1,p;p,1] + [p,p;1,1]
  \\
  t_1(p) &= x_1 + x_1^{-1} + x_2 + x_2^{-1} = [p^2,p;1,p] +
  [1,p;p^2,p] + [p,p^2;p,1] +
  [p,1,;p,p^2]\\
  t_2(p) &= x_1x_2 + x_1x_2^{-1} + x_1^{-1}x_2 + x_1^{-1}x_2^{-1} =
  [p^2,p^2;1,1] + [p^2,1;1,p^2] + [1,p^2;p^2,1] +
  [1,1;p^2,p^2]\\
\end{align*}

Thus at least restricted to the apartment we have the following
natural correspondences:

\begin{align*}
  t_0(p) &= \sum_{\Gamma M \Gamma = \Gamma \diag(1,1;p,p) \Gamma} [M]\longleftrightarrow T(p)\\
  t_1(p) &= \sum_{\Gamma M \Gamma = \Gamma \diag(1,p;p^2,p) \Gamma}
  [M]
  \longleftrightarrow T_1(p^2)\\
\end{align*}

and in general $t_k(p) \longleftrightarrow T_{n-k}^n(p^2)$ for $1 \le
k < n$ ($T_n(p^2) \longleftrightarrow x_0^2x_1 \cdots x_n$).
\end{remark}

To see the significance of these operators we need to provide a little
background.  Recall that associated to a simultaneous Hecke
eigenfunction $F$ of weight $k$ for $Sp_n(\Z)$ are the Satake
$p$-parameters $(\alpha_0, \ldots, \alpha_n) = (\alpha_0(p), \ldots,
\alpha_n(p)) \in \C^{n+1}/W_n$ for each prime $p$, generalizing the
Hecke eigenvalues.  The Satake parameters satisfy
$\alpha_0(p)^2\alpha_1(p) \cdots\alpha_n(p) = p^{nk-n(n+1)/2}$ and are
used to define the spinor and standard zeta functions.

The standard zeta function is defined by $D_F(s) = \prod_p
D_{F,p}(p^{-s})^{-1}$ ($\Re(s) > 1$), where
\[D_{F,p}(v) = (1 -
v)\prod_{m=1}^n (1 - \alpha_m v)(1 - \alpha_m^{-1} v),
\] while
the spinor zeta function is defined by $Z_F(s) = \prod_p
Z_{F,p}(p^{-s})^{-1}$ ($\Re(s) > nk/2 - n(n+1)/4 + 1$), where
\[
Z_{F,p}(v) = (1 - \alpha_0 v) \prod_{m = 1}^n\ \ \prod_{1\le i_1 <
  \cdots < i_m\le n} (1 - \alpha_0\alpha_{i_1}\cdots\alpha_{i_m} v).
\]


For $Sp_2$, Andrianov and Zhuravlev \cite{AZbook} define a family of
Hecke operators $T^2(p^\ell)$ whose images under the (Satake)
spherical map $\Omega$ (from $\Hp$ to $\Q[x_0^{\pm1}, x_1^{\pm1},
x_2^{\pm1}]^{W_2}$) satisfy
      \[\ds \sum_{\ell \ge 0}\Omega(T^2(p^\ell)) v^\ell = \frac{(1 -
        p^{-1}x_0^2x_1x_2 v^2)}
      {(1-x_0v)(1-x_0x_1v)(1-x_0x_2v)(1-x_0x_1x_2v)},\]
      the denominator of which is essentially that of the spinor zeta function for $Sp_2$:
      
      $\ds Z_F(s) = \prod Z_{F,p}(p^{-s})^{-1}$, where $\ds Z_{F,p}(v)
      = (1 - \alpha_0 v)(1 - \alpha_0\alpha_1 v)(1 - \alpha_0\alpha_2
      v)(1 - \alpha_0\alpha_1\alpha_2 v)$.  
We have taken the opposite
      perspective here.  We have defined interesting operators in the
      polynomial ring and will determine what they correspond to in the Hecke algebra.

\begin{prop}
  \label{prop:Hecke-generating-series}
  The operators $t_0(p^\ell)$ and $t_k(p^\ell)$ have a generating
  series which is a rational function:


\[\sum_{\ell \ge 0} t_0^n(p^\ell) v^\ell =
\left[ (1 - x_0 v) \prod_{m = 1}^n\ \ \prod_{1\le i_1 < \cdots <
    i_m\le n} (1 - x_0 x_{i_1}\cdots x_{i_m} v) \right]^{-1}
\]
which clearly corresponds to the local factor of the spinor zeta
function, and
\[\sum_{\ell \ge 0} t_k^n(p^\ell) v^\ell = \left[
  \prod_{\substack{1\le i_1 < \cdots < i_k \le n,\ \delta_{i_j} = \pm
      1}}(1 - x_{i_1}^{\delta_{i_1}} \cdots x_{i_k}^{\delta_{i_k}} v)
\right]^{-1}
\]
which, when $k = 1$, is simply $\ds \sum_{\ell \ge 0} t_1^n(p^\ell)
v^\ell = \left[ \prod_{m=1}^n (1 - x_m v)(1 - x_m^{-1} v)
\right]^{-1}$ which in turn (up to an initial ``zeta'' factor)
corresponds to the local factor of the standard zeta function:
\newline $\ds D_{F,p}(v) = (1 - v)\prod_{m=1}^n (1 - \alpha_m v)(1 -
\alpha_m^{-1} v)$.

\end{prop}


\begin{proof}
  The proof is immediate from Proposition~\ref{prop:basiceuler} and
  the computation of orbits in Lemma~\ref{lem:orbits}.
\end{proof}


\begin{remark}
  Except for $k = 0$ and $k = 1$, these are new Hecke operators whose
  generating series are rational functions which likely correspond to
  new zeta functions which can be studied in the context of Siegel
  modular forms.


  
  We note that for the case of $n=2$, the operator $t_0(p^\ell)$ has a
  generating function with the same denominator as
  $\Omega(T^2(p^\ell))$, but with numerator 1.
\end{remark}



\section{Connections with the global Hecke algebra}

 
Recall that we have fixed a $2n$-dimensional symplectic space $(V,
\langle * ,* \rangle)$ over $\Q$. Let $\Delta_n$ the Bruhat-Tits
building for $Sp_n(\Q_p)$.  Then the vertices of the building,
$\textrm{Vert}(\Delta_n)$, are in one-to-one correspondence with
homothety classes of lattices in $V$.  Let ${\mathcal B}$ be the
rational vector space with basis $\textrm{Vert}(\Delta_n)$. In this
section, we define two natural (essentially) faithful representations
of the local Hecke algebra $\Hp$, and a connecting homomorphism which
adds the necessary structure to our intuitive definitions given
earlier.  We will produce the following commutative diagram of algebra
homomorphisms.

 \begin{center}
   $\xymatrix{
%%%%%  row one
     {\H_p} \ar@{->}[drr]_\Psi\ar@{->}[rr]^(.3)\Phi &&{\Q}[x_0^{\pm1},x_1^{\pm1},\dots,x_n^{\pm1}]^{W_n}\ar@{->}[d]^\rho\\
%%%%%  row two
     && \textrm{End}({\mathcal B})\\
   }$
\end{center}
\subsection {An action of the Hecke algebra on the building}
 
For $\xi \in GSp_n^+(\Q)$, any double coset $\Gamma\xi\Gamma$ can be
represented by a diagonal element, so we assume that henceforth. For
$\xi = \diag(p^{a_1},\dots,p^{a_n};p^{b_1},\dots,p^{b_n}) \in
GSp_n^+(\Q)$, we have that the double coset $\Gamma\xi\Gamma$
determines a collection of right cosets $\{\Gamma \xi_\nu\}$ which by
Proposition~\ref{prop:one-to-one-corresp} are in one-to-one
correspondence with the collection of lattices $\{ M \}$ with $\{L:M\}
= \{p^{a_1},\dots,p^{a_n};p^{b_1},\dots,p^{b_n} \}$. Using this
identification, it is natural to define the operator $T_{\mathcal
  B}(p^{a_1},\dots,p^{a_n};p^{b_1},\dots,p^{b_n})\in \textrm{End}(\B)$
induced by:

\[
T_{\mathcal B}(p^{a_1},\dots,p^{a_n};p^{b_1},\dots,p^{b_n})([L]) =
\sum_{\{L:M\} = \{p^{a_1},\dots,p^{a_n};p^{b_1},\dots,p^{b_n} \}} [M]
\]
where the sum is over all vertices in the building with prescibed
``elementary divisors''.

For brevity, we shall often simply write $T_\B(\xi)([L]) =
\sum_{\{L:M\} = \xi} [M]$.  The map is clearly well-defined and (by
definition) linear.

In the usual notation, let $\xi_1 =
\diag(p^{a_1},\dots,p^{a_n},p^{b_1},\dots,p^{b_n})$, $\xi_2 =
\diag(p^{c_1},\dots,p^{c_n},p^{d_1},\dots,p^{d_n})$ be elements of
$GSp_n^+(\Q)$ and write $\G \xi_1 \G$ as the disjoint union $\cup \G
\alpha_i$, and write $\G \xi_2 \G$ as the disjoint union $\cup \G
\beta_j$.  In the Hecke algebra $\Hp$, the multiplication law is
defined by (e.g., see section 3.1 of \cite{Shimurabook}):

\[
(\G\xi_1\G)(\G\xi_2\G) = \G\xi_1\G\xi_2\G = \sum_{i,j}
\G\alpha_i\beta_j
\]
where the right cosets are not necessarily distinct.  More precisely,

\[
(\G\xi_1\G)(\G\xi_2\G) = \sum_{i,j} \G\alpha_i\beta_j = \sum_{\G\xi\G}
c(\xi)\G\xi\G
\]
where the sum is over all double cosets $\G\xi\G \subset
\G\xi_1\G\xi_2\G$, and where $c(\xi)$ is the number of pairs $(i,j)$
for which $\G\alpha_i\beta_j = \G \xi$.


\begin{theorem}
  \label{thm:lattice-representation}
  The correspondence $\G \xi \G \mapsto T_\B(\xi)$ induces a representation
  $\Psi:\Hp \to \End(\B)$, whose kernel consists of double cosets of
  the form $\G \xi \G$ with $\xi = p^\mu I_{2n}$, $\mu \in \Z$.
\end{theorem}


\begin{proof}
   We first verify that $\Psi$ is a ring homomorphism.
   Using the notation above, we have

   \begin{align*}
     T_\B(\xi_1) T_\B(\xi_2)([L]) &=
     T_\B(\xi_1)(\sum_{\{L:M\}=\xi_2}[M])\\
     &= \sum_{\{L:M\}=\xi_2} \sum_{\{M:N\}=\xi_1} [N]\\
   \end{align*}
   
   By Proposition~\ref{prop:one-to-one-corresp}, each lattice $M$ for
   which $\{L:M\}=\xi_2$ is of the form $M = L\beta_j$.  Now

\[
\{M:N\} = \xi_1 \iff \{L\beta_j:N\} = \xi_1 \iff \{L:N\beta_j^{-1}\} = \xi_1
\]

Now let $P$ be such that $\{L:P\} = \xi_1$.  Then again by
Proposition~\ref{prop:one-to-one-corresp}, $P = L\alpha_i$ for some
$i$.  But then $P = N\beta_j^{-1}$, so $N = P\beta_j =
L\alpha_i\beta_j$.

Thus, $\ds T_\B(\xi_1) T_\B(\xi_2)([L]) 
     = \sum_{\{L:M\}=\xi_2} \sum_{\{M:N\}=\xi_1} [N]
     = \sum_{i,j} [L\alpha_i\beta_j]$.
From the discussion preceding the theorem (and once again
     Proposition~\ref{prop:one-to-one-corresp}), this last sum is
     exactly $\sum_{\G\xi\G} c(\xi)T_\B(\xi)([L])$ which is the image of 
$(\G\xi_1\G)(\G\xi_2\G)$.

To compute the kernel of $\Psi$, suppose $\sum_{\G\xi\G}
c(\xi)T_\B(\xi)$ is the trivial map.  Then 
\[
\sum_{\G\xi\G} c(\xi)T_\B(\xi)([L]) = \sum_{\G\xi\G}
 c(\xi)\sum_{\{L:M\}=\xi}[M] = [L]
\]
for all vertices $[L] \in \textrm{Vert}(\Delta_n)$.
But the elements $[M]\in \textrm{Vert}(\Delta_n)$ are a basis for
$\B$, we have all the only one $\xi$, and for that $\xi$, $c(\xi) =
1$. Thus we have $\sum_{\{L:M\}=\xi}[M] = [L]$ for all $[L]$.  Now if
$\G\xi \G = \cup \xi_\nu$, then by
proposition~\ref{prop:one-to-one-corresp}, 
$\sum_{\{L:M\}=\xi}[M] = \sum_\nu [L\xi_\nu] = [L]$, so there can be
only one right coset:  $\G \xi \G = \G \xi$, and $[L\xi] = [L]$.
Since $\{L : L\xi\} = \xi$, we must have $\xi = p^\mu I_{2n}$ for some
integer $\mu$.
\end{proof}



Now to define the representation $\Phi:\Hp \to
\Q[x_0^{\pm1},x_1^{\pm1},\dots,x_n^{\pm1}]^{W_n}$, we begin by
defining certain symmetric polynomials.  As before, let
$\xi = \diag(p^{a_1},\dots,p^{a_n},p^{b_1},\dots,p^{b_n}) \in
GSp_n^+(\Q)$.  Recall this means there is an integer $\mu$ so that
$\mu = a_i + b_i$ for all $i$.  Define $t(\xi)$ to be the sum of the
images of $(x_0^\mu x_1^{a_1}\cdots x_n^{a_n})$ under the action of
the (spherical Weyl) group $W_n$.  We denote this as

\[
t(\xi) = \sum_{w \in W_n} (x_0^\mu x_1^{a_1}\cdots x_n^{a_n})^w
\]

Once again, we remind the reader that this sum is really over the
classes of monomials $x_0^\mu x_1^{a_1}\cdots x_n^{a_n}$ (modulo
$x_0^2x_1\cdots x_n=1$) under the action of the Weyl group $W_n$.
Since $x_0^2x_1\cdots x_n$ is fixed by all elements of $W_n$, this definition
is well-defined.

Loosely, speaking $\Phi$ should map $\G \xi \G$ to a multiple of
$t(\xi)$.  The key to defining this homomorphism is the following
identification.  We know that $W_n \cong C_n$ is simply signed a
permutation group (permuting the $n$ pairs of lines defining a frame),
and so can be identified with a subgroup of $\G$ (see 10.1 of
\cite{Garrett}).

\begin{remark}
  
  The isomorphism $W_n \cong C_n$ can be seen quite explicitly if we
  identify the generators of the affine Weyl group $C_n$ with
  generators of $W_n$.  In the previous section, we wrote down
  reflections $s_j$ which generated the Weyl groups: $s_1, \dots, s_n$
  generate the spherical Weyl group $C_n$, while $s_1, \dots, s_{n+1}$
  generate the affine Weyl group $\widetilde C_n$.
  
  The group $W_n$ is generated by the $\tau_i$ and the permutations in
  the symmetric group $S_n$.  With obvious identifications, we see
  that $\tau_j = (i\ j)\tau_i (i\ j)$ and since $S_n$ is generated by
  the transpositions $\{ (1\ 2), (2\ 3), \dots, (n-1\ n)\}$, $W_n =
  \la \tau_1, (1\ 2), (2\ 3), \dots, (n-1\ n)\ra$.  One identifies
  $\tau_1$ with $s_1$, and $(j-1\ j)$ with $s_j$ ($2 \le j \le n$) and
  verifies they satisfy the same Coxeter relations as the $s_i$, and so
  satisfy the deletion condition, which by Tit's theorem (see Chapter
  1 of \cite{Garrett}) gives a Coxeter system.
  
\end{remark}


By definition, 

\begin{equation*}
  T_\B(\xi)([L]) = \sum_{\gamma_\nu \in (\xi^{-1}\G \xi \cap
  \G)\backslash \G} [L\xi\gamma_\nu]
\end{equation*}

So identifying $W_n$ with a subgroup of $\G$, we may write $\G$ as a
union of double cosets: $\G = \cup (\xi^{-1}\G \xi \cap\G)\delta_\mu
W_n$, so the expression above becomes

\begin{equation*}
  T_\B(\xi)([L]) = \sum_{\delta_\mu \in (\xi^{-1}\G \xi \cap
  \G)\backslash \G/W_n} \sum_{w\in W_n}[L\xi\delta_\mu w]
\end{equation*}

Given that $W_n$ acts invariantly on any given apartment, we see that
for a fixed $\mu$, the collection of vertices $[L\xi\delta_\mu w]$ as
$w$ runs over $W_n$ consists (with multiplicity $[W_n : \Stab(L\xi\delta_\mu)]$)
of all the vertices $[M]$ in a given apartment with $\{L:M\} = \xi$.
Moreover, since every apartment in the building is isomorphic (with
structure determined by the Coxeter group $\widetilde C_n$), all of the
multiplicities $[W_n : \Stab(L\xi\delta_\mu)]$ are the same,
independent of $\mu$.




From the remark above it is now clear that 
after an appropriate change of frame (in particular a change of a basis),
$t(\xi) = \sum_{w \in W_n} (x_0^\mu x_1^{a_1}\cdots x_n^{a_n})^w$ can
be naturally identified with summand $\sum_{w\in W_n}[L\xi\delta_\mu w]$
occuring in 

\begin{equation*}
  T_\B(\xi)([L]) = \sum_{\delta_\mu \in (\xi^{-1}\G \xi \cap
  \G)\backslash \G/W_n} \sum_{w\in W_n}[L\xi\delta_\mu w]
\end{equation*}

If we define $\eta(\xi) = \#(\xi^{-1}\G \xi \cap
  \G)\backslash \G/W_n$, we can identify 
$T_\B(\xi)$ with $\eta(\xi)t(\xi)$, thus we define 
 $\Phi(\G\xi\G) = \eta(\xi)t(\xi)$.
 
 To define the connecting homomorphism $\rho:
 \Q[x_0^{\pm1},x_1^{\pm1},\dots,x_n^{\pm1}]^{W_n} \to \End(\B)$, we
 observe that any polynomial $p \in
 \Q[x_0^{\pm1},x_1^{\pm1},\dots,x_n^{\pm1}]^{W_n}$, can be written as
 a sum $\sum_\xi c(\xi) t(\xi)$ where the $\xi$ are a set of
 representatives of the monomials occuring in $p$ modulo $W_n$, so we
 only have to define $\rho$ on a basis.
We define $\rho(t(\xi)) =
\eta(\xi)^{-1}T_\B(\xi)$, so that $\rho\circ\Phi = \Psi$.


To prove that $\Phi$ is a ring homomorphism, it is convenient to think
of $\Phi$ as the restriction to $\H_p$ of a linear map $\Phi: L(\G,G)\to
\Q[x_0^{\pm1},x_1^{\pm1},\dots,x_n^{\pm1}]^{W_n}$.  Recall that in
this setting we view the double coset $\G \xi \G$ as the sum of right
cosets $\sum \G \xi_\nu$ ($\G \xi \G = \cup \G \xi_\nu$) which is
right invariant under $\G$.  While the map $\Phi$ (with domain
$L(\G,G)$) will only be a vector space homomorphism, its restriction to
$\H_p$ will be a ring homomorphism.  

What we have said above is that we want $\Phi$ to take $\G \xi \G$ to
$\eta(\xi)t(\xi)$ so that there are as many monomials (counting
multiplicities) as right cosets $\G\xi \G = \cup \G \xi_\nu$.  Since
any right coset $\G \xi_\nu$ determines the same double coset, we must
map any right coset to the same polynomial.  From this, the definition
is clear:  $\Phi(\G \xi_\nu) = |W_n|^{-1} \sum_{w \in W_n} (x^{\xi
  w})$. Then extending linearly yields the desired 
$\Phi(\G \xi \G) =  \eta(\xi)t(\xi)$.

\begin{theorem}
  \label{thm:polynomial-representation}
  The correspondence $\G \xi \G \mapsto \eta(\xi)t(\xi)$ is a
  surjective homomorphism $\Phi:\Hp \to
  \Q[x_0^{\pm1},x_1^{\pm1},\dots,x_n^{\pm1}]^{W_n}$, whose kernel
  consists of double cosets of the form $\G \xi \G$ with $\xi = p^\mu
  I_{2n}$, $\mu \in \Z$.
\end{theorem}

\begin{proof}
  First note that since
  $\Q[x_0^{\pm1},x_1^{\pm1},\dots,x_n^{\pm1}]^{W_n} = 
\Q[x_0, \dots, x_n]^{W_n}[(x_0^2x_1\cdots x_n)^{-1}]$ (and we are
  working modulo $x_0^2x_1\cdots x_n = 1$), the codomain is spanned as
  a vector space by elements of the form $t(\xi)$, from which the
  surjectivity is immediate.
  Let $\G \alpha\G, \G\beta\G \in \H_p$, and write (as a disjoint union)
  \begin{align*}
    \G\alpha\G &= \mathop{\bigcup}_{\alpha_i \in (\alpha^{-1}\G\alpha \cap
    \G)\backslash \G / W_n} \mathop{\bigcup}_{w\in W_n} \G \alpha \alpha_i w
    \textrm{ and }\\
    \G\beta\G &= \mathop{\bigcup}_{\beta_j \in (\beta^{-1}\G\beta \cap
    \G)\backslash \G / W_n} \mathop{\bigcup}_{w'\in W_n} \G\beta \beta_j w'.\\
  \end{align*}

Without loss of generality, we are assuming that $\alpha$ and $\beta$
are diagonal.  Then in $\H_p$, $(\G\alpha\G)(\G\beta\G) = \sum_{i,j}\G \alpha\alpha_i
w\beta\beta_j w' = \sum_\xi c(\xi) \G \xi \G$.  Our first task is to
determine the $\xi$ which can occur.  Since $\beta_j, w' \in \G$, it is
clear that $\G\alpha\alpha_i w\beta\beta_j w'\G =
\G\alpha\alpha_i w\beta\G$, however it it also true that 
$\G\alpha\alpha_i w\beta\G = \G\alpha w\beta\G$.  To see the latter we
deal with symplectic elementary divisors
(Lemma~\ref{lem:symplectic-divisors}).  Recall that elementary
divisors $\{L : L\xi\}$ are uniquely determined by the double coset $\G
\xi \G$.  We have that
\begin{equation*}
  \{L : L \alpha \alpha_i w \beta \} = 
  \{L\beta^{-1} w^{-1} : L \alpha \alpha_i\},
\end{equation*}
and we note that $L = L\alpha_i = L\alpha_i^{-1}$ since $\alpha_i \in
\G$.  Moreover, $\alpha_i^{-1}\alpha \alpha_i$ and $\alpha$ are simply
two matrix representations of the same linear transformation acting on
$L$.  Thus
\begin{equation*}
  \{L : L \alpha \alpha_i w \beta \} = 
%  \{L\beta^{-1} w^{-1} : L \alpha \alpha_i\}=
  \{L\beta^{-1} w^{-1} : L \alpha_i^{-1}\alpha \alpha_i\}=
\{L\beta^{-1} w^{-1} : L \alpha\}=
\{L : L \alpha w \beta \}
\end{equation*}
Thus  $(\G\alpha\G)(\G\beta\G) = \sum_{w\in W_n} d(w) \G \alpha w
\beta \G$ (where the double cosets may not be distinct).  Here $d(w)$
is easily computed from the above observations:

\[d(w) \deg(\G \alpha w \beta) = \eta(\alpha)\eta(\beta) |W_n|,\]
where $\deg(\G \alpha w \beta)$ is the number of right cosets in the
double coset, that is,  $\deg(\G \xi \G) =\# \left( \xi^{-1}\G \xi \cap \G\right)\backslash \G$.

It follows that 
\begin{align*}
\Phi\left( (\G\alpha\G) (\G \beta \G)\right) &= 
\sum_w d(w) \eta(\alpha w \beta) t(\alpha w \beta)\\
&= \sum_w \eta(\alpha)\eta(\beta)|W_n| (\deg(\G \alpha w \beta\G))^{-1}
\eta(\alpha w \beta) t(\alpha w \beta) \\
&=  \eta(\alpha)\eta(\beta) \sum_w t(\alpha w \beta).
\end{align*}
We note that since $\alpha$ and $\beta$ are diagonal and $w$ is a
``signed'' permutation, the product $\alpha w \beta$ is diagonal so
that $t(\alpha w \beta)$ does indeed make sense.
On the other hand 
\begin{align*}
  \Phi(\G\alpha\G) \Phi(\G \beta \G)& =
  \eta(\alpha)\eta(\beta)t(\alpha)t(\beta)\\
&= \eta(\alpha)\eta(\beta) (\sum_w x^{\alpha w}) (\sum_{w'} x^{\beta
  w'})\\
&= \eta(\alpha)\eta(\beta) (\sum_w \sum_{w'} x^{\alpha w \beta w'})
  \textrm{ (see remark \ref{remark:monomials})}\\
&= \eta(\alpha)\eta(\beta) \sum_w (\sum_{w'} x^{(\alpha w \beta)w'})\\
&= \eta(\alpha)\eta(\beta) \sum_w t(\alpha w \beta)
\end{align*}

Thus $\Phi$ is a ring homomorphism.  To compute its kernel, suppose
that $\Phi(\sum_\xi c(\xi) \G \xi \G) = \sum_\xi
c(\xi)\eta(\xi)t(\xi)$ is trivial, that is to say (as polynomials)
\begin{equation*}
  \sum_\xi c(\xi)\eta(\xi)\sum_{w\in W_n} x^{\xi w} = (x_0^2x_1\cdots
  x_n)^\ell \textrm{ for some integer } \ell
\end{equation*}
From this we immediately infer that there can only be one $\xi$ and
that $x^\xi = x^{\xi w}$ for all $w \in W_n$.  It is trivial to check
that if $\xi$ corresponds to the class of the lattice 
$[p^{a_1}, \dots, p^{a_n}; p^{b_1}, \dots, p^{b_n}]$ with $\mu =
a_i+b_i$ for all $i$, then the monomial $x^\xi = x_0^\mu x_1^{a_1} \cdots
x_n^{a_n}$ is fixed by all elements of $W_n$ iff $x^\xi = (x_0^2x_1\cdots
  x_n)^\ell$ for some integer $\ell$.  Then
$\sum_\xi c(\xi)\eta(\xi)\sum_{w\in W_n} x^{\xi w} = 
c(\xi)\eta(\xi) |W_n| x^\xi = x^\xi$ and $\xi = p^\ell I_{2n}$.  Since
$\deg(\G p^\ell I_{2n}\G) = 1$, $\eta(\xi) = |W_n|^{-1}$, so $c(\xi) =
1$ which establishes the result.
\end{proof}


\begin{remark}
  The image  under $\Phi$ of $\G p I \G$ is (the class of) is $x_0^2x_1\cdots x_n$.
  For comparison, one should note that for a simultaneous Hecke
  eigenform of weight $k$ for $\G = \G_n$, the associated Satake $p$-parameters $(\alpha_0,
  \alpha_1, \dots, \alpha_n) \in \C^{n+1}/{W_n}$ satisfy 
  $\alpha_0^2\alpha_1\cdots \alpha_n = p^{nk - n(n+1)/2}$, and also
  that under the Satake isomorphism (Andrianov's spherical map
  $\Omega$), the image of $\G p I \G$ is $p^{-n(n+1)/2}x_0^2x_1\cdots x_n$.
\end{remark}



\begin{corollary}
  The connecting map $\rho: \Q[x_0^{\pm1}, \dots, x_n^{-1}]^{W_n} 
  \to \End(\B)$
  defined by $\rho( t(\xi)) = \eta(\xi)^{-1}T_\B(\xi)$ is an injective
  ring homomorphism. satisfying $\rho \circ \Phi = \Psi$.
\end{corollary}


\begin{proof}
  Both $\Phi$ and $\Psi$ have the same kernel $K$, so induce injective
  homomorphisms from $\widetilde\Phi: \H_p/K \to \Q[x_0^{\pm1}, \dots,
  x_n^{-1}]^{W_n}$, and $\widetilde\Psi: \H_p/K \to \End(\B)$.  Since,
  $\Phi$ and hence $\widetilde\Phi$ is surjective, $\rho =
  \widetilde\Psi \circ \widetilde \Phi^{-1}$ has the desired properties.
\end{proof}

\bigskip

\begin{remark}
  We remark that while closely related, there is a difference between the spherical map
  $\Omega$ defined by Andrianov \cite{AZbook} and our map $\Phi$.  

  Let $\xi_0 = \begin{pmatrix}I_2&0\\0&pI_2\end{pmatrix}$ and
  $\xi_k = \begin{pmatrix}
  I_{n-k}&0&0&0\\0&pI_k&0&0\\0&0&p^2I_{n-k}&0\\0&0&0&pI_k
\end{pmatrix}$, $1 \le k \le n$.
By Lemma~\ref{lem:orbits},
\begin{equation*}t(\xi_0) = \sum_{w\in W_n} x^{\xi_0 w} = |W_n| [W_n:
\Stab(x_0)]^{-1} \sum_{\varepsilon_i = 0,1} x_0 x_1^{\varepsilon_1}\cdots
x_n^{\varepsilon_n} 
= \frac{|W_n|}{2^n} t_0(p), 
\end{equation*}
so that 
\[
\Phi(T(p)) = \eta(\xi_0)t(\xi_0) = \frac{\deg(\xi_0)}{|W_n|}  \frac{|W_n|}{2^n}\sum_{\varepsilon_i = 0,1} x_0 x_1^{\varepsilon_1}\cdots
x_n^{\varepsilon_n} =  \frac{\deg(\xi_0)}{2^n} \Omega(T(p)) = \frac{\deg(\xi_0)}{2^n}t_0(p).
\]

\noindent As mentioned earlier for $1 \le k < n$, $T_{n-k}(p^2)$ corresponds to $t_k(p)$, so we have
\begin{equation*}
t(\xi_k) = \sum_{w\in W_n} x^{\xi_k w} = |W_n| [W_n:
\Stab(\xi_k)]^{-1} \sum_{1\le i_1\le \cdots \le i_k\le n,\
  \delta_{i_j} = \pm 1}
x_{i_1}^{\delta_{i_1}} \cdots
x_{i_k}^{\delta_{i_k}} = \frac{|W_n|}{2^k \binom{n}{k}}t_k(p),
\end{equation*}
so that
\[
\Phi(T_{n-k}(p^2)) = \eta(\xi_k)t(\xi_k) =
\frac{\deg(\xi_k)}{|W_n|}\frac{|W_n|}{2^k \binom{n}{k}}t_k(p) = 
\frac{\deg(\xi_k)}{2^k \binom{n}{k}}t_k(p)
\]

Computations with the spherical map are far less trivial.  For
comparison purposes we compute $\Omega(T_1^2(p^2))$.  We have already
shown that
\[
 \Phi(T_1^2(p^2)) = \frac{\deg(\xi_1)}{4} t_1(p) = \frac{\deg(\xi_1)}{4}(x_1 + x_1^{-1} + x_2 + x_2^{-1})
\]
while
\[
\Omega(T_2^2(p^2)) = \Omega(p I_4) = p^{-3}x_0^2 x_1 x_2
\]
and
\begin{align*} 
    \Omega(T_1^2(p^2)) &= 
    p^{-1}(x_0^2x_1 + x_0^2x_2 + x_0^2x_1^2x_2 + x_0^2x_1x_2^2) +
    p^{-2}(p^2-1)(x_0^2x_1x_2)\\
    &= (x_0^2 x_1 x_2)[p^{-1}(x_1 + x_1^{-1} + x_2 + x_2^{-1}) + p^{-2}(p^2-1)]\\
    &= (x_0^2 x_1 x_2)[p^{-1}t_1(p) + p^{-2}(p^2-1)]\\
    &= \Omega(T_2^2(p^2)) [p^2 t_1(p) + (p^2-1)]
  \end{align*}

So it is clear that the action of $\Omega$ is somewhat more
complicated, intertwining the Hecke operators in comparison to the
action of $\Phi$.
\end{remark}


\newpage

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\address{Department of Mathematics, 6188 Bradley Hall, Dartmouth
  College, Hanover, NH 03755}

\fax{(603) 646-1312}

\email {thomas.r.shemanske@dartmouth.edu}

\URL{http://www.math.dartmouth.edu/\symbol{126}trs/}

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