Hecke Operators for the Symplectic Group and Buildings

Thomas R. Shemanske

June 21, 2001

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}(ℚ_{p}), 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.

1 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 ([7]), or correlations between Fourier coefficients and Hecke eigenvalues in degree 2 ([4]). 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}(ℚ_{p}), 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}.

1991 Mathematics Subject Classification. Primary 11F60; Secondary 11F46 20E42

Key Words and Phrases. Hecke Operators, Siegel modular forms, Symplectic Group, Bruhat–Tits Building

It is well-known (see e.g., Cartier [2], 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 [1], 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}(ℚ_{p}). 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}(ℚ_{p}), 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.

2 Preliminaries

2.1 The Classical Hecke Algebra

Much of this material can be found in Chapter 3 of [1]; we state it here to set the notation. Let Γ = Γ_{n} = Sp_{n}(ℤ) ⊂ SL_{2n}(ℤ), and let G = GSp_{n}^{+}(ℚ) ⊂ GL_{2n}(ℚ) be the group of symplectic similitudes with scalar factor r(M) ∈ ℚ_{+}^{×}:

GSp_{n}^{+}(ℚ) = \{M = ( \begin{matrix}A & B \\ C & D \end{matrix}) ∈ M_{2n}(ℚ) | A^{t}C = C^{t}A, B^{t}D = D^{t}B, A^{t}D − C^{t}B = r(M)I_{2n}\}

= \{M = ( \begin{matrix}A & B \\ C & D \end{matrix}) ∈ M_{2n}(ℚ) | AB^{t} = B^{t}A, CD^{t} = DC^{t}, AD^{t} − BC^{t} = r(M)I_{2n}\}.

Let H denote the rational Hecke algebra associated to the pair Γ and G. That is, as a vector space, H is generated by all double cosets ΓξΓ (ξ ∈ G), and we turn H into an algebra by defining the multiplication law as follows: Given ξ_{1}, ξ_{2} ∈ G, define

Γξ_{1}Γ · Γξ_{2}Γ = ∑_{ξ} c(ξ)ΓξΓ,  (2.1)

where the sum is over all double cosets ΓξΓ ⊆ Γξ_{1}Γξ_{2}Γ, and the c(ξ) are nonnegative integers (see [6]). There is an alternate characterization of the Hecke algebra which will be convenient as well. Let L(Γ, G) be the rational vector space with basis consisting of right cosets Γξ for ξ ∈ G. The Hecke algebra can be thought of as those elements of L(Γ, G) which are right invariant under the action of Γ. Thus we can and will think of a double coset ΓξΓ = ∪Γξ_{ν} as the disjoint union of right cosets and as the sum of those right cosets ∑ Γξ_{ν} ∈ L(Γ, 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 ∩ GL_{2n}(ℤ[p^{−1}]) in the above construction. H_{p} is generated by double cosets ΓξΓ with ξ of the form diag(p^{a_{1}}, …, p^{a_{n}}, p^{b_{1}}, …, p^{b_{n}}) where a_{1} ≤ ⋯ ≤ a_{n} ≤ b_{n} ≤ ⋯ ≤ b_{1} are integers with p^{a_{i}+b_{i}} = r(ξ) for all i. It is occasionally useful to consider the “integral” Hecke algebra \underline{H}_{p} generated by all ξ as above with ξ = diag(p^{a_{1}}, …, p^{a_{n}}, p^{b_{1}}, …, p^{b_{n}}) ∈ 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 Γ_{p} = Sp_{n}(ℤ_{p}) and G_{p} = GSp_{n}(ℚ_{p}). To be explicit we set H_{p} to be the Hecke algebra generated as a ℚ-vector space by double cosets Γ_{p}ξΓ_{p} with ξ = diag(p^{a_{1}}, …, p^{a_{n}}, p^{b_{1}}, …, 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:

Lemma 2.1. Let ξ ∈ GSp_{n}^{+}(ℚ) ∩ GL_{2n}(ℤ[p^{−1}]). If ΓξΓ is the disjoint union ∪Γα_{i}, then Γ_{p}ξΓ_{p} is the disjoint union ∪Γ_{p}α_{i}.

Proof. Without loss of generality, we may assume ξ ∈ M_{2n}(ℤ) since the general case follows by multiplying by a power of p. First, it is clear that ∪Γ_{p}α_{i} ⊆ Γ_{p}ξΓ_{p}, and that the α_{i} have integer entries. Next, we see that the cosets Γ_{p}α_{i} are disjoint since if not then α_{i}α_{j}^{−1} ∈ Γ_{p} ∩ GL_{2n}(ℤ[p^{−1}]) ⊆ Γ. To show that the union is all of Γ_{p}ξΓ_{p}, we need only show that ξΓ_{p} ⊆ Γ_{p}ξΓ, for then any element γ̃_{1}ξγ̃_{2} = γ̃_{3}ξγ (γ̃_{i} ∈ Γ_{p}, γ ∈ Γ), so ξγ ∈ Γα_{i} for some i, hence γ̃_{1}ξγ̃_{2} ∈ Γ_{p}α_{i}. To see ξΓ_{p} ⊆ Γ_{p}ξΓ is really just a density argument: Let q = r(ξ) be the similitude factor associated to ξ. Recall (see [1]) that there is a natural surjective homomorphism Sp_{n}(R) → Sp_{n}(R/qR) with R = ℤ or ℤ_{p}. Denote by Γ_{p}(q) the kernel of the homomorphism Sp_{n}(ℤ_{p}) → Sp_{n}(ℤ_{p}/qℤ_{p}). From Chapter 2, §3.3 of [1], we have that Γ_{p}(q) ⊂ Γ_{p} ∩ξ^{−1}Γ_{p}ξ. Using the fact that Sp_{n}(ℤ_{p}/qℤ_{p}) ≅ Sp_{n}(ℤ/qℤ) (and that Sp_{n}(ℤ) → Sp_{n}(ℤ/qℤ) is surjective), we may write Γ_{p} = ∪Γ_{p}(q)δ_{i} with the δ_{i} ∈ Γ. Let γ̃ ∈ Γ_{p}, and write γ̃ = γ̃_{0}δ_{k} for some γ̃_{0} ∈ Γ_{p}(q) and some k. Then

ξγ̃ = ξγ̃_{0}δ_{k} = ξγ̃_{0}δ_{k}(δ_{k}^{−1}ξ^{−1})ξδ_{k} = (ξγ̃_{0}ξ^{−1})ξδ_{k} ∈ Γ_{p}ξΓ

since ξγ̃_{0}ξ^{−1} ∈ ξΓ_{p}(q)ξ^{−1} ⊂ ξ(Γ_{p} ∩ ξ^{−1}Γ_{p}ξ)ξ^{−1} ⊂ Γ_{p}.

The integral Hecke algebra \underline{H}_{p} is generated by the (n + 1) Hecke operators

T (p) = Γ_{p} (\begin{matrix}I_{n} & 0 \\ 0 & pI_{n} \end{matrix}) Γ_{p}

and for k = 1, …, n,

T_{k}^{n}(p^{2}) = T_{k}(p^{2}) = Γ_{p} (\begin{matrix}I_{n−k} & 0 & 0 & 0 \\ 0 & pI_{k} & 0 & 0 \\ 0 & 0 & p^{2}I_{n−k} & 0 \\ 0 & 0 & 0 & pI_{k} \end{matrix}) Γ_{p},

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:

\underline{H}_{p} ≅ ℚ[x_{0}, …, x_{n}]^{W_{n}}
H_{p} ≅ ℚ[x_{0}^{±1}, …, x_{n}^{±1}]^{W_{n}} ≅ ℚ[x_{0}, …, x_{n}]^{W_{n}}[(x_{0}^{2}x_{1} ⋯ x_{n})^{−1}]

where W_{n} is the group of ℚ-automorphisms of the rational function field ℚ(x_{0}, …, x_{n}) generated by all permutations of the variables x_{1}, …, x_{n} and by the automorphisms τ_{1}, …, τ_{n} which are given by:

τ_{i}(x_{0}) = x_{0}x_{i},  τ_{i}(x_{i}) = x_{i}^{−1},  τ_{i}(x_{j}) = x_{j} (0 < j ≠ i).

W_{n} is a signed permutation group, in particular, W_{n} = ⟨τ_{i}⟩⋊S_{n} ≅ (ℤ/2ℤ)^{n} ⋊S_{n} ≅ C_{n} where C_{n} is Coxeter group associated to the spherical building for Sp_{n}(ℚ_{p}).

2.2 Symplectic lattices and elementary divisors

Let E be the field ℚ or ℚ_{p}, O its ring of integers, and (V, ⟨∗, ∗⟩) a 2n-dimensional symplectic space over E. Fix a symplectic basis B = {u_{1}, …, u_{n}, v_{1}, …, v_{n}} of V satisfying ⟨u_{i}, v_{j}⟩ = δ_{ij} (Kronecker delta), ⟨u_{i}, u_{j}⟩ = ⟨v_{i}, v_{j}⟩ = 0. Let Γ = Sp_{n}(O), and let G = GSp_{n}(E) be the group of symplectic similitudes with scalar factor r(M) ∈ E^{×}. Note that Γ is a normal subgroup of GSp_{n}(E), being the kernel of the determinant map to E^{×}.

Notation 2.2. 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^{×}/O^{×}. For E = ℚ, we let S = ℚ_{+}^{×}, the positive rationals, while for E = ℚ_{p}, we let S = {p^{ν} | ν ∈ ℤ}. We will denote by GSp_{n}^{S}(E) = {M ∈ GSp_{n}(E) | r(M) ∈ S}, so when E = ℚ, GSp_{n}^{S}(E) is the classical GSp_{n}^{+}(ℚ). It is worth noting that Sp_{n}(E) ⊂ GSp_{n}^{S}(E).

With obvious modification to the proof, the following is Lemma 3.6 of [1].

Lemma 2.3. Let ξ ∈ GSp_{n}^{S}(E), then every double coset ΓξΓ has a unique representative of the form sd(ξ) = diag(d_{1}, …, d_{n}, e_{1}, …, e_{n}) where d_{i}, e_{i} ∈ S and d_{i} | d_{i+1}, d_{n} | e_{n}, e_{i+1} | e_{i}, and d_{i}e_{i} = r(ξ)

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.

Proposition 2.4. Let L be a symplectic lattice. Then Γ = {A ∈ GSp_{n}^{S}(E) | LA = 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.

Proof. Let Γ̃ = {A ∈ GSp_{n}^{S}(E) | LA = L }. It is clear that Γ ⊂ Γ̃, so we need only establish the other inclusion. Let A ∈ Γ̃. Since Γ̃ is a group, and Γ ⊂ Γ̃, any element of the double coset ΓAΓ is also an element of Γ̃. By Lemma 2.3, we have that sd(A) = diag(d_{1}, …, d_{n}; e_{1}, …, e_{n}) ∈ Γ̃, where the d_{i}, e_{i} ∈ S satisfy e_{1}O ⊂ ⋯ ⊂ e_{n}O ⊂ d_{n}O ⊂ ⋯ ⊂ d_{1}O and e_{i}d_{i} = r(A). It is trivial to see that the elementary divisors of Lsd(A) in L are {L : Lsd(A)} = {d_{1}, …, d_{n}, e_{1}, … e_{n}}. On the other hand, LA = L, so Lsd(A) = L and we must have d_{i} = e_{i} = 1 for all i. Thus ΓAΓ = Γ, so A ∈ Γ.

Fix a symplectic lattice L and put R = R_{L} = {LA | A ∈ GSp_{n}^{S}(E)}.

Lemma 2.5. Let M and N be lattices in R. Then there exists a symplectic basis {u_{1}, …, u_{n}, v_{1}, …, v_{n}} of V, and elements d_{i}, e_{i} ∈ S satisfying e_{1}O ⊂ ⋯ ⊂ e_{n}O ⊂ d_{n}O ⊂

⋯ ⊂ d_{1}O and e_{i}d_{i} = r ∈ S such that M = ⨁_{i=1}^{n} Ou_{i} ⊕ ⨁_{i=1}^{n} Ov_{i} and N = ⨁_{i=1}^{n} Od_{i}u_{i} ⊕ ⨁_{i=1}^{n} Oe_{i}v_{i}.

Remark 2.6. 1. Note that in the general linear case, GSp_{n}^{S} would be replaced by GL_{2n}^{+}, and R would be the set of all lattices of full rank in V.

2. 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 Γ ⊂ SL_{2n}(O). That is, if we choose two lattices from 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}, …, d_{n}, e_{1}, …, e_{n}} to mean there exist bases of M and N as in the lemma.

Proof. Since M and N are in R, there exists an A ∈ GSp_{n}^{S}(E) with N = MA. Assume that Γ is identified with the stabilizer of M. By Lemma 2.3, sd(A) = diag(d_{1}, …, d_{n}, e_{1}, …, e_{n}) = γ_{1}Aγ_{2} for some γ_{i} ∈ Γ, where sd(A) is the “symplectic divisor” matrix of A. Finally, it is clear that since Mγ_{i} = M, that {M : N } = {Mγ_{1} : Mγ_{1}A} = {Mγ_{1}γ_{2} : Mγ_{1}Aγ_{2}} = {M : Msd(A)} = {d_{1}, …, d_{n}, e_{1}, …, e_{n}}, from which the lemma follows.

Lemma 2.7. For A and B in GSP_{n}^{S}(E), ΓA = ΓB if and only if LA = LB.

Proof. ΓA = ΓB if and only if AB^{−1} ∈ Γ, which by Proposition 2.4 is true if and only if L = LAB^{−1}.

Lemma 2.8. Let M and N be lattices in R. The elementary divisors of M and N in L satisfy {L : M} = {L : N } if and only if there exists an A ∈ Γ such that MA = N.

Proof. The result is clear if there exists an A ∈ Γ such that MA = N. To prove the converse, we note that by definition of the symplectic elementary divisors, there exist elements d_{i}, e_{i} ∈ S satisfying e_{1}O ⊂ ⋯ ⊂ e_{n}O ⊂ d_{n}O ⊂ ⋯ ⊂ d_{1}O and e_{i}d_{i} = r ∈ S and symplectic O-bases

{u_{1}^{(j)}, …, u_{n}^{(j)}; v_{1}^{(j)}, …, v_{n}^{(j)}} (j = 1, 2)

of L such that

L = ⨁_{i=1}^{n} Ou_{i}^{(1)} ⊕ ⨁_{i=1}^{n} Ov_{i}^{(1)} M = ⨁_{i=1}^{n} Od_{i}u_{i}^{(1)} ⊕ ⨁_{i=1}^{n} Oe_{i}v_{i}^{(1)}
L = ⨁_{i=1}^{n} Ou_{i}^{(2)} ⊕ ⨁_{i=1}^{n} Ov_{i}^{(2)} N = ⨁_{i=1}^{n} Od_{i}u_{i}^{(2)} ⊕ ⨁_{i=1}^{n} Oe_{i}v_{i}^{(2)}

Let A be the matrix of the linear transformation (with respect to either basis) taking u_{i}^{(1)} ↦ u_{i}^{(2)}, and v_{i}^{(1)} ↦ v_{i}^{(2)}. Clearly A ∈ Sp_{n}(E) ⊂ GSp_{n}^{S}(E) as it maps one symplectic basis to another. Since LA = L, A ∈ Γ by Proposition 2.4 above. Since A obviously maps M to N, the proof is complete.

Proposition 2.9. Let A ∈ GSp_{n}^{S}(E), and

ΓAΓ = Γsd(A)Γ = Γ diag(d_{1}, …, d_{n}, e_{1}, …, e_{n})Γ.

Then Γξ ↦ Lξ gives a one-to-one correspondence between the cosets Γξ in ΓAΓ and lattices M ∈ R with {L : M} = {d_{1}, …, d_{n}, e_{1}, … e_{n}}.

Proof. We may assume that A = diag(d_{1}, …, d_{n}, e_{1}, …, e_{n}). If Γξ = ΓAδ with δ ∈ Γ, then Lξ ∈ R and we have {L : Lξ} = {L : LAδ} = {L : LA} = {d_{1}, …, d_{n}, e_{1}, … e_{n}}. Conversely, if M ∈ R and {L : M} = {d_{1}, …, d_{n}, e_{1}, … e_{n}}, then by Lemma 2.8 there exists an element B ∈ Γ such that M = LAB. Clearly ΓAB ⊂ ΓAΓ. The correspondence is one-to-one since by Lemma 2.7, Γξ = Γζ if and only if Lξ = Lζ.

3 The Bruhat-Tits Building for Sp_{n}(ℚ_{p})

3.1 Lattices

The Bruhat-Tits building for Sp_{n}(ℚ_{p}) 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 [5]; 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, π a uniformizing parameter, and k = O/πO the residue field. Let (V, ⟨∗, ∗⟩) be a symplectic (non-degenerate alternating) space of dimension 2n.

Definition 3.1. An O-lattice Λ is primitive if ⟨Λ, Λ⟩ ⊆ O and ⟨∗, ∗⟩ induces a nondegenerate form on the alternating space Λ/πΛ over k.

We describe an apartment system for the building as follows. A frame is an unordered n-tuple {λ_{1}^{1}, λ_{1}^{2}}, …, {λ_{n}^{1}, λ_{n}^{2}} of pairs of lines {λ_{i}^{1}, λ_{i}^{2}} so that V = ∑_{1}^{n}(λ_{i}^{1} + λ_{i}^{2}), (λ_{i}^{1} + λ_{i}^{2}) is orthogonal to (λ_{j}^{1} + λ_{j}^{2}) for i ≠ j, and each (λ_{i}^{1} + λ_{i}^{2}) is a hyperbolic plane. We say that the frame determines the apartment Σ. Vertices in Σ are homothety classes of lattices, denoted [Λ]. A vertex [Λ] lies in Σ (determined by the above frame), if there are free Omodules M_{i}^{j} ⊂ λ_{i}^{j} so that Λ = ⊕_{i,j}M_{i}^{j} for some (and hence every) representative Λ of the homothety class. More concretely, vertices of the building are homothety classes of lattices [Λ] which possess a representative Λ such that

There exists a lattice Λ_{0} with π^{−1}Λ_{0} primitive, Λ_{0} ⊆ Λ ⊆ π^{−1}Λ_{0}, and ⟨Λ, Λ⟩ ⊆ πO

or equivalently, Λ/Λ_{0} is a totally isotropic k-subspace of the non-degenerate alternating space π^{−1}Λ_{0}/Λ_{0}.

The maximal simplicies (chambers) are unordered (n + 1)-tuples [Λ_{0}], [Λ_{1}] …, [Λ_{n}] of homothety classes of lattices with representatives Λ_{i} satisfying:

π^{−1}Λ_{0} is primitive, Λ_{0} ⊆ Λ_{i} ⊆ π^{−1}Λ_{0}, and Λ_{1}/Λ_{0} ⊂ Λ_{2}/Λ_{0} ⊂ ⋯ ⊂ Λ_{n}/Λ_{0} is a maximal isotropic flags of k-subspaces in π^{−1}Λ_{0}/Λ_{0}.

With respect to a fixed symplectic basis {e_{1}, f_{1}, …, e_{n}, f_{n}} (⟨e_{i}, f_{i}⟩ = 1, ⟨e_{i}, e_{i}⟩ = ⟨f_{i}, f_{i}⟩ = 0), let Λ be the O-lattice Λ = Oπ^{a_{1}}e_{1} ⊕ ⋯ Oπ^{a_{n}}e_{n} ⊕ Oπ^{b_{1}}f_{1} ⊕ ⋯ ⊕ Oπ^{b_{n}}f_{n}. With the basis fixed, we often denote this lattice as (π^{a_{1}}, … π^{a_{n}}; π^{b_{1}}, …, π^{b_{n}}).

We note that ⟨Λ, Λ⟩ ⊆ O iff ⟨π^{a_{i}}e_{i}, π^{b_{i}}f_{i}⟩ = π^{a_{i}+b_{i}} ∈ O which is true iff a_{i} + b_{i} ≥ 0. Given a_{i} + b_{i} ≥ 0, the induced alternating form on Λ/πΛ is non-degenerate over k = O/πO iff a_{i} + b_{i} = 0 for all i.

Example 3.2. Let {e_{1}, …, e_{n}, f_{1}, …, f_{n}} be a symplectic basis for V (with ⟨e_{i}, f_{i}⟩ = 1), and put λ_{i}^{1} = Ke_{i} and λ_{i}^{2} = Kf_{i}. The frame {λ_{i}^{1}, λ_{i}^{2}} defines an apartment Σ. Let Λ_{0} = π(⊕Oe_{i} ⊕ Of_{i}). Then π^{−1}Λ_{0} is primitive. Denote by [π^{a_{1}}, …, π^{a_{n}}; π^{b_{1}}, …, π^{b_{n}}] the class of the lattice Oπ^{a_{1}}e_{1}⊕⋯⊕Oπ^{a_{n}}e_{n}⊕Oπ^{b_{1}}f_{1}⊕⋯⊕Oπ^{b_{n}}f_{n}. Then the following flags determine (fundamental) chambers in Σ.

[Λ_{0}] = [π, …, π; π, …, π] ⊂ [Λ_{1}] = [1, π, …, π; π, …, π] ⊂ [Λ_{2}] = [1, 1, π, …, π; π, …, π] ⊂ ⋯ ⊂ [Λ_{n}] = [1, 1, …, 1; π, …, π].

[Λ_{0}] = [π, …, π; π, …, π] ⊂ [Λ_{1}] = [π, …, π; 1, π, …, π] ⊂ [Λ_{2}] = [π, …, π; 1, 1, π …, π] ⊂ ⋯ ⊂ [Λ_{n}] = [π, …, π; 1, 1, …, 1].

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′ ∈ S. We say s ∼ s′ if there are lattices Λ_{s} ∈ s and Λ_{s′} ∈ s′ and a lattice Λ_{0} such that π^{−1}Λ_{0} is primitive, Λ_{0} ⊆ Λ_{s} ⊆ π^{−1}Λ_{0}, πΛ_{0} ⊆ Λ_{s′} ⊆ π^{−1}Λ_{0}, and either Λ_{s} ⊂ Λ_{s′} or Λ_{s′} ⊂ Λ_{s}. The associated flag complex yields the building.

Remark 3.3. This definition is somewhat subtle. For example, consider the case n = 2. Then it would appear if Λ_{0} = π(Oe_{1}⊕Oe_{2}⊕Of_{1}⊕Of_{2}) that any lattices Λ_{s} and Λ_{s′} satisfying πΛ_{0} ⊂ Λ_{s′} ⊂ Λ_{s} ⊂ π^{−1}Λ_{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 ν and a lattice Λ_{1} with π^{−1}Λ_{1} primitive and Λ_{1} ⊂ π^{ν}Λ_{s′} ⊂ π^{−1}Λ_{1} which eliminates some of the possibilities.

Example 3.4. Consider the case n = 2, with Λ_{0} = π(Oe_{1} ⊕ Oe_{2} ⊕ Of_{1} ⊕ Of_{2}), and let [Λ_{0}] denote any special vertex in the apartment Σ (a vertex with the most hyperplanes through it)

We restrict our attention to the apartment Σ and hence to the frame determined by the lines spanned by the e_{i} and f_{i}. As above, any vertex in Σ is the class of a lattice of the form Oπ^{a_{1}}e_{1} ⊕ ⋯ ⊕ Oπ^{a_{n}}e_{n} ⊕ Oπ^{b_{1}}f_{1} ⊕ ⋯ ⊕ Oπ^{b_{n}}f_{n}, denoted [π^{a_{1}}, …, π^{a_{n}}; π^{b_{1}}, …, π^{b_{n}}].

For n = 2, we need to provide pairs of lattices Λ_{1} and Λ_{2} with Λ_{1}/Λ_{0} ⊂ Λ_{2}/Λ_{0} a maximal isotropic flag in π^{−1}Λ_{0}/Λ_{0}, so we only list the pairs of lattices Λ_{1} and Λ_{2}.

Since Sp_{n} is of type C_{n}, the Weyl group is isomorphic to (ℤ/2ℤ)^{n} ⋊ S_{n} (the signed permutation group) and has order 2^{n}n!, so for n = 2 we expect 8 chambers containing the given vertex = [Λ_{0}] = [1, 1, 1, 1] = [π, π, π, π]. The other pairs of vertices defining the chambers are:

1. [1, π, π, π] ⊂ [1, 1, π, π]

2. [π, 1, π, π] ⊂ [1, 1, π, π]

3. [π, π, 1, π] ⊂ [π, π, 1, 1] 4. [π, π, π, 1] ⊂ [π, π, 1, 1]

5. [1, π, π, π] ⊂ [1, π, π, 1]

6. [π, π, π, 1] ⊂ [1, π, π, 1]

7. [π, 1, π, π] ⊂ [π, 1, 1, π]

8. [π, π, 1, π] ⊂ [π, 1, 1, π]

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 C̃_{n} which has Coxeter diagram:

•_{1}  •_{2}  •_{3}  •_{n−1}  •_{n}  •_{n+1}

with (n + 1) vertices, and the two endpoints being “special” vertices in the sense of [5]. 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_{i}s_{j} has order m_{ij} indicated by the Coxeter diagram (m_{12} = m_{n(n+1)} = 4, m_{i(i+1)} = 3, i ≠ 1, n, and m_{ij} = 2 otherwise).

Acting on the symplectic basis {e_{1}, …, e_{n}, f_{1}, …, f_{n}}, define the reflections (any basis vector not specified is fixed):

• s_{1}: Interchange e_{1} and f_{1}

• s_{j} (2 ≤ j ≤ n): Interchange e_{j−1} ↔ e_{j} and f_{j−1} ↔ f_{j}

• s_{n+1}: e_{n} ↦ π^{−1}f_{n}, f_{n} ↦ πe_{n}

That is, acting on a vertex [π^{a_{1}}, …, π^{a_{n}}; π^{b_{1}}, …, π^{b_{n}}],

• s_{1} takes [π^{a_{1}}, …, π^{a_{n}}; π^{b_{1}}, …, π^{b_{n}}] to [π^{b_{1}}, π^{a_{2}}, …, π^{a_{n}}; π^{a_{1}}, π^{b_{2}} …, π^{b_{n}}]

• s_{j} (2 ≤ j ≤ n): takes [π^{a_{1}}, …, π^{a_{n}}; π^{b_{1}}, …, π^{b_{n}}] to [π^{a_{1}}, …, π^{a_{j−1}}, π^{a_{j}}, …, π^{a_{n}}; π^{b_{1}}, …, π^{b_{j−1}}, π^{b_{j}}, …, π^{b_{n}}]

• s_{n+1}: takes [π^{a_{1}}, …, π^{a_{n}}; π^{b_{1}}, …, π^{b_{n}}] to [π^{a_{1}}, …, π^{a_{n−1}}, π^{b_{n}+1}; π^{b_{1}}, …, π^{b_{n−1}}, π^{a_{n}−1}]

Now let’s proceed to label the apartment Σ. Label a special vertex v_{0} = [π, …, π; π, …, π], and pick a fundamental chamber containing it. Label each of the codimension 1 faces in the chamber containing the fixed vertex s_{1}, …, 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}, …, s_{n}), and the other is fixed by s_{2}, …, s_{n+1}. From above, we see that the vertex [π^{a_{1}}, …, π^{a_{n}}; π^{b_{1}}, …, π^{b_{n}}] is fixed by s_{j} (2 ≤ j ≤ 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 [π^{a_{1}}, …, π^{a_{n}}; π^{b_{1}}, …, π^{b_{n}}] = [π, …, π; 1, …, 1] is the other special vertex in the fundamental chamber. Now using the reflections, we may label the apartment.

Example 3.5. For Sp_{2}(ℚ_{p}) we have the following (partial) labeling of an apartment by classes of lattices:

[figure: [p^{2}, 1; 1, p^{2}]; [p^{2}, 1; 1; p]; [p^{2}, p; 1, p]; [p^{2}, p; 1, 1]; [p^{2}, p^{2}; 1, 1]; •; •; [p, 1; 1, p]; [p, p; 1, p]; [p, p; 1, 1]; [p, p^{2}; 1, 1]; [p^{2}, p^{3}; p, 1]; [p, 1; p, p^{2}]; [p, 1; p, p]; [p, p; p, p]; [p, p; p, 1]; [p, p^{2}; p, 1]; [p^{2}, p^{3}; p^{2}, 1]; •; [1, 1; p, p]; [1, p; p, p]; [1, p; p, 1]; [1, p^{2}; p, 1]; [p, p^{3}; p^{2}, 1]; •; [1, 1; p^{2}, p]; [1, p; p^{2}, p]; [1, p; p^{2}, 1]; [1, p^{2}; p^{2}, 1]; •]

Let Λ = Oπ^{a_{1}}e_{1} ⊕ ⋯ ⊕ Oπ^{a_{n}}e_{n} ⊕ Oπ^{b_{1}}f_{1} ⊕ ⋯ ⊕ Oπ^{b_{n}}f_{n}. The dual lattice Λ^{♯} is defined to be {v ∈ V | ⟨v, Λ⟩ ⊆ O}. It too is a lattice, and it it easily seen from the bilinearity of the alternating form that Λ^{♯} = Oπ^{−b_{1}}e_{1} ⊕ ⋯ ⊕ Oπ^{−b_{n}}e_{n} ⊕ Oπ^{−a_{1}}f_{1} ⊕ ⋯ ⊕ Oπ^{−a_{n}}f_{n}. It is also clear that (π^{ν}Λ)^{♯} = π^{−ν}Λ^{♯}, so [Λ^{♯}] depends only on [Λ], and in particular [Λ] = [Λ^{♯}] iff π^{μ}Λ^{♯} = Λ for some μ ∈ ℤ.

Proposition 3.6. Let Λ = Oπ^{a_{1}}e_{1} ⊕ ⋯ ⊕ Oπ^{a_{n}}e_{n} ⊕ Oπ^{b_{1}}f_{1} ⊕ ⋯ ⊕ Oπ^{b_{n}}f_{n}. Then [Λ] = [Λ^{♯}] iff there exists an integer μ, so that for all i, a_{i} + b_{i} = μ. In this case we call the vertex self-dual.

Proof. Using our explicit characterization of the dual lattice, [Λ] = [Λ^{♯}] iff there exists an integer μ so that π^{μ}Λ^{♯} = Λ, that is iff π^{μ}π^{−b_{i}} = π^{a_{i}} and π^{μ}π^{−a_{i}} = π^{b_{i}} which is iff μ = a_{i} + b_{i} for all i.

Proposition 3.7. If Λ = Oπ^{a_{1}}e_{1} ⊕ ⋯ ⊕ Oπ^{a_{n}}e_{n} ⊕ Oπ^{b_{1}}f_{1} ⊕ ⋯ ⊕ Oπ^{b_{n}}f_{n}, and the vertex [Λ] is self-dual, then it image under the Weyl group is again a self-dual lattice.

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.

Remark 3.8. Actually we can say a little more about the action. The two special vertices in our fundamental chamber ([π, …, π; π, …, π] and [π, …, π; 1, …, 1]) have the value of μ equal (modulo 2) to 0 and 1 respectively. This value of μ 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).

3.2 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}(ℚ_{p}) by monomials in ℚ[x_{0}^{±1}, …, x_{n}^{±1}] 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}^{±1}, …, x_{n}^{±1}] which are invariant under the Weyl group W_{n}. Actually, our labelling will be in ℚ[x_{0}^{±1}, …, x_{n}^{±1}]^{W_{n}} modulo the relation x_{0}^{2}x_{1} ⋯ x_{n} = 1. Since H_{p} ≅ ℚ[x_{0}^{±1}, …, x_{n}^{±1}]^{W_{n}} ≅ ℚ[x_{0}, …, x_{n}]^{W_{n}}[(x_{0}^{2}x_{1} ⋯ x_{n})^{−1}] (see [1]), reducing by the relation x_{0}^{2}x_{1} ⋯ x_{n} = 1 produces a subring of ℚ[x_{0}, x_{1}, …, x_{n}]^{W_{n}} ≅ \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 Γp^{ν}I_{2n}Γ 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 ΓpI_{2n}Γ which distinguishes H_{p} for \underline{H}_{p}.

Fix a (fundamental) apartment Σ in the building by means of a frame and symplectic basis {e_{1}, …, e_{n}, f_{1}, …, f_{n}} as in the previous section. Let [Λ_{0}] be the class of the lattice Λ_{0} = ℤ_{p}e_{1} ⊕ ⋯ ⊕ ℤ_{p}e_{n} ⊕ ℤ_{p}f_{1} ⊕ ⋯ ⊕ ℤ_{p}f_{n}, labeling a fixed special vertex in the apartment Σ. From the previous section, we saw that a typical vertex [Λ] in Σ is special iff the vertex is self-dual, that is Λ = ℤ_{p}p^{a_{1}}e_{1} ⊕ ⋯ ⊕ ℤ_{p}p^{a_{n}}e_{n} ⊕ ℤ_{p}p^{b_{1}}f_{1} ⊕ ⋯ ⊕ ℤ_{p}p^{b_{n}}f_{n} for which there is an integer μ with μ = 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}^{±1}, …, x_{n}^{±1}]:

[p^{a_{1}}, …, p^{a_{n}}; p^{b_{1}}, …, p^{b_{n}}] ⟷ x_{0}^{μ}x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}}

modulo the relation x_{0}^{2}x_{1} ⋯ x_{n} = 1 which corresponds to the class [p, …, p; p, …, p] = [Λ_{0}]. That is, if Λ is replaced by p^{c}Λ, x_{0}^{μ}x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}} is replaced by (x_{0}^{2}x_{1} ⋯ x_{n})^{c}x_{0}^{μ}x_{1}^{a_{1}} ⋯ 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}^{μ}x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}} rather than something like [x_{0}^{μ}x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}}]. This avoids obvious confusion in statements like ℚ[x_{0}^{±1}, …, x_{n}^{±1}]^{W_{n}} = ℚ[x_{0}, …, x_{n}]^{W_{n}}[(x_{0}^{2}x_{1} ⋯ x_{n})^{−1}].

Remark 3.9. We also note one more important property of this labelling. Any special vertex v = [p^{a_{1}}, …, p^{a_{n}}; p^{b_{1}}, …, p^{b_{n}}] in the apartment can be viewed as an element of GSp_{n}^{+}(ℚ) via v ↔ diag(p^{a_{1}}, …, p^{a_{n}}, p^{b_{1}}, …, p^{b_{n}}). As such, there is a natural multiplication of the vertices. If (with v as above), we denote x_{0}^{μ}x_{1}^{a_{1}} ⋯ 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.

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}}, …, p^{a_{n}}; p^{b_{1}}, …, p^{b_{n}}] ⟷ x_{0}^{μ}x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}} with μ = a_{i}+b_{i}, we immediately note that diag(p^{a_{1}}, …, p^{a_{n}}; p^{b_{1}}, …, p^{b_{n}}) ∈ GSp_{n}^{+}(ℚ), so that Γ_{n} diag(p^{a_{1}}, …, p^{a_{n}}; p^{b_{1}}, …, p^{b_{n}})Γ_{n} is in the local Hecke algebra H_{p}. Thus there is a clear connection between the Hecke operator Γ_{n} diag(p^{a_{1}}, …, p^{a_{n}}; p^{b_{1}}, …, p^{b_{n}})Γ_{n} and the monomial x_{0}^{μ}x_{1}^{a_{1}} ⋯ 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 Σ for Sp_{2}, corresponding to our previous labelling by classes of lattices:

Example 3.10. A partial labeling of the special vertices in an apartment for Sp_{2}(ℚ_{p}) by monomials

[figure: x_{1}x_{2}^{−1}; •; x_{1}; •; x_{1}x_{2}; •; •; x_{0}x_{1}; •; x_{0}x_{1}x_{2}; •; x_{0}x_{1}x_{2}^{2}; x_{2}^{−1}; •; 1; •; x_{2}; •; •; x_{0}; •; x_{0}x_{2}; •; x_{0}x_{2}^{2}; x_{0}^{2}; •; x_{1}^{−1}; •; x_{1}^{−1}x_{2}; •]

Remark 3.11. 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_{1}x_{2} is two vertices to the right of x_{1}).

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 2.9 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 ℓ, define h^{r}(ℓ) = ∑_{\begin{matrix}∑ j_{k}=ℓ \\ j_{k}≥0 \end{matrix}} z_{1}^{j_{1}}z_{2}^{j_{2}} ⋯ z_{r}^{j_{r}}. Note that

h^{r}(ℓ) is a symmetric polynomial in the r variables z_{1}, …, z_{r}, and in particular, h^{r}(0) = 1 and h^{r}(1) = z_{1} + ⋯ + z_{r}.

Proposition 3.12. The generating series associated to the h^{r}(ℓ) satisfies

∑_{ℓ≥0} h^{r}(ℓ)u^{ℓ} = [(1 − uz_{1}) ⋯ (1 − uz_{r})]^{−1}

Proof. This is essentially obvious:

[(1 − uz_{1}) ⋯ (1 − uz_{r})]^{−1} = (∑_{a_{1}≥0}(uz_{1})^{a_{1}}) ⋯ (∑_{a_{r}≥0}(uz_{r})^{a_{r}})

= ∑_{ℓ≥0} u^{ℓ} · [ ∑_{\begin{matrix}∑ a_{i}=ℓ \\ a_{i}≥0 \end{matrix}} z_{1}^{a_{1}} ⋯ z_{r}^{a_{r}}]

It is clear from the definitions above that the coefficient of u^{ℓ} in the given expression is h^{r}(ℓ).

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.

Lemma 3.13. Under the action of W_{n}, we obtain the following orbits:

1. Orbit_{W_{n}}(x_{0}) = {x_{0}x_{1}^{ε_{1}} ⋯ x_{n}^{ε_{n}} | ε_{i} = 0, 1}.

2. Orbit_{W_{n}}(x_{1} ⋯ x_{k}) = {x_{i_{1}}^{δ_{i_{1}}} ⋯ x_{i_{k}}^{δ_{i_{k}}} | 1 ≤ i_{1} ≤ ⋯ ≤ i_{k} ≤ n, δ_{i_{j}} = ±1}.

In particular, the orbits have size 2^{n} and 2^{k}\binom{n}{k} respectively.

Proof. Recall that W_{n} is the group of ℚ-automorphisms of the rational function field ℚ(x_{0}, …, x_{n}) generated by all permutations of the variables x_{1}, …, x_{n} and by the automorphisms τ_{1}, …, τ_{n} which are given by:

τ_{i}(x_{0}) = x_{0}x_{i},  τ_{i}(x_{i}) = x_{i}^{−1},  τ_{i}(x_{j}) = x_{j} (0 < j ≠ i).

In particular, W_{n} = ⟨τ_{i}⟩ ⋊ S_{n} ≅ (ℤ/2ℤ)^{n} ⋊ S_{n}. Since τ_{k_{s}}τ_{k_{s−1}} ⋯ τ_{k_{1}}(x_{0}) = x_{0}x_{k_{1}} ⋯ x_{k_{s}} for distinct k_{j} ≥ 1, it is clear that Orbit_{W_{n}}(x_{0}) ⊇ {x_{0}x_{1}^{ε_{1}} ⋯ x_{n}^{ε_{n}} | ε_{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})] ≤ [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} ⋯ x_{k} to any monomial x_{i_{1}} ⋯ x_{i_{k}} with 1 ≤ i_{1} ≤ ⋯ ≤ i_{k} ≤ n. Applying τ_{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.

With these orbits determined, the following definitions become less mysterious. We start with h^{r}(ℓ) 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:

t_{0}^{n}(p^{ℓ}) = h^{2^{n}}(ℓ)|\begin{matrix}z_{i}↦σ_{i}(x_{0}) \\ σ_{i}∈W_{n}/ Stab(x_{0}) \end{matrix}.

and for 1 ≤ k ≤ n,

t_{k}^{n}(p^{ℓ}) = h^{2^{k}\binom{n}{k}}(ℓ)|\begin{matrix}z_{i}↦σ_{i}(x_{1}···x_{k}) \\ σ_{i}∈W_{n}/ Stab(x_{1}···x_{k}) \end{matrix}.

In particular,

t_{0}^{n}(p) = ∑_{ε_{i}=0,1} x_{0}x_{1}^{ε_{1}} ⋯ x_{n}^{ε_{n}}. (2^{n} summands)

and

t_{k}^{n}(p) = ∑_{\begin{matrix}1≤i_{1}<···<i_{k}≤n \\ δ_{i_{j}} =±1 \end{matrix}} x_{i_{1}}^{δ_{i_{1}}} ⋯ x_{i_{k}}^{δ_{i_{k}}} . (2^{k}\binom{n}{k} summands)

Here once again, we are suppressing the fact that we are looking at classes of monomials, and that really we have defined (for example)

t_{0}^{n}(p) = ∑_{ε_{i}=0,1}[x_{0}x_{1}^{ε_{1}} ⋯ x_{n}^{ε_{n}}].

Since x_{0}^{2}x_{1} ⋯ x_{n} is fixed by every element of W_{n}, these operators are well-defined.

Remark 3.14. 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 3.5 and 3.10.

t_{0}(p) = x_{0} + x_{0}x_{1} + x_{0}x_{2} + x_{0}x_{1}x_{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_{1}x_{2} + x_{1}x_{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}]

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

t_{0}(p) = ∑_{ΓMΓ=Γ diag(1,1;p,p)Γ}[M] ⟷ T (p)
t_{1}(p) = ∑_{ΓMΓ=Γ diag(1,p;p^{2},p)Γ}[M] ⟷ T_{1}(p^{2})

and in general t_{k}(p) ⟷ T_{n−k}^{n}(p^{2}) for 1 ≤ k < n (T_{n}(p^{2}) ⟷ x_{0}^{2}x_{1} ⋯ x_{n}).

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 (α_{0}, …, α_{n}) = (α_{0}(p), …, α_{n}(p)) ∈ ℂ^{n+1}/W_{n} for each prime p, generalizing the Hecke eigenvalues. The Satake parameters satisfy α_{0}(p)^{2}α_{1}(p) ⋯ α_{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) = ∏_{p} D_{F,p}(p^{−s})^{−1} (ℜ(s) > 1), where

D_{F,p}(v) = (1 − v) ∏_{m=1}^{n}(1 − α_{m}v)(1 − α_{m}^{−1}v),

while the spinor zeta function is defined by Z_{F} (s) = ∏_{p} Z_{F,p}(p^{−s})^{−1} (ℜ(s) > nk/2 − n(n + 1)/4 + 1), where

Z_{F,p}(v) = (1 − α_{0}v) ∏_{m=1}^{n} ∏_{1≤i_{1}<···<i_{m}≤n}(1 − α_{0}α_{i_{1}} ⋯ α_{i_{m}}v).

For Sp_{2}, Andrianov and Zhuravlev [1] define a family of Hecke operators T^{2}(p^{ℓ}) whose images under the (Satake) spherical map Ω (from H_{p} to ℚ[x_{0}^{±1}, x_{1}^{±1}, x_{2}^{±1}]^{W_{2}}) satisfy

∑_{ℓ≥0} Ω(T^{2}(p^{ℓ}))v^{ℓ} = \frac{(1 − p^{−1}x_{0}^{2}x_{1}x_{2}v^{2})}{(1 − x_{0}v)(1 − x_{0}x_{1}v)(1 − x_{0}x_{2}v)(1 − x_{0}x_{1}x_{2}v)},

the denominator of which is essentially that of the spinor zeta function for Sp_{2}:

Z_{F} (s) = ∏ Z_{F,p}(p^{−s})^{−1}, where Z_{F,p}(v) = (1 − α_{0}v)(1 − α_{0}α_{1}v)(1 − α_{0}α_{2}v)(1 − α_{0}α_{1}α_{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.

Proposition 3.15. The operators t_{0}(p^{ℓ}) and t_{k}(p^{ℓ}) have a generating series which is a rational function:

∑_{ℓ≥0} t_{0}^{n}(p^{ℓ})v^{ℓ} = [(1 − x_{0}v) ∏_{m=1}^{n} ∏_{1≤i_{1}<···<i_{m}≤n}(1 − x_{0}x_{i_{1}} ⋯ x_{i_{m}}v)]^{−1}

which clearly corresponds to the local factor of the spinor zeta function, and

∑_{ℓ≥0} t_{k}^{n}(p^{ℓ})v^{ℓ} = [ ∏_{1≤i_{1}<···<i_{k}≤n, δ_{i_{j}} =±1}(1 − x_{i_{1}}^{δ_{i_{1}}} ⋯ x_{i_{k}}^{δ_{i_{k}}} v)]^{−1}

which, when k = 1, is simply ∑_{ℓ≥0} t_{1}^{n}(p^{ℓ})v^{ℓ} = [∏_{m=1}^{n}(1 − x_{m}v)(1 − x_{m}^{−1}v)]^{−1} which in turn (up

to an initial “zeta” factor) corresponds to the local factor of the standard zeta function:

D_{F,p}(v) = (1 − v) ∏_{m=1}^{n}(1 − α_{m}v)(1 − α_{m}^{−1}v).

Proof. The proof is immediate from Proposition 3.12 and the computation of orbits in Lemma 3.13.

Remark 3.16. 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^{ℓ}) has a generating function with the same denominator as Ω(T^{2}(p^{ℓ})), but with numerator 1.

4 Connections with the global Hecke algebra

Recall that we have fixed a 2n-dimensional symplectic space (V, ⟨∗, ∗⟩) over ℚ. Let Δ_{n} the Bruhat-Tits building for Sp_{n}(ℚ_{p}). Then the vertices of the building, Vert(Δ_{n}), are in one-to-one correspondence with homothety classes of lattices in V. Let B be the rational vector space with basis Vert(Δ_{n}). In this section, we define two natural (essentially) faithful representations of the local Hecke algebra H_{p}, 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.

\xymatrix{H_{p} \ar[dr]_{Ψ} \ar[r]^{Φ} & ℚ[x_{0}^{±1}, x_{1}^{±1}, …, x_{n}^{±1}]^{W_{n}} \ar[d]^{ρ} \\  & End(B)}

4.1 An action of the Hecke algebra on the building

For ξ ∈ GSp_{n}^{+}(ℚ), any double coset ΓξΓ can be represented by a diagonal element, so we assume that henceforth. For ξ = diag(p^{a_{1}}, …, p^{a_{n}}; p^{b_{1}}, …, p^{b_{n}}) ∈ GSp_{n}^{+}(ℚ), we have that the double coset ΓξΓ determines a collection of right cosets {Γξ_{ν}} which by Proposition 2.9 are in one-to-one correspondence with the collection of lattices {M} with {L : M} = {p^{a_{1}}, …, p^{a_{n}}; p^{b_{1}}, …, p^{b_{n}}}. Using this identification, it is natural to define the operator T_{B}(p^{a_{1}}, …, p^{a_{n}}; p^{b_{1}}, …, p^{b_{n}}) ∈ End(B) induced by:

T_{B}(p^{a_{1}}, …, p^{a_{n}}; p^{b_{1}}, …, p^{b_{n}})([L]) = ∑_{{L:M}={p^{a_{1}},...,p^{an};p^{b_{1}},...,p^{bn}}}[M]

where the sum is over all vertices in the building with prescibed “elementary divisors”.

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

In the usual notation, let ξ_{1} = diag(p^{a_{1}}, …, p^{a_{n}}, p^{b_{1}}, …, p^{b_{n}}), ξ_{2} = diag(p^{c_{1}}, …, p^{c_{n}}, p^{d_{1}}, …, p^{d_{n}}) be elements of GSp_{n}^{+}(ℚ) and write Γξ_{1}Γ as the disjoint union ∪Γα_{i}, and write Γξ_{2}Γ as the disjoint union ∪Γβ_{j}. In the Hecke algebra H_{p}, the multiplication law is defined by (e.g., see section 3.1 of [6]):

(Γξ_{1}Γ)(Γξ_{2}Γ) = Γξ_{1}Γξ_{2}Γ = ∑_{i,j} Γα_{i}β_{j}

where the right cosets are not necessarily distinct. More precisely,

(Γξ_{1}Γ)(Γξ_{2}Γ) = ∑_{i,j} Γα_{i}β_{j} = ∑_{ΓξΓ} c(ξ)ΓξΓ

where the sum is over all double cosets ΓξΓ ⊂ Γξ_{1}Γξ_{2}Γ, and where c(ξ) is the number of pairs (i, j) for which Γα_{i}β_{j} = Γξ.

Theorem 4.1. The correspondence ΓξΓ ↦ T_{B}(ξ) induces a representation Ψ : H_{p} → End(B), whose kernel consists of double cosets of the form ΓξΓ with ξ = p^{μ}I_{2n}, μ ∈ ℤ.

Proof. We first verify that Ψ is a ring homomorphism. Using the notation above, we have

T_{B}(ξ_{1})T_{B}(ξ_{2})([L]) = T_{B}(ξ_{1})( ∑_{{L:M}=ξ_{2}}[M])

= ∑_{{L:M}=ξ_{2}} ∑_{{M:N}=ξ_{1}}[N]

By Proposition 2.9, each lattice M for which {L : M} = ξ_{2} is of the form M = Lβ_{j}. Now

{M : N} = ξ_{1} ⟺ {Lβ_{j} : N} = ξ_{1} ⟺ {L : Nβ_{j}^{−1}} = ξ_{1}

Now let P be such that {L : P } = ξ_{1}. Then again by Proposition 2.9, P = Lα_{i} for some i. But then P = Nβ_{j}^{−1}, so N = P β_{j} = Lα_{i}β_{j}.

Thus, T_{B}(ξ_{1})T_{B}(ξ_{2})([L]) = ∑_{{L:M}=ξ_{2}} ∑_{{M:N}=ξ_{1}}[N] = ∑_{i,j}[Lα_{i}β_{j}]. From the discus-

sion preceding the theorem (and once again Proposition 2.9), this last sum is exactly ∑_{ΓξΓ} c(ξ)T_{B}(ξ)([L]) which is the image of (Γξ_{1}Γ)(Γξ_{2}Γ).

To compute the kernel of Ψ, suppose ∑_{ΓξΓ} c(ξ)T_{B}(ξ) is the trivial map. Then

∑_{ΓξΓ} c(ξ)T_{B}(ξ)([L]) = ∑_{ΓξΓ} c(ξ) ∑_{{L:M}=ξ}[M] = [L]

for all vertices [L] ∈ Vert(Δ_{n}). But the elements [M] ∈ Vert(Δ_{n}) are a basis for B, we have all the only one ξ, and for that ξ, c(ξ) = 1. Thus we have ∑_{{L:M}=ξ}[M] = [L] for all [L]. Now if ΓξΓ = ∪ξ_{ν}, then by proposition 2.9, ∑_{{L:M}=ξ}[M] = ∑_{ν}[Lξ_{ν}] = [L], so there can be only one right coset: ΓξΓ = Γξ, and [Lξ] = [L]. Since {L : Lξ} = ξ, we must have ξ = p^{μ}I_{2n} for some integer μ.

Now to define the representation Φ : H_{p} → ℚ[x_{0}^{±1}, x_{1}^{±1}, …, x_{n}^{±1}]^{W_{n}}, we begin by defining certain symmetric polynomials. As before, let ξ = diag(p^{a_{1}}, …, p^{a_{n}}, p^{b_{1}}, …, p^{b_{n}}) ∈ GSp_{n}^{+}(ℚ). Recall this means there is an integer μ so that μ = a_{i} + b_{i} for all i. Define t(ξ) to be the sum of the images of (x_{0}^{μ}x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}}) under the action of the (spherical Weyl) group W_{n}. We denote this as

t(ξ) = ∑_{w∈W_{n}}(x_{0}^{μ}x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}})^{w}

Once again, we remind the reader that this sum is really over the classes of monomials x_{0}^{μ}x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}} (modulo x_{0}^{2}x_{1} ⋯ x_{n} = 1) under the action of the Weyl group W_{n}. Since x_{0}^{2}x_{1} ⋯ x_{n} is fixed by all elements of W_{n}, this definition is well-defined.

Loosely, speaking Φ should map ΓξΓ to a multiple of t(ξ). The key to defining this homomorphism is the following identification. We know that W_{n} ≅ 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 Γ (see 10.1 of [3]).

Remark 4.1. The isomorphism W_{n} ≅ 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}, …, s_{n} generate the spherical Weyl group C_{n}, while s_{1}, …, s_{n+1} generate the affine Weyl group C̃_{n}.

The group W_{n} is generated by the τ_{i} and the permutations in the symmetric group S_{n}. With obvious identifications, we see that τ_{j} = (i j)τ_{i}(i j) and since S_{n} is generated by the transpositions {(1 2), (2 3), …, (n − 1 n)}, W_{n} = ⟨τ_{1}, (1 2), (2 3), …, (n − 1 n)⟩. One identifies τ_{1} with s_{1}, and (j − 1 j) with s_{j} (2 ≤ j ≤ 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 [3]) gives a Coxeter system.

By definition,

T_{B}(ξ)([L]) = ∑_{γ_{ν}∈(ξ^{−1}Γξ∩Γ)\Γ}[Lξγ_{ν}]

So identifying W_{n} with a subgroup of Γ, we may write Γ as a union of double cosets: Γ = ∪(ξ^{−1}Γξ ∩ Γ)δ_{μ}W_{n}, so the expression above becomes

T_{B}(ξ)([L]) = ∑_{δ_{μ}∈(ξ^{−1}Γξ∩Γ)\Γ/W_{n}} ∑_{w∈W_{n}}[Lξδ_{μ}w]

Given that W_{n} acts invariantly on any given apartment, we see that for a fixed μ, the collection of vertices [Lξδ_{μ}w] as w runs over W_{n} consists (with multiplicity [W_{n} : Stab(Lξδ_{μ})]) of all the vertices [M] in a given apartment with {L : M} = ξ. Moreover, since every apartment in the building is isomorphic (with structure determined by the Coxeter group C̃_{n}), all of the multiplicities [W_{n} : Stab(Lξδ_{μ})] are the same, independent of μ.

From the remark above it is now clear that after an appropriate change of frame (in particular a change of a basis), t(ξ) = ∑_{w∈W_{n}}(x_{0}^{μ}x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}})^{w} can be naturally identified with summand ∑_{w∈W_{n}}[Lξδ_{μ}w] occuring in

T_{B}(ξ)([L]) = ∑_{δ_{μ}∈(ξ^{−1}Γξ∩Γ)\Γ/W_{n}} ∑_{w∈W_{n}}[Lξδ_{μ}w]

If we define η(ξ) = #(ξ^{−1}Γξ ∩ Γ)\Γ/W_{n}, we can identify T_{B}(ξ) with η(ξ)t(ξ), thus we define Φ(ΓξΓ) = η(ξ)t(ξ).

To define the connecting homomorphism ρ : ℚ[x_{0}^{±1}, x_{1}^{±1}, …, x_{n}^{±1}]^{W_{n}} → End(B), we observe that any polynomial p ∈ ℚ[x_{0}^{±1}, x_{1}^{±1}, …, x_{n}^{±1}]^{W_{n}}, can be written as a sum ∑_{ξ} c(ξ)t(ξ) where the ξ are a set of representatives of the monomials occuring in p modulo W_{n}, so we only have to define ρ on a basis. We define ρ(t(ξ)) = η(ξ)^{−1}T_{B}(ξ), so that ρ ∘ Φ = Ψ.

To prove that Φ is a ring homomorphism, it is convenient to think of Φ as the restriction to H_{p} of a linear map Φ : L(Γ, G) → ℚ[x_{0}^{±1}, x_{1}^{±1}, …, x_{n}^{±1}]^{W_{n}}. Recall that in this setting we view the double coset ΓξΓ as the sum of right cosets ∑ Γξ_{ν} (ΓξΓ = ∪Γξ_{ν}) which is right invariant under Γ. While the map Φ (with domain L(Γ, 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 Φ to take ΓξΓ to η(ξ)t(ξ) so that there are as many monomials (counting multiplicities) as right cosets ΓξΓ = ∪Γξ_{ν}. Since any right coset Γξ_{ν} determines the same double coset, we must map any right coset to the same polynomial. From this, the definition is clear: Φ(Γξ_{ν}) = |W_{n}|^{−1} ∑_{w∈W_{n}}(x^{ξw}). Then extending linearly yields the desired Φ(ΓξΓ) = η(ξ)t(ξ).

Theorem 4.2. The correspondence ΓξΓ ↦ η(ξ)t(ξ) is a surjective homomorphism Φ : H_{p} → ℚ[x_{0}^{±1}, x_{1}^{±1}, …, x_{n}^{±1}]^{W_{n}}, whose kernel consists of double cosets of the form ΓξΓ with ξ = p^{μ}I_{2n}, μ ∈ ℤ.

Proof. First note that since ℚ[x_{0}^{±1}, x_{1}^{±1}, …, x_{n}^{±1}]^{W_{n}} = ℚ[x_{0}, …, x_{n}]^{W_{n}}[(x_{0}^{2}x_{1} ⋯ x_{n})^{−1}] (and we are working modulo x_{0}^{2}x_{1} ⋯ x_{n} = 1), the codomain is spanned as a vector space by elements of the form t(ξ), from which the surjectivity is immediate. Let ΓαΓ, ΓβΓ ∈ H_{p}, and write (as a disjoint union)

ΓαΓ = ⋃_{α_{i}∈(α^{−1}Γα∩Γ)\Γ/W_{n}} ⋃_{w∈W_{n}} Γαα_{i}w and
ΓβΓ = ⋃_{β_{j}∈(β^{−1}Γβ∩Γ)\Γ/W_{n}} ⋃_{w′∈W_{n}} Γββ_{j}w′.

Without loss of generality, we are assuming that α and β are diagonal. Then in H_{p}, (ΓαΓ)(ΓβΓ) = ∑_{i,j} Γαα_{i}wββ_{j}w′ = ∑_{ξ} c(ξ)ΓξΓ. Our first task is to determine the ξ which can occur. Since β_{j}, w′ ∈ Γ, it is clear that Γαα_{i}wββ_{j}w′Γ = Γαα_{i}wβΓ, however it it also true that Γαα_{i}wβΓ = ΓαwβΓ. To see the latter we deal with symplectic elementary divisors (Lemma 2.3). Recall that elementary divisors {L : Lξ} are uniquely determined by the double coset ΓξΓ. We have that

{L : Lαα_{i}wβ} = {Lβ^{−1}w^{−1} : Lαα_{i}},

and we note that L = Lα_{i} = Lα_{i}^{−1} since α_{i} ∈ Γ. Moreover, α_{i}^{−1}αα_{i} and α are simply two matrix representations of the same linear transformation acting on L. Thus

{L : Lαα_{i}wβ} = {Lβ^{−1}w^{−1} : Lα_{i}^{−1}αα_{i}} = {Lβ^{−1}w^{−1} : Lα} = {L : Lαwβ}

Thus (ΓαΓ)(ΓβΓ) = ∑_{w∈W_{n}} d(w)ΓαwβΓ (where the double cosets may not be distinct). Here d(w) is easily computed from the above observations:

d(w) deg(Γαwβ) = η(α)η(β)|W_{n}|,

where deg(Γαwβ) is the number of right cosets in the double coset, that is, deg(ΓξΓ) = # (ξ^{−1}Γξ ∩ Γ) \Γ.

It follows that

Φ ((ΓαΓ)(ΓβΓ)) = ∑_{w} d(w)η(αwβ)t(αwβ)

= ∑_{w} η(α)η(β)|W_{n}|(deg(ΓαwβΓ))^{−1}η(αwβ)t(αwβ)

= η(α)η(β) ∑_{w} t(αwβ).

We note that since α and β are diagonal and w is a “signed” permutation, the product αwβ is diagonal so that t(αwβ) does indeed make sense. On the other hand

Φ(ΓαΓ)Φ(ΓβΓ) = η(α)η(β)t(α)t(β)

= η(α)η(β)(∑_{w} x^{αw})(∑_{w′} x^{βw′})

= η(α)η(β)(∑_{w} ∑_{w′} x^{αwβw′}) (see remark 3.9)

= η(α)η(β) ∑_{w}(∑_{w′} x^{(αwβ)w′})

= η(α)η(β) ∑_{w} t(αwβ)

Thus Φ is a ring homomorphism. To compute its kernel, suppose that Φ(∑_{ξ} c(ξ)ΓξΓ) = ∑_{ξ} c(ξ)η(ξ)t(ξ) is trivial, that is to say (as polynomials)

∑_{ξ} c(ξ)η(ξ) ∑_{w∈W_{n}} x^{ξw} = (x_{0}^{2}x_{1} ⋯ x_{n})^{ℓ} for some integer ℓ

From this we immediately infer that there can only be one ξ and that x^{ξ} = x^{ξw} for all w ∈ W_{n}. It is trivial to check that if ξ corresponds to the class of the lattice [p^{a_{1}}, …, p^{a_{n}}; p^{b_{1}}, …, p^{b_{n}}] with μ = a_{i}+b_{i} for all i, then the monomial x^{ξ} = x_{0}^{μ}x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}} is fixed by all elements of W_{n} iff x^{ξ} = (x_{0}^{2}x_{1} ⋯ x_{n})^{ℓ} for some integer ℓ. Then ∑_{ξ} c(ξ)η(ξ) ∑_{w∈W_{n}} x^{ξw} = c(ξ)η(ξ)|W_{n}|x^{ξ} = x^{ξ} and ξ = p^{ℓ}I_{2n}. Since deg(Γp^{ℓ}I_{2n}Γ) = 1, η(ξ) = |W_{n}|^{−1}, so c(ξ) = 1 which establishes the result.

Remark 4.2. The image under Φ of ΓpIΓ is (the class of) is x_{0}^{2}x_{1} ⋯ x_{n}. For comparison, one should note that for a simultaneous Hecke eigenform of weight k for Γ = Γ_{n}, the associated Satake p-parameters (α_{0}, α_{1}, …, α_{n}) ∈ ℂ^{n+1}/W_{n} satisfy α_{0}^{2}α_{1} ⋯ α_{n} = p^{nk−n(n+1)/2}, and also that under the Satake isomorphism (Andrianov’s spherical map Ω), the image of ΓpIΓ is p^{−n(n+1)/2}x_{0}^{2}x_{1} ⋯ x_{n}.

Corollary 4.3. The connecting map ρ : ℚ[x_{0}^{±1}, …, x_{n}^{−1}]^{W_{n}} → End(B) defined by ρ(t(ξ)) = η(ξ)^{−1}T_{B}(ξ) is an injective ring homomorphism. satisfying ρ ∘ Φ = Ψ.

Proof. Both Φ and Ψ have the same kernel K, so induce injective homomorphisms from Φ̃ : H_{p}/K → ℚ[x_{0}^{±1}, …, x_{n}^{−1}]^{W_{n}}, and Ψ̃ : H_{p}/K → End(B). Since, Φ and hence Φ̃ is surjective, ρ = Ψ̃ ∘ Φ̃^{−1} has the desired properties.

Remark 4.4. We remark that while closely related, there is a difference between the spherical map Ω defined by Andrianov [1] and our map Φ.

Let ξ_{0} = (\begin{matrix}I_{2} & 0 \\ 0 & pI_{2} \end{matrix}) and ξ_{k} = (\begin{matrix}I_{n−k} & 0 & 0 & 0 \\ 0 & pI_{k} & 0 & 0 \\ 0 & 0 & p^{2}I_{n−k} & 0 \\ 0 & 0 & 0 & pI_{k} \end{matrix}), 1 ≤ k ≤ n. By Lemma 3.13,

t(ξ_{0}) = ∑_{w∈W_{n}} x^{ξ_{0}w} = |W_{n}|[W_{n} : Stab(x_{0})]^{−1} ∑_{ε_{i}=0,1} x_{0}x_{1}^{ε_{1}} ⋯ x_{n}^{ε_{n}} = \frac{|W_{n}|}{2^{n}}t_{0}(p),

so that

Φ(T (p)) = η(ξ_{0})t(ξ_{0}) = \frac{deg(ξ_{0})}{|W_{n}|} \frac{|W_{n}|}{2^{n}} ∑_{ε_{i}=0,1} x_{0}x_{1}^{ε_{1}} ⋯ x_{n}^{ε_{n}} = \frac{deg(ξ_{0})}{2^{n}}Ω(T (p)) = \frac{deg(ξ_{0})}{2^{n}}t_{0}(p).

As mentioned earlier for 1 ≤ k < n, T_{n−k}(p^{2}) corresponds to t_{k}(p), so we have

t(ξ_{k}) = ∑_{w∈W_{n}} x^{ξ_{k}w} = |W_{n}|[W_{n} : Stab(ξ_{k})]^{−1} ∑_{1≤i_{1}≤···≤i_{k}≤n, δ_{i_{j}} =±1} x_{i_{1}}^{δ_{i_{1}}} ⋯ x_{i_{k}}^{δ_{i_{k}}} = \frac{|W_{n}|}{2^{k}\binom{n}{k}}t_{k}(p),

so that

Φ(T_{n−k}(p^{2})) = η(ξ_{k})t(ξ_{k}) = \frac{deg(ξ_{k})}{|W_{n}|} \frac{|W_{n}|}{2^{k}\binom{n}{k}}t_{k}(p) = \frac{deg(ξ_{k})}{2^{k}\binom{n}{k}}t_{k}(p)

Computations with the spherical map are far less trivial. For comparison purposes we compute Ω(T_{1}^{2}(p^{2})). We have already shown that

Φ(T_{1}^{2}(p^{2})) = \frac{deg(ξ_{1})}{4}t_{1}(p) = \frac{deg(ξ_{1})}{4}(x_{1} + x_{1}^{−1} + x_{2} + x_{2}^{−1})

while

Ω(T_{2}^{2}(p^{2})) = Ω(pI_{4}) = p^{−3}x_{0}^{2}x_{1}x_{2}

and

Ω(T_{1}^{2}(p^{2})) = p^{−1}(x_{0}^{2}x_{1} + x_{0}^{2}x_{2} + x_{0}^{2}x_{1}^{2}x_{2} + x_{0}^{2}x_{1}x_{2}^{2}) + p^{−2}(p^{2} − 1)(x_{0}^{2}x_{1}x_{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)]
= Ω(T_{2}^{2}(p^{2}))[p^{2}t_{1}(p) + (p^{2} − 1)]

So it is clear that the action of Ω is somewhat more complicated, intertwining the Hecke operators in comparison to the action of Φ.

References

[1] A. N. Andrianov and V. G. Zhuravlëv, Modular forms and Hecke operators, American Mathematical Society, Providence, RI, 1995, Translated from the 1990 Russian original by Neal Koblitz. MR 96d:11045

[2] P. Cartier, Representations of p-adic groups: a survey, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–155. MR 81e:22029

[3] Paul Garrett, Buildings and classical groups, Chapman & Hall, London, 1997. MR 98k:20081

[4] J. Hafner and L. Walling, Siegel modular forms and Hecke operators in degree 2, Millennial Conference on Number Theory, (to appear).

[5] Mark Ronan, Lectures on buildings, Academic Press Inc., Boston, MA, 1989. MR 90j:20001

[6] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971, Kanô Memorial Lectures, No. 1. MR 47 #3318

[7] T. Veenstra, Siegel modular forms, l-functions, and Satake parameters, J. Number Th. (to appear).

Department of Mathematics, 6188 Bradley Hall, Dartmouth College, Hanover, NH 03755

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E-mail address: thomas.r.shemanske@dartmouth.edu
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