BRUHAT-TITS BUILDINGS AND HECKE OPERATORS

JOHN A. RHODES AND THOMAS R. SHEMANSKE June 12, 1998 10:52

Abstract. Serre introduced the notion of a Hecke operator acting on the vertices of a tree (the Bruhat-Tits building for SL_{2}(ℚ_{p}) ). In this paper we give a natural characterization of an algebra of Hecke operators acting on the vertices of the Bruhat-Tits building for SL_{n}(ℚ_{p}). This algebra of operators arises as a representation of an abstract Hecke algebra which, from work of Andrianov, is known to be isomorphic to the ring of symmetric polynomials in n variables. We show that this ring of symmetric polynomials acts in a completely natural way on the apartments of the building. Moreover, the abstract Hecke ring gives rise to n families of Hecke operators whose generating series produce “Euler factors” for various exterior powers. As a byproduct of our work, we give a natural characterization of Andrianov’s spherical map by which he established this isomorphism between the abstract Hecke ring and the ring of symmetric polynomials.

1. Introduction

We are interested in characterizing an algebra of Hecke operators which act on the free abelian group generated by the vertices in the Bruhat-Tits building for SL_{n}(ℚ_{p}). This algebra of operators not only acts naturally in terms of the geometry of the building, but also is a representation space for the (p-part of the) standard Hecke algebra given in terms of double cosets. As a consequence, we obtain a geometrically motivated characterization of Andrianov’s spherical map through which it is shown that the p-part of the Hecke algebra is isomorphic to the ring of symmetric polynomials ℚ[x_{1}, …, x_{n}]^{sym}. Everything we do here will work just as well over any nonarchimedean local field, but for simplicity we restrict to ℚ_{p}.

We begin with some notation concerning buildings and a summary of Serre’s description of the Hecke operators when the building is a tree. Let V be an ndimensional vector space over ℚ_{p}, and Δ_{n} be the Bruhat-Tits building for SL_{n}(ℚ_{p}) (see [3] or [4]). The vertices of Δ_{n} are in one-to-one correspondence with homothety classes of lattices (free ℤ_{p}-modules) in V of rank n. By the elementary divisor theorem, given two lattices L and M, there exists a basis {e_{1}, …, e_{n}} of V and rational integers a_{1}, …, a_{n}, so that

L = ℤ_{p}e_{1} ⊕ ⋯ ⊕ ℤ_{p}e_{n}, M = ℤ_{p}p^{a_{1}}e_{1} ⊕ ⋯ ⊕ ℤ_{p}p^{a_{n}}e_{n}.

Date: June 12, 1998.

1991 Mathematics Subject Classification. Primary 11F25, 20E42; Secondary 11F60.

Key words and phrases. Bruhat–Tits Building, Hecke Operators.

Given this notation, Serre [6], in the case of n = 2, defines the distance between the two vertices represented by the lattices L and M as |a_{1} − a_{2}|. Generalizations for n > 2 appear in [5]. Using this distance function, Serre defines a graph by placing an edge between any two vertices which are unit distance apart. It then follows that the resulting graph (which is the Bruhat-Tits building for SL_{2}(ℚ_{p})) is a (p + 1)- regular tree, and that vertices which are distance m from a given vertex are simply the endpoints of walks without backtracking of length m.

Serre defines a family of ℤ-linear operators θ_{ℓ} which act on the free abelian group generated by the vertices of the tree by setting

θ_{ℓ}(v) = ∑_{d(v,w)=ℓ} w,

that is, a vertex is mapped to the sum of its neighbors at distance ℓ. From the structure of the tree, he notes that

θ_{1}θ_{1} = θ_{2} + (p + 1)θ_{0} and θ_{1}θ_{ℓ} = θ_{ℓ+1} + pθ_{ℓ−1} for ℓ ≥ 2.

Compared to the recursions satisfied by classical Hecke operators T (p^{ℓ}) (or merely in terms of simplicity of expression) the recursion is slightly off in the base case. To correct this, Serre defines

T_{0} = θ_{0}, T_{1} = θ_{1}, and T_{ℓ} = θ_{ℓ} + T_{ℓ−2} for ℓ ≥ 2,

which yields (for all ℓ ≥ 1) the relation

T_{1}T_{ℓ} = T_{ℓ+1} + pT_{ℓ−1}.

While a generating series for the θ_{ℓ} operators is not quite as simple, the T_{ℓ} operators satisfy

∑_{ℓ≥0} T_{ℓ}u^{ℓ} = [1 − T_{1}u + pu^{2}]^{−1} .

For obvious reasons, the operators T_{ℓ} are called Hecke operators. We observe that, while the Hecke operators satisfy a cleaner recursion than the θ_{ℓ}s, their definition is in some ways less satisfying, since the θ_{ℓ}s appear to be a more natural family of operators with respect to the structure of the tree.

In our generalizations to SL_{n}(ℚ_{p}) for n > 2, we provide motivation for why the operators T_{ℓ} are in fact quite natural.

2. Higher Rank Hecke Operators on Δ_{n}

With Serre’s characterization of the Hecke operators on SL_{2}(ℚ_{p}) so readily in hand, we give the generalization to the case of SL_{n}(ℚ_{p}). While the definition given here is, in and of itself, a natural generalization of Serre’s, we do not show until later how the corresponding algebra of these operators arises as the image of a representation of the abstract Hecke algebra given by double cosets. In later sections we also indicate a number of interesting “Euler” factors which arise by considering various generating series involving Hecke operators. On the other hand, we do take this opportunity to give an action of symmetric polynomials on the vertices of the building which will motivate the characterization of the isomorphism of the local Hecke algebra and the ring of symmetric polynomials established by Andrianov using the spherical map.

We begin with a little notation. As in the previous section, let V be an ndimensional vector space over ℚ_{p}, and let Δ_{n} be the Bruhat-Tits building for SL_{n}(ℚ_{p}). The vertices of Δ_{n} are in one-to-one correspondence with homothety classes of lattices in V of rank n, and we let B_{0} denote the free abelian group generated by the vertices of Δ_{n} and B = ℚ ⊗ B_{0} the corresponding vector space over ℚ.

For positive integers m and n, denote by P_{n}(m) the set of partitions of m into n pieces such that a given partition satisfies: m ≥ i_{1} ≥ ⋯ ≥ i_{n} ≥ 0 (and ∑ i_{k} = m). Endow P_{n}(m) with the lexicographic ordering. Fix a lattice L in V whose homothety class [L] is a vertex in Δ_{n}, and let Γ_{p} = GL_{n}(ℤ_{p}).

With the lattice L fixed as above, we recall [7] that there is a 1 – 1 correspondence between right cosets Γ_{p}ξ_{ν} in Γ_{p} diag(p^{a_{1}}, …, p^{a_{n}})Γ_{p} and lattices M in V = L ⊗ ℚ_{p} with elementary divisors {L : M} = {p^{a_{1}}, …, p^{a_{n}}}.

Thus, analogous to the action of a double coset on functions, it is natural to define the Hecke operator T_{B}(p^{a_{1}}, …, p^{a_{n}}) (acting on B_{0} (or B)) by

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

Also with i = (i_{1}, …, i_{n}) ∈ P_{n}(ℓ), we define

T_{B}(p^{ℓ}) = ∑_{i∈P_{n}(ℓ)} T_{B}(p^{i_{1}}, …, p^{i_{n}}).

To foreshadow a bit, it is important to note that since the ring structure of the local Hecke algebra is characterized [7] precisely in terms of elementary divisors of lattices on V, we will have for free that the generating series for the operators T_{B}(p^{ℓ}) is expressible as the same rational function as the generating series for the classical Hecke operators T (p^{ℓ}) (see Theorem 3.21 of [7]), namely

∑_{ℓ≥0} T_{B}(p^{ℓ})u^{ℓ} = [∑_{k=0}^{n}(−1)^{k}p^{k(k−1)/2}t̃_{k}^{n}(p)u^{k}]^{−1} ,

with the t̃_{k}^{n}(p) “elementary symmetric polynomials” defined in a subsequent section.

There is a corresponding definition of the Hecke operator on an apartment (in terms of classes of lattices):

T_{A}(p^{a_{1}}, …, p^{a_{n}})([L]) =\begin{matrix}∑ & ∗ \\ \{L:M\}=\{p^{a_{1}},... ,p^{an}\} \end{matrix}[M]

where the sum is over all lattices relative to the (unordered) basis {e_{1}, …, e_{n}} of L. As we shall see, it is actually the Hecke operator restricted to the apartment which is the most natural from the point of view of symmetric polynomials, and represented our starting point in this investigation.

The operators defined above which act on the building are perfectly natural. Moreover, we show as an example, that in the case n = 2, the operators defined above are precisely the operators T_{ℓ} defined by Serre. The notation is as in the introduction.

Example 2.1. For n = 2, we have T_{B}(p^{ℓ}) = ∑_{k=0}^{[ℓ/2]} T_{B}(p^{k}, p^{ℓ−k}).

Recall that Serre defined two operators

θ_{ℓ}([L]) = ∑_{d([L],[M])=ℓ}[M], ℓ ≥ 1; θ_{0} = 1
T_{ℓ}([L]) = θ_{ℓ} + T_{ℓ−2}, ℓ ≥ 2; T_{0} = θ_{0}, T_{1} = θ_{1}

By Serre’s definition of distance,

d([L], [M]) = ℓ iff {L : M} = {p^{a}, p^{b}} with ℓ = |a − b|

Since [M] = [p^{k}M] we have that

θ_{ℓ}([L]) = ∑_{{L:M}={1,p^{ℓ}}}[M]

Claim: T_{ℓ} = T_{B}(p^{ℓ}) for all ℓ ≥ 0. We proceed by induction on ℓ. For ℓ = 0

this is trivial. For ℓ = 1, we observe that T_{1}([L]) = θ_{1}([L]) = ∑_{{L:M}={1,p}}[M] = T_{B}(1, p)([L]) = T_{B}(p)([L]).

Now assume that ℓ ≥ 2.

T_{ℓ}([L]) = θ_{ℓ}([L]) + T_{ℓ−2}([L])

= ∑_{{L:M}={1,p^{ℓ}}}[M] + T_{B}(p^{ℓ−2})([L]) by induction

= T_{B}(1, p^{ℓ})([L]) + ∑_{k=0}^{[(ℓ−2)/2]} T_{B}(p^{k}, p^{ℓ−2−k})([L])

= T_{B}(1, p^{ℓ})([L]) + ∑_{k=0}^{[(ℓ−2)/2]} T_{B}(p, p)T_{B}(p^{k}, p^{ℓ−2−k})([L])

= T_{B}(1, p^{ℓ})([L]) + ∑_{k=0}^{[(ℓ−2)/2]} T_{B}(p^{k+1}, p^{ℓ−1−k})([L])

= T_{B}(1, p^{ℓ})([L]) + ∑_{k=1}^{[ℓ/2]} T_{B}(p^{k}, p^{ℓ−k})([L])

= ∑_{k=0}^{[ℓ/2]} T_{B}(p^{k}, p^{ℓ−k})([L]) = T_{B}(p^{ℓ})([L]).

Now we introduce a labeling of the apartments in a building which will make natural an action of symmetric polynomials on the apartment, and hence provide the insight for connecting the local Hecke algebra with the ring of symmetric polynomials.

To specify an apartment in Δ_{n}, choose an unordered set of n one-dimensional subspaces V_{1}, V_{2}, …, V_{n}, such that V = V_{1} ⊕ V_{2} ⊕ ⋯ ⊕ V_{n}. The vertices in the apartment can be viewed as homothety classes of lattices L = ℤ_{p}v_{1} ⊕ ⋯ ⊕ ℤ_{p}v_{n} with v_{i} ∈ V_{i} for each i. We denote by A_{0} the free abelian group generated by the vertices in this apartment, and A the corresponding vector space over ℚ.

Now fix an ordering of the chosen subspaces V_{i}, which we will refer to as an orientation of the apartment. This orientation will allow us define first a labeling of the vertices in the apartment by monomials, and then a polynomial action on A. To begin, fix a vertex v in the apartment, and let v correspond to the class of the lattice L_{0} = ℤ_{p}e_{1} ⊕ ⋯ ⊕ ℤ_{p}e_{n}. Then the vertices of the apartment are in one-to-one correspondence with the classes of lattices ℤ_{p}p^{a_{1}}e_{1} ⊕ ⋯ ⊕ ℤ_{p}p^{a_{n}}e_{n}, where the a_{i} run over ℤ. Focusing attention on the ordered n-tuple (a_{1}, a_{2}, …, a_{n}), the vertices of the apartment are thus in one-to-one correspondence with the elements of ℤ^{n}/ℤ · (1, 1, …, 1).

Following [3], if u = (u_{1}, …, u_{n}) and v = (v_{1}, …, v_{n}) are in ℤ^{n}, write

u ⪯ v if u_{i} ≤ v_{i} ≤ u_{i} + 1 for all i.

Calling two elements of ℤ^{n}/ℤ · (1, 1, …, 1) incident if they admit representatives u and v with u ⪯ v then produces a flag complex which defines the full simplicial structure of the apartment. It is clear that for every coset in ℤ^{n}/ℤ · (1, 1, …, 1), there is a unique representative (a_{1}, …, a_{n}) in which all entries are nonnegative and at least one is zero. Using this representative, we label the corresponding vertex with the monomial x_{1}^{a_{1}}x_{2}^{a_{2}} … x_{n}^{a_{n}}. As indicated by the structure of a flag complex, this labelling is far from arbitrary.

Consider our fixed vertex v represented by the lattice L_{0} = ℤ_{p}e_{1} ⊕ ⋯ ⊕ ℤ_{p}e_{n}, and hence by the monomial 1. Any chamber (i.e., (n − 1)-simplex) containing v corresponds to a maximal flag of lattices:

pL_{0} = L_{n} ⊊ L_{n−1} ⊊ ⋯ ⊊ L_{1} ⊊ L_{0}.

Fix such a (fundamental) chamber, by choosing

L_{i} = ℤ_{p}pe_{1} ⊕ ⋯ ⊕ ℤ_{p}pe_{i} ⊕ ℤ_{p}e_{i+1} ⊕ ⋯ ⊕ ℤ_{p}e_{n}.

This chamber then has vertices labeled 1, x_{1}, x_{1}x_{2}, …, x_{1}x_{2} ⋯ x_{n−1}.

To illustrate the labeling, consider an apartment for SL_{2}:

…  • y^{3}  • y^{2}  • y  • 1  • x  • x^{2}  • x^{3}  …

We fix a vertex and label it 1, and then label a fundamental chamber (determining vertex x). Using the incidence relation defined above, it is clear that x is incident to 1, x^{2} to x, x^{3} to x^{2}, and so on. This labels the right hand side of the apartment. It is also clear that y is incident to 1, y^{2} to y, and so on, which uniquely determines the left hand side.

Next consider a piece of a labeled apartment for SL_{3}:

[figure: y^{2}; xy^{2}; x^{2}y^{2}; x^{3}y^{2}; y^{2}z; y; xy; x^{2}y; x^{3}y; yz; 1; x; x^{2}; yz^{2}; z; xz; x^{2}z; x^{3}z; z^{2}; xz^{2}; x^{2}z^{2}; x^{3}z^{2}]

As before, we fix a vertex and label it 1. Choosing a fundamental chamber fixes vertices labeled x and xy. It is clear that x and y are the only two vertices incident to both 1 and xy, hence the vertex labeled y is determined. z is also incident to 1, but not to y or x, which determines its placement. Loosely speaking, we see that there are three “principal directions” indicated by a motion from 1 to x, y or z. We label the vertices as follows: starting from 1 and moving in the x-direction, we label the vertices x, x^{2}, x^{3}, …. Similarly for movements in the y and z directions. The incidence relation says that the vertex xy is labeled as such because to get there from 1 we move one unit in the x direction and by one unit in the y direction.

For an apartment for SL_{n}, we choose an orientation and a vertex to be labeled 1, and thus a fundamental chamber in the manner described above. This has fixed the labelling of 1, x_{1}, x_{1}x_{2}, …, x_{1}x_{2} ⋯ x_{n−1}. From here the rest of the labelling can be deduced from incidence relations. For consider the residue of the vertex 1, that is the collection of chambers which contain the vertex 1. It is well known that this is a spherical building of type A_{n} containing n! chambers. Within the residue consider the n − 1 chambers sharing a common codimension 1 face with the given chamber. Each such face is determined by n − 1 vertices, and hence the nth vertex defining such a chamber is uniquely determined, and hence so is the class of the lattice which represents that vertex. Using the incidence relation, we may continue in this way to label the entire apartment.

Now we note that there is a natural action of monomials on the vertices of the apartment. For nonnegative integers a_{i} and indeterminates x_{i}, we say that a vertex w is a x_{1}^{a_{1}}x_{2}^{a_{2}} ⋯ x_{n}^{a_{n}}-translate of v if there exists lattices L_{1} ∈ v and L_{2} ∈ w with L_{2} ⊂ L_{1} and {L_{1} : L_{2}} = {p^{a_{1}}, p^{a_{2}} …, p^{a_{n}}}, where the latter notation represents the elementary divisors of L_{2} relative to L_{1}. Note that since vertices are defined by classes of lattices, any vertex v is an x_{1} ⋯ x_{n} - translate of itself, i.e. the x_{1}x_{2} ⋯ x_{n} - translate of v = [L] is [pL] = v. It is also obvious that the composition of translates corresponding to n-tuples of nonnegative integers (a_{1}, a_{2} …, a_{n}) and (b_{1}, b_{2} …, b_{n}) is equal to the translate corresponding to (a_{1} + b_{1}, a_{2} + b_{2}, …, a_{n} + b_{n}), and that translates exist and are unique. However, if w is an x_{1}^{a_{1}}x_{2}^{a_{2}} ⋯ x_{n}^{a_{n}}-translate of v, the n-tuple (a_{1}, a_{2}, …, a_{n}) is determined only up to multiples of (1, 1, …, 1).

Using the monomial labeling of the apartment, if w is an x_{1}^{a_{1}}x_{2}^{a_{2}} ⋯ x_{n}^{a_{n}}-translate of v, then to go from v to w along the edges of the apartment, proceed a_{i} units in each of the x_{i} directions starting from v. Alternately the x_{1}^{a_{1}}x_{2}^{a_{2}} ⋯ x_{n}^{a_{n}}-translate of the vertex labeled x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}} is labeled by the product of these two monomials, modulo the relation x_{1}x_{2} ⋯ x_{n} = 1.

This notion of translation extends to an action of the polynomial ring ℚ[x_{1}, x_{2}, …, x_{n}] on A. Notice that this is independent of our choice of which vertex is labeled 1, since a change in that choice merely multiplies all labels by a fixed monomial. However, the action does depend on our choice of an orientation. We can however pass to a ring of orientation-independent operators by restricting to a smaller class of polynomials.

Since an orientation is defined by an ordering of the subspaces V_{1}, V_{2}, …, V_{n}, possible orientations correspond to elements of S_{n}. One readily sees that changing orientations affects our monomial labeling by σ ∈ S_{n} acting on the variables x_{1}, x_{2}, …, x_{n}. Thus orientation-independent operators are simply symmetric polynomials, so the connection to Andrianov’s work is becoming visible. To proceed, we need to work through some of the formalism of the local Hecke algebras.

3. Abstract Hecke Algebras

One goal of our work is to show that the p-part of the global Hecke algebra is naturally isomorphic to a ring of symmetric polynomials, and that the algebra of Hecke operators which act on the vertices of the Bruhat-Tits building is simply a representation of this standard Hecke algebra.

To begin, we review the forms of the Hecke theory given by Shimura [7] and by Andrianov and Zhuravlev [2], and then relate these to that of the Hecke algebra defined over a local field. Since these connections are mostly straightforward, detail will be minimized.

3.1. Global Hecke Algebras. The Hecke algebra described by Shimura [7] is given in terms of double cosets with respect to the discrete group SL_{n}(ℤ), while the algebra described by Andrianov and Zhuravlev [2] is given in terms of formal sums of right cosets which are invariant under right action by the discrete group GL_{n}(ℤ).

Let Γ be a subgroup of a group G and S ⊃ Γ, a semigroup contained in the commensurator of Γ in G. For ξ ∈ S, let

ΓξΓ = ∪Γξ_{ν}

be the decomposition of the double coset into disjoint right cosets. Andrianov denotes by (ξ) the formal sum ∑ Γξ_{ν}. In Shimura’s definition the Hecke algebra H = H(Γ, S) is the free ℤ-module generated by such ΓξΓ, while for Andrianov and Zhuravlev the generators are the (ξ). Since multiplication in the respective Hecke rings is defined in exactly the same way, obviously the map ΓξΓ ↦ (ξ) induces an isomorphism between the respective rings. We will therefore feel free to use either of the notations ΓξΓ or (ξ) as is most convenient in what follows.

In [7] Shimura studies a Hecke ring H with respect to Γ = SL_{n}(ℤ), and S = M_{n}^{+}(ℤ) (n × n integral matrices with positive determinant), while Andrianov focuses on Γ = GL_{n}(ℤ) and S = GL_{n}(ℚ). However, the map H(SL_{n}(ℤ), M_{n}^{+}(ℤ)) → H(GL_{n}(ℤ), GL_{n}(ℚ)) defined by (ξ) ↦ (ξ) factors as

H(SL_{n}(ℤ), M_{n}^{+}(ℤ)) → H(SL_{n}(ℤ), GL_{n}^{+}(ℚ)) → H(GL_{n}(ℤ), GL_{n}(ℚ)),

where the map on the left is an injection, and that on the right an isomorphism.

Draft Remark 1. The left hand map being an injection is clear. For the right, we use Proposition III.1.9 of [2].

Lemma 3.1. The rational Hecke algebra R(SL_{n}(ℤ), GL_{n}^{+}(ℚ)) is isomorphic to R(GL_{n}(ℤ), GL_{n}(ℚ)).

Proof. In the notation of [2], let Γ_{0} = SL_{n}(ℤ), Γ = GL_{n}(ℤ), S_{0} = GL_{n}^{+}(ℚ) and S = GL_{n}(ℚ). We note that the conditions in equation (1.26) are satisfied:

1. Γ_{0} ⊂ Γ

2. S ⊂ ΓS_{0}

3. Γ ∩ (S_{0} · S_{0}^{−1}) ⊂ Γ_{0} the last since the elements of (S_{0} · S_{0}^{−1}) have positive determinant. Given a right coset Γs (s ∈ S), we have that Γs = Γs_{0} for some s_{0} ∈ S_{0} by (2). It follows (see [2] that the map ΓsΓ ↦ Γ_{0}s_{0}Γ_{0} is a monomorphism of rings. Since S_{0} ⊂ S, this map will be an isomorphism if we show that the degree of Γs_{0}Γ is the same as the degree of Γ_{0}s_{0}Γ_{0} for all s_{0} ∈ S_{0}. That is, we must show that

[Γ : Γ ∩ s_{0}^{−1}Γs_{0}] = [Γ_{0} : Γ_{0} ∩ s_{0}^{−1}Γ_{0}s_{0}]

for all s_{0} ∈ S_{0}. Consider the map

Γ_{0} ↪ Γ → (Γ ∩ s_{0}^{−1}Γs_{0})\Γ given by A ↦ (Γ ∩ s_{0}^{−1}Γs_{0}) A

If A is in the kernel of this homomorphism, then A ∈ Γ_{0} ∩ s_{0}^{−1}Γs_{0}, so A = s_{0}γs_{0}^{−1} for some γ ∈ Γ. But then det(γ) = det(A) = 1, hence γ ∈ Γ_{0} which yields that A ∈ Γ_{0} ∩ s_{0}^{−1}Γ_{0}s_{0}. It follows immediately that the kernel is equal to Γ_{0} ∩ s_{0}^{−1}Γ_{0}s_{0}, so we have an injection of (Γ_{0} ∩ s_{0}^{−1}Γ_{0}s_{0})\Γ_{0} into (Γ ∩ s_{0}^{−1}Γs_{0})\Γ. To show surjectivity, let B ∈ Γ. We must show that there exists an A ∈ Γ_{0} so that (Γ∩s_{0}^{−1}Γs_{0})B = (Γ∩s_{0}^{−1}Γs_{0})A. We are concerned with the degrees of the double cosets Γs_{0}Γ and Γ_{0}s_{0}Γ_{0}, so without loss we may assume that s_{0} is diagonal. Thus s_{0} diag(−1, 1, …, 1)s_{0}^{−1} = diag(−1, 1, …, 1), from which we deduce that diag(−1, 1, …, 1) ∈ (Γ ∩ s_{0}^{−1}Γs_{0}) and hence if B ∉ Γ_{0}, we may take A = diag(−1, 1, …, 1)B. This completes the proof.

A refinement of this leads to

Proposition 3.2. H(GL_{n}(ℤ), GL_{n}(ℚ)) is generated, as a ring, by H(SL_{n}(ℤ), M_{n}^{+}(ℤ)) together with the elements (p^{−1}I_{n}) for all primes p. (I_{n} denotes the n × n identity matrix.)

Draft Remark 2. something similar is lemma 2.15 of [2].

Setting

H^{n} = H(GL_{n}(ℤ), GL_{n}(ℚ))
\underline{H}^{n} = H(SL_{n}(ℤ), M_{n}^{+}(ℤ)),

we therefore view \underline{H}^{n} as the integral subring of H^{n}.

While we have defined Hecke algebras H = H(Γ, S) only as ℤ-modules, obviously we can choose coefficients from any ring. In particular, the associated rational algebra is H_{ℚ} = ℚ ⊗ H. In an abuse of notation we may simply denote this H as well, with the context making clear that scalars are in ℚ.

Draft Remark 3. ******** we need to watch to see if we ever do this **********

3.2. Local Hecke Algebras. Henceforth, we shall be concerned with the p-part of the Hecke algebra H^{n}. In the notation of [2], this is defined by

H_{p}^{n} = H(GL_{n}(ℤ), GL_{n}(ℤ[p^{−1}])) ⊂ H^{n}.

The subrings H_{p}^{n} for all primes p generate H^{n}.

The integral subring \underline{H}_{p}^{n} of H_{p}^{n} is

\underline{H}_{p}^{n} = ⟨(ξ) | ξ ∈ M_{n}(ℤ), det(ξ) = ±p^{λ}⟩
= ⟨(diag(p^{i_{1}}, …, p^{i_{n}})) | 0 ≤ i_{1} ≤ ⋯ ≤ i_{n}⟩

where the angle brackets enclose generators as a ℤ-module. As a ring, H_{p}^{n} is generated by \underline{H}_{p}^{n} together with the single element (p^{−1}I_{n}).

Thus the study of the the global Hecke ring H^{n} reduces to study of the local Hecke rings H_{p}^{n}, which in turn can be understood through their integral subrings \underline{H}_{p}^{n}. However, the remainder of this paper will focus on the p-adic Hecke ring

H_{p}^{n} = H(GL_{n}(ℤ_{p}), GL_{n}(ℚ_{p})),

and its integral subring

\underline{H}_{p}^{n} = ⟨(diag(p^{i_{1}}, …, p^{i_{n}})) | 0 ≤ i_{1} ≤ ⋯ ≤ i_{n}⟩,

which together with the single element (p^{−1}I_{n}) generates H_{p}^{n}. This is sufficient for studying the above algebras, though, since H_{p}^{n} (resp. \underline{H}_{p}^{n}) is isomorphic to H_{p}^{n} (resp. \underline{H}_{p}^{n}).

To see this isomorphism, let Γ = GL_{n}(ℤ), Γ_{p} = GL_{n}(ℤ_{p}), S = GL_{n}(ℤ[p^{−1}]), and S_{p} = GL_{n}(ℚ_{p}). Then

H_{p}^{n} = H(Γ, S) = ⟨Γ diag(p^{i_{1}}, p^{i_{2}}, …, p^{i_{n}})Γ | i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{n}⟩,

while

H_{p}^{n} = H(Γ_{p}, S_{p}) = ⟨Γ_{p} diag(p^{i_{1}}, …, p^{i_{n}})Γ_{p} | i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{n}⟩.

Now consider the obvious map H_{p}^{n} → H_{p}^{n} induced by taking ΓξΓ ↦ Γ_{p}ξΓ_{p} for ξ = diag(p^{i_{1}}, …, p^{i_{n}}). It is immediate that this is a vector space isomorphism, so we need only show it preserves multiplication. Since multiplication in the Hecke algebras is defined in terms of multiplication of right coset representatives, it suffices to establish the following:

Lemma 3.3. If α ∈ GL_{n}(ℤ[p^{−1}]) and ΓαΓ = ∪Γα_{i} (disjoint union), then

(3.1) Γ_{p}αΓ_{p} = ∪Γ_{p}α_{i} (disjoint union).

Proof. We may assume α has entries in ℤ, since the general case then follows by multiplication of α by p^{k} for some k ∈ ℤ. Note that for such an α each coset representative α_{i} has coefficients in ℤ and has determinant ±p^{λ}. Suppose that Γ_{p}α_{i} ∩ Γ_{p}α_{j} is non-empty. Then α_{i}α_{j}^{−1} ∈ Γ_{p}, has entries in ℤ[1/p] ∩ ℤ_{p}, and has determinant ±1, thus α_{i}α_{j}^{−1} ∈ Γ. Thus Γα_{i} = Γα_{j}, and so α_{i} = α_{j}. Therefore the union in (3.1) is disjoint.

To establish the equality in (3.1), note first that Γ_{p}α_{i} ⊆ Γ_{p}αΓ_{p} is clear. Thus we need only show that if β ∈ Γ_{p}αΓ_{p}, then β ∈ Γ_{p}α_{i} for some i. Without loss of generality, we may assume α is diagonal, since it may be brought into that form by multiplication on the left and right by elements of Γ.

Let β = γ_{1}αγ_{2} with γ_{1}, γ_{2} ∈ Γ_{p}. We may assume det(γ_{2}) = 1, since if otherwise, we can set ϵ = diag(det(γ_{2}), 1, 1, …, 1) ∈ Γ_{p} and write

γ_{1}αγ_{2} = γ_{1}αϵϵ^{−1}γ_{2} = (γ_{1}ϵ)α(ϵ^{−1}γ_{2}),

where we’ve used the fact that diagonal matrices commute. The expressions in parentheses then can be taken as the new γ_{1} and γ_{2}.

Choose a positive integer m such that p^{m}α^{−1} ∈ M_{n}(pℤ). By a density argument, pick γ_{3} ∈ Γ, δ ∈ M_{n}(ℤ_{p}), with γ_{2} = γ_{3} + p^{m}δ. Then

β = γ_{1}αγ_{2} = γ_{1}α(γ_{3} + p^{m}δ) = γ_{1}(I_{n} + p^{m}αδγ_{3}^{−1}α^{−1})αγ_{3}.

By the choice of m, the last expression in parentheses is in Γ_{p}. Thus β ∈ Γ_{p}αγ_{3}. Since Γαγ_{3} = Γα_{i} for some i, this means β ∈ Γ_{p}α_{i} .

Draft Remark 4. Here we give the required density argument. Let Γ_{p}^{±1} denote the subgroup of Γ_{p} consisting of matrices with determinant ±1. We show that Γ is dense in Γ_{p}^{±1}. Let γ_{1} ∈ Γ_{p}^{±1}. Then using elementary row and column operations, we can express γ_{1} as a product of permutation matrices, matrices E which look like the identity but with one off-diagonal entry in ℤ_{p}, and a diagonal matrix with determinant ±1. For any positive integer m we need to find a matrix γ_{2} ∈ Γ with γ_{1} ≡ γ_{2} mod p^{m}. To do this, it is sufficient to do so for each matrix mentioned above in the product expressing γ_{1}. Permutation matrices are already in Γ, and handling the E matrices is easy. The diagonal matrix is a little more involved. If D = diag(d_{1}, d_{2}, … d_{n}) where d_{i} ∈ ℤ_{p}^{×}, we first find a 2 × 2 matrix

in GL_{2}(ℤ) congruent to (\begin{matrix}d_{1} & 0 \\ 0 & d_{1}^{−1} \end{matrix}) mod p^{m}. We do this by picking

a ∈ ℤ, a ≡ d_{1} mod p^{m}, and then picking c, d ∈ ℤ so that ad − p^{2m}c = 1.

Then (\begin{matrix}a & p^{m} \\ p^{m}c & d \end{matrix}) ∈ GL_{2}(ℤ). Let D_{1} be the n × n matrix with this

2 × 2 block in the upper left and 1’s on the diagonal. Then, D_{1}^{−1}D ≡ diag(1, d_{1}^{−1}d_{2}, d_{3}, … d_{n}) mod p^{m}. Continuing ‘in this way’, we ultimately get D ≡ D_{1}D_{2} … D_{n} mod p^{n} where D_{i} ∈ Γ.

3.3. Generating Series. For motivational purposes we recall a generating series for certain elements of the local Hecke ring. As in the previous section, let H_{p}^{n} denote the p-part of the Hecke algebra H^{n}.

Let d_{k}^{n}(p) be the diagonal matrix diag(\underbrace{1, …, 1}_{n−k}, \underbrace{p, …, p}_{k}) and denote by T_{k}^{n}(p) the

double coset Γd_{k}^{n}(p)Γ in H_{p}^{n}. In the notation of [2] this is π_{k}^{n}(p) = (d_{k}^{n}(p)). Let

T (p^{ℓ}) = ∑_{\begin{matrix}α∈Γ\M_{n}(ℤ)/Γ \\ det α=±p^{ℓ} \end{matrix}} Γ α Γ ∈ H_{p}^{n}

The following is Theorem 3.21 of [7] and Proposition III.2.22 of [2]:

thm:globalhecke Theorem 3.4. The formal Hecke series ∑_{ℓ≥0} T (p^{ℓ})u^{ℓ} is a rational function given by:

∑_{ℓ≥0} T (p^{ℓ})u^{ℓ} = [∑_{k=0}^{n}(−1)^{k}p^{\frac{k(k−1)}{2}} T_{k}^{n}(p)u^{k}]^{−1} .

Further light is shed on this identity by exploring connections of the local Hecke algebra to an algebra of symmetric polynomials. Andrianov and Zhuravlev [2] give a map ω between the Hecke rings H_{p}^{n}, \underline{H}_{p}^{n} and certain rings of symmetric polynomials in n variables. Known as the spherical map, ω is first defined on right cosets and, after extension by linearity, is then restricted to the Hecke algebra. While its definition is more complicated than we wish to we state here, we do want to describe the action of the spherical map on certain Hecke operators as a means of motivating why one would study Hecke operators through symmetric polynomial rings.

Let s_{k} = s_{k}(x_{1}, x_{2}, …, x_{n}) = ∑_{1≤i_{1}<···<i_{k}≤n} x_{i_{1}}x_{i_{2}} ⋯ x_{i_{k}} be the kth elementary sym-

metric polynomial. Then under the spherical map ω, we have (Lemma 2.21 of [2] (Chapter 3)) that

ω(π_{k}^{n}(p)) = p^{\frac{−k(k+1)}{2}} s_{k}(x_{1}, …, x_{n}).

so that at least formally (and again identifying π_{k}^{n}(p) with T_{k}^{n}(p))

\begin{aligned}∑_{ℓ≥0} ω(T (p^{ℓ}))u^{ℓ} & = [∑_{k=0}^{n}(−1)^{k}p^{\frac{k(k−1)}{2}} ω(π_{k}^{n}(p))u^{k}]^{−1} \\ & = [∑_{k=0}^{n}(−1)^{k}p^{−k}s_{k}(x_{1}, …, x_{n})u^{k}]^{−1} \\ & = ∏_{i=1}^{n} (1 − p^{−1}x_{i}u)^{−1} . \end{aligned} (3.2)

This shows how the structure of the “Euler factor” of this generating series of Hecke operators is most easily expressed through the spherical map. Using the spherical map, Andrianov also shows that the p-part of the Hecke algebra, \underline{H}_{p}^{n}, is isomorphic to the full ring of symmetric polynomials in n variables. With such an intimate connection between Hecke operators and symmetric polynomials we are led to a more comprehensive study.

4. Symmetric polynomials

Based upon the nice action of the spherical map on the elements π_{k}^{n}(p), one might be inclined to investigate the image of double cosets under the spherical map as a means of characterizing “natural” Hecke operators in the ring of symmetric polynomials. In fact the image of most double cosets is not particularly attractive, so we take another approach.

First, we define a family of symmetric polynomials, denoted t_{k}^{n}(p^{ℓ}) and which we will call Hecke operators, whose associated generating series are rational functions — in particular kth exterior powers. We shall see that one subcollection of these operators, the t_{1}^{n}(p^{ℓ}), are essentially formal analogs of the classical Hecke operators T (p^{ℓ}). In particular, the generating series for both collections of operators give rise to essentially the same rational functions.

This section deals only with symmetric polynomials without explicit connections to the Hecke algebra. However, later in the paper we shall see that these polynomials (actually via a representation of the associated algebra) act naturally on the Bruhat-Tits building for SL_{n}, and reduce to the operators which Serre introduced on trees in the case n = 2.

The symmetric group S_{n} acts naturally on polynomials in n variables. Thus for a polynomial P it makes sense to refer to the stabilizer Stab(P) in S_{n} of P. For a polynomial P in n variables, define the symmetrized polynomial associated to P, Sym_{n}(P), by

Sym_{n}(P) = ∑_{σ∈S_{n}/ Stab(P)} σ(P)

We understand that if P is a constant, that Sym_{n}(P) = P.

For our later convenience, we observe the following.

Lemma 4.1. Let M be a monomial, and F = Sym_{n}(M). If M′ is another monomial occurring in F, then F = Sym_{n}(M′).

Draft Remark 5. Proof. Let H = Stab(M) in S_{n}. Then F = ∑_{σ∈S_{n}/H} σ(M). If M′ i s another monomial in F, then M′ = τM for some τ ∈ S_{n}. Since Stab(M′) = τ Stab(M)τ^{−1}, by definition, we have

Sym_{n}(M′) = ∑_{φ∈S_{n}/τHτ^{−1}} φ(M′) = ∑_{φ∈S_{n}/τHτ^{−1}} φτ(M)

We need only establish that as φ runs through a set of coset representatives of S_{n}/τHτ^{−1}, φτ runs through a set of coset representatives for S_{n}/H. We have that φτH = φ′τH if and only if τ^{−1}φ^{−1}φ′τ ∈ H from which we see the correspondence is 1 − 1. As there are the same number of cosets in both cases, we are done.

Let m and n be positive integers and denote by P_{n}(m) the set of partitions of m into n pieces such that a given partition satisfies: m ≥ i_{1} ≥ ⋯ ≥ i_{n} ≥ 0 (and ∑ i_{k} = m).

Introduce the lexicographic ordering on P_{n}(m), and let i = (i_{1}, i_{2}, …, i_{n}) ∈ P_{n}(m). For indeterminates z_{1}, z_{2}, …, z_{n} define hp(0, …, 0) = 1,

hp(i) = hp(i_{1}, i_{2}, …, i_{n}) = ∑_{\begin{matrix}j≤i \\ j∈P_{n}(m) \end{matrix}} Sym_{n}(z_{1}^{j_{1}}z_{2}^{j_{2}} ⋯ z_{n}^{j_{n}}), and

h^{n}(ℓ) = hp(\underbrace{ℓ, 0, …, 0}_{n}) = ∑_{j∈P_{n}(ℓ)} Sym_{n}(z_{1}^{j_{1}}z_{2}^{j_{2}} ⋯ z_{n}^{j_{n}}) = ∑_{\begin{matrix}∑ j_{k}=ℓ \\ j_{k}≥0 \end{matrix}} z_{1}^{j_{1}}z_{2}^{j_{2}} ⋯ z_{n}^{j_{n}}

Note that the hp(i) form a basis for the ℚ-algebra of symmetric polynomials. However, the polynomials h^{n}(ℓ) appear in a generating series with a particularly simple rational expression.

Proposition 4.2. The generating series associated to the h^{r}(ℓ) satisfies prop:basiceuler

∑_{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).

Let x_{1}, x_{2}, …, x_{n} be indeterminates. We now define our family of Hecke operators as symmetric polynomials in ℚ[x_{1}, x_{2}, …, x_{n}]: For 1 ≤ k ≤ n, define

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

We observe that the index [S_{n} : Stab(x_{1}x_{2} ⋯ x_{k})] is the size of the orbit of x_{1}x_{2} ⋯ x_{k} under the group action, that is, the number of distinct monomials in k different variables which one can form with n variables. This is clearly \binom{n}{k}, the binomial coefficient. Thus there is one element σ_{i} ∈ S_{n}/ Stab(x_{1}x_{2} ⋯ x_{k}) to correspond to each of the variables z_{1}, …, z\binom{n}{k}.

The above definition is sufficiently complex to warrant an example, however we note before giving detail that t_{k}^{n}(p) is nothing more than the k^{th} elementary symmetric polynomial in the variables x_{1}, x_{2}, …, x_{n}.

Example 4.3. We begin by computing

h^{r}(1) = hp(\underbrace{1, 0, …, 0}_{r}) = Sym_{r}(z_{1}) = z_{1} + z_{2} + ⋯ + z_{r}.

Then for example,

t_{1}^{n}(p) = h^{n}(1)|\begin{matrix}z_{i}↦σ_{i}(x_{1}) \\ σ_{i}∈S_{n}/ Stab(x_{1}) \end{matrix} = x_{1} + x_{2} + ⋯ + x_{n} = s_{1}(x_{1}, x_{2}, …, x_{n}),

the first elementary symmetric polynomial, and in general

t_{k}^{n}(p) = h^{\binom{n}{k}}(1)|\begin{matrix}z_{i}↦σ_{i}(x_{1}x_{2}···x_{k}) \\ σ_{i}∈S_{n}/ Stab(x_{1}x_{2}···x_{k}) \end{matrix} = ∑_{1≤i_{1}<···<i_{k}≤n} x_{i_{1}} ⋯ x_{i_{k}} = s_{k}(x_{1}, x_{2}, …, x_{n}),

the k^{th} elementary symmetric polynomial.

Remark 4.4. Notice that the polynomials t_{k}^{n}(p^{ℓ}) are symmetric polynomials in the variables x_{1}, x_{2}, …, x_{n}. As such,

t_{k}^{n}(p^{ℓ}) ∈ ℚ[x_{1}, …, x_{n}]^{sym} = ℚ[s_{1}, s_{2}, …, s_{n}].

Since t_{k}^{n}(p) = s_{k}, we have that the algebra generated by these operators is actually equal to the full ring of symmetric polynomials:

⟨t_{k}^{n}(p^{ℓ}) | ℓ ≥ 0, k = 1, 2, …, n⟩ = ⟨t_{k}^{n}(p) | k = 1, 2, …, n⟩ = ℚ[s_{1}, s_{2}, …, s_{n}]

Turning to generating series, we find

prop:generaleuler Proposition 4.5. For k = 1, 2, …, n, the generating series for the Hecke operators t_{k}^{n}(p^{ℓ}) is a rational function:

∑_{ℓ≥0} t_{k}^{n}(p^{ℓ})u^{ℓ} = [ ∏_{σ∈S_{n}/ Stab(x_{1}x_{2}···x_{k})}(1 − uσ(x_{1}x_{2} ⋯ x_{k}))]^{−1}

Remark 4.6. The Euler factors which appear here also appeared in work of Andrianov [1]. He defined a zeta function which encapsulated the algebra structure of the full local Hecke algebra and whose rational expression had denominator equal to the product of all of the denominators above, i.e.,

∏_{k=1}^{n} ∏_{σ∈S_{n}/ Stab(x_{1}x_{2}···x_{k})}(1 − uσ(x_{1}x_{2} ⋯ x_{k})).

However, the numerator of that rational expression was non-trivial and not explicitly given. In this paper, we are producing a family of operators whose generating series are rational functions whose denominators are the individual factors of this product, with numerators 1.

Proof. From Proposition 4.2, we know that

∑_{ℓ≥0} h^{\binom{n}{k}}(ℓ)u^{ℓ} = [ \begin{matrix}( & {n \atop k} & ) \\ ∏ \\ i=1 \end{matrix}(1 − uz_{i})]^{−1}.

But

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

which completes the proof.

Example 4.7. Consider the case of n = 4. We have the four Hecke series:

∑_{ℓ≥0} t_{1}^{4}(p^{ℓ})u^{ℓ} = [(1 − ux_{1})(1 − ux_{2})(1 − ux_{3})(1 − ux_{4})]^{−1}
∑_{ℓ≥0} t_{2}^{4}(p^{ℓ})u^{ℓ} = [(1 − ux_{1}x_{2})(1 − ux_{1}x_{3})(1 − ux_{1}x_{4})(1 − ux_{2}x_{3})(1 − ux_{2}x_{4})(1 − ux_{3}x_{4})]^{−1}
∑_{ℓ≥0} t_{3}^{4}(p^{ℓ})u^{ℓ} = [(1 − ux_{1}x_{2}x_{3})(1 − ux_{1}x_{2}x_{4})(1 − ux_{1}x_{3}x_{4})(1 − ux_{2}x_{3}x_{4})]^{−1}
∑_{ℓ≥0} t_{4}^{4}(p^{ℓ})u^{ℓ} = [(1 − ux_{1}x_{2}x_{3}x_{4})]^{−1}

Remark 4.8. Later in this paper we establish a connection between these t_{k}^{n} and classical Hecke operators. Accepting such a connection for now, note that the Euler factor corresponding to t_{k}^{n} has degree \binom{n}{k} and the monomials are the kth exterior powers of the x_{i}s. Then, observing that the Euler factor corresponding to ∑ t_{1}^{n}(p^{ℓ})u^{ℓ} seems most basic to GL_{n} and that the Euler factor corresponding to ∑ t_{k}^{n}(p^{ℓ})u^{ℓ} has degree \binom{n}{k} suggests a correspondence between forms on GL_{n} and forms on GL\binom{n}{k}.

We now illustrate parallels between the polynomial Hecke operators, t_{1}^{n}(p^{ℓ}), and those defined through double cosets, T (p^{ℓ}). We first note the following

Lemma 4.9. Let s_{j} = s_{j}(x_{1}, x_{2}, …, x_{n}). Then

(1 − ux_{1}) ⋯ (1 − ux_{n}) = 1 − s_{1}u + s_{2}u^{2} + ⋯ + (−1)^{n}s_{n}u^{n}.

Draft Remark 6. Proof.
(1 − ux_{1}) ⋯ (1 − ux_{n}) = (−1)^{n}(x_{1} ⋯ x_{n})(u − x_{1}^{−1}) ⋯ (u − x_{n}^{−1})

\begin{matrix}= (−1)^{n}(x_{1} ⋯ x_{n})[u^{n} − s_{1}(x_{1}^{−1}, …, x_{n}^{−1})u^{n−1} + s_{2}(x_{1}^{−1}, …, x_{n}^{−1})u^{n−2} \\ + ⋯ + (−1)^{n}s_{n}(x_{1}^{−1}, …, x_{n}^{−1})u^{0}] \end{matrix}

= (x_{1} ⋯ x_{n})[s_{n}(x_{1}^{−1}, …, x_{n}^{−1}) − us_{n−1}(x_{1}^{−1}, …, x_{n}^{−1}) + ⋯ + (−1)^{n}u^{n}]

Finally, observe that since \binom{n}{j} = \binom{n}{n−j} is the number of monomials in s_{j}(x_{1}^{−1}, …, x_{n}^{−1}), we have that (x_{1} ⋯ x_{n})s_{j}(x_{1}^{−1}, …, x_{n}^{−1}) = s_{n−j}(x_{1}, …, x_{n}) for j ≥ 1 which completes the proof.

From Proposition 4.2 and the lemma above, we have that

∑_{ℓ≥0} t_{1}^{n}(p^{ℓ})u^{ℓ} = [ ∏_{i=1}^{n}(1 − ux_{i})]^{−1}

= [∑_{k=0}^{n}(−1)^{k}s_{k}(x_{1}, …, x_{n})u^{k}]^{−1} ,

which is similar to (3.2) above. In fact, if we apply the algebra automorphism of ℚ[s_{1}, …, s_{n}] induced by taking s_{k}(x_{1}, x_{2}, …, x_{n}) to p^{−k}s_{k}(x_{1}, x_{2}, …, x_{n}), the resulting “Euler” factor would coincide with the earlier one.

prop:sympolys Proposition 4.10. The only extensions to ℚ[x_{1}, x_{2}, … x_{n}] of the automorphism of ℚ[s_{1}, s_{2} …, s_{n}] which maps s_{j} ↦ p^{−j}s_{j} are those which map x_{j} ↦ p^{−1}x_{σ(j)} for some σ in the symmetric group S_{n}.

Remark 4.11. 1. The proposition says that the automorphism of ℚ[s_{1}, s_{2}, …, s_{n}] can be thought of as the restriction (to the subalgebra of symmetric polynomials) of the automorphism induced by x_{i} ↦ p^{−1}x_{σ(i)} for any σ ∈ S_{n}.

2. We could have hidden this automorphism by defining our Hecke operator t_{k}^{n}(p^{ℓ}) as

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

but the introduction of the power of p into the expression z_{i} ↦ p^{−k}σ_{i}(x_{1}x_{2} ⋯ x_{k}) would have hardly been motivated at that point. Moreover, as we shall see in the next section, we want to define a representation of this algebra on an apartment associated to SL_{n}(ℚ_{p}) and then on the building. Since the structure of the apartments are independent of p, the dependence upon p in the operators should appear only at the level of the building.

Proof. (of proposition) Let φ : ℚ[x_{1}, x_{2}, …, x_{n}] → ℚ[x_{1}, …, x_{n}] be a ℚ-algebra isomorphism such that φ(s_{j}(x_{1}, …, x_{n})) = p^{−j}s_{j}(x_{1}, x_{2}, …, x_{n}). We want to show that φ(x_{j}) = p^{−1}x_{σ(j)} for some permutation σ in S_{n}. Let ξ_{j} = φ(x_{j}). Then

s_{j}(ξ_{1}, …, ξ_{n}) = φ(s_{j}(x_{1}, x_{2}, …, x_{n})) = p^{−j}s_{j}(x_{1}, x_{2}, …, x_{n}) = s_{j}(\frac{x_{1}}{p} , …, \frac{x_{n}}{p}).

Since the polynomials

∏_{k=1}^{n}(u − ξ_{k}) = ∑_{k=0}^{n}(−1)^{k}s_{k}(ξ_{1}, …, ξ_{n})u^{n−k}

= ∑_{k=0}^{n}(−1)^{k}s_{k}(\frac{x_{1}}{p} , …, \frac{x_{n}}{p})u^{n−k} = ∏_{k=1}^{n}(u − \frac{x_{k}}{p})

have the same roots, the proof is complete.

5. Action on Buildings and Apartments

There are at least two reasons we defined the Hecke operators in the ring of polynomials. The first is that we produce a natural way of characterizing Andrianov’s spherical map — which is instrumental in showing that the local Hecke algebra is isomorphic to the ring of symmetric polynomials. The second is that there is a natural action of this polynomial Hecke algebra on the apartments of the Bruhat-Tits building for SL_{n} over a local field. It was actually the latter reason which was provided the initial impetus for this investigation. In the first section we give an alternate characterization of the spherical map.

5.1. A natural characterization of the spherical map. As in the previous section, x_{1}, x_{2}, …, x_{n} are indeterminates and s_{k} = s_{k}(x_{1}, x_{2}, …, x_{n}) is the kth elementary symmetric polynomial in the x_{i}. We let ω : \underline{H}_{p}^{n} → ℚ[s_{1}, …, s_{n}] denote Andrianov’s spherical map([2]).

We have previously established a canonical isomorphism between the (integral subrings of the) p-part of the global Hecke algebra \underline{H}_{p}^{n} and the local Hecke algebra (\underline{H}_{p}^{n}).

Recalling that Γ_{p} = GL_{n}(ℤ_{p}) and d_{k}^{n}(p) is the diagonal matrix diag(\underbrace{1, …, 1}_{n−k}, \underbrace{p, …, p}_{k}),

we may justifiably denote by T_{k}^{n}(p) the double coset Γ_{p}d_{k}^{n}(p)Γ_{p} in H_{p}^{n}.

Define maps φ_{A}, φ_{B}, ψ_{A}, ψ_{B} corresponding the the commutative diagrams which follow and which are induced by:

φ_{A} : H_{p}^{n} → ℚ[s_{1}, …, s_{n}] : T_{k}^{n}(p) ↦ p^{−k(k−1)/2}t_{k}^{n}(p) = p^{−k(k−1)/2}s_{k}
φ_{B} : H_{p}^{n} → ℚ[s_{1}, …, s_{n}] : T_{k}^{n}(p) ↦ t_{k}^{n}(p) = s_{k}
ψ_{A} : ℚ[s_{1}, …, s_{n}] → ℚ[s_{1}, …, s_{n}] : t_{k}^{n}(p) ↦ p^{−k}t_{k}^{n}(p) = p^{−k}s_{k}
ψ_{B} : ℚ[s_{1}, …, s_{n}] → ℚ[s_{1}, …, s_{n}] : t_{k}^{n}(p) ↦ t̂_{k}^{n}(p) = p^{−k(k+1)/2}t_{k}^{n}(p) = p^{−k(k+1)/2}s_{k}

We note that ψ_{A} ∘ φ_{A} = ψ_{B} ∘ φ_{B}, and that the four maps are ring isomorphisms, mapping well-known (algebraically independent) generators to (algebraically independent) generators. Each composition (together with the canonical isomorphism of \underline{H}_{p}^{n} and \underline{H}_{p}^{n}) gives an alternate and quite natural interpretation of the spherical map without reference to right cosets. To reveal the significance of these maps, we characterize their action on various Hecke operators.

We note that the diagrams which follow contain information about the action of the Hecke algebra on the Bruhat-Tits building which will be developed fully in the next section. For now, we make only the following remarks. Let Δ_{n} be the Bruhat-Tits building for SL_{n}(ℚ_{p}), and let B (resp. A) denote ℚ-vector space with basis the vertices of Δ_{n} (resp. by the vertices of a fixed apartment in Δ_{n}). The map ρ is a representation of the polynomial Hecke algebra in the building (apartment), defined by the diagrams below and

ρ(t_{k}^{n}(p)) = t̃_{k}^{n}(p) = \begin{cases}t_{k}^{n}(p) & if k < n \\ 1 & if k = n \end{cases} .

[figure: \underline{H}_{p}^{n}; φ_{A}; φ_{B}; ≅; ℚ[s_{1}, …, s_{n}]; ρ; ψ_{A}; ℚ[s_{1}, …, s_{n}]; ρ; ψ_{B}; _{\underline{H}n p}; ω; End(A); End(B); ℚ[s_{1}, …, s_{n}]]

[figure: T_{k}^{n}(p); φ_{A}; φ_{B}; ≅; p^{−k(k−1)/2}t_{k}^{n}(p); ρ; ψ_{A}; t_{k}^{n}(p); ρ; ψ_{B}; π_{k}^{n}(p); ω; p^{−k(k−1)/2}t̃_{k}^{n}(p); t̃_{k}^{n}(p); ω(π_{k}^{n}(p)) = p^{−k(k+1)/2}t_{k}^{n}(p)]

We want to show how the maps defined above act on our various Hecke operators: T (p^{ℓ}), t(p^{ℓ}), t_{1}^{n}(p^{ℓ}). The summary is contained in the commutative diagram below, but we take our time to explore these relations.

[figure: ∑_{ℓ≥0} T (p^{ℓ})u^{ℓ} = [∑_{k=0}^{n}(−1)^{k}p^{\frac{k(k−1)}{2}} T_{k}^{n}(p)u^{k}]^{−1}; φ_{A}; φ_{B}; ≅; ∑_{ℓ≥0} t_{1}^{n}(p^{ℓ})u^{ℓ} = [∑_{k=0}^{n}(−1)^{k}s_{k}u^{k}]^{−1}; ρ; ψ_{A}; ∑_{ℓ≥0} t̂_{1}^{n}(p^{ℓ})u^{ℓ} = [∑_{k=0}^{n}(−1)^{k}p^{k(k−1)/2}s_{k}u^{k}]^{−1}; ρ; ψ_{B}; ∑_{ℓ≥0} t(p^{ℓ})u^{ℓ} = [∑_{k=0}^{n}(−1)^{k}p^{\frac{k(k−1)}{2}} π_{k}^{n}(p)u^{k}]^{−1}; ω; ∑_{ℓ≥0} T_{A}(p^{ℓ})u^{ℓ} = [∑_{k=0}^{n}(−1)^{k}t̃_{k}^{n}(p)u^{k}]^{−1}; ∑_{ℓ≥0} T_{B}(p^{ℓ})u^{ℓ} = [∑_{k=0}^{n}(−1)^{k}p^{k(k−1)/2}t̃_{k}^{n}(p)u^{k}]^{−1}; ∑_{ℓ≥0} ω(t(p^{ℓ}))u^{ℓ} = [∑_{k=0}^{n}(−1)^{k}p^{−k}s_{k}u^{k}]^{−1}]

Consider the rational functions associated to the two Hecke series:

∑_{ℓ≥0} T (p^{ℓ})u^{ℓ} = [∑_{k=0}^{n}(−1)^{k}p^{\frac{k(k−1)}{2}} T_{k}^{n}(p)u^{k}]^{−1} and ∑_{ℓ≥0} t_{1}^{n}(p^{ℓ})u^{ℓ} = [∑_{k=0}^{n}(−1)^{k}s_{k}u^{k}]^{−1} .

By comparing the rational functions which represent the generating functions, we see that it is quite natural to map (inducing φ_{A}) p^{\frac{k(k−1)}{2}} T_{k}^{n}(p) to s_{k} = t_{k}^{n}(p), i.e., T_{k}^{n}(p) to p^{\frac{−k(k−1)}{2}} t_{k}^{n}(p).

It is equally natural to simply map (inducing φ_{B}) T_{k}^{n}(p) to t_{k}^{n}(p). Here the new operators t̂_{1}^{n}(p^{ℓ}) are defined by the rational function, the image of which under the representation ρ will be a natural Hecke operator on the Bruhat-Tits building for SL_{n}.

The maps ψ_{A} and ψ_{B} are simply elementary automorphisms of the symmetric polynomial ring: ψ_{A} taking s_{k} ↦ p^{−k}s_{k} (discussed in Proposition 4.10), and ψ_{B} taking s_{k} ↦ p^{−k(k+1)/2}s_{k}.

Since the spherical map ω takes π_{k}^{n}(p) to p^{−k(k+1)/2}s_{k} = p^{−k(k+1)/2}t_{k}^{n}(p), we see that the composition of the φs with the ψs gives the spherical map in a completely natural way.

5.2. Buildings, Apartments and Symmetric Polynomials. In this section we finally associate Hecke operators characterized by symmetric polynomials with operators acting on the building.

In an earlier section, we saw how to label the vertices of an apartment in the Bruhat-Tits building for SL_{n}(ℚ_{p}) in such a way as to admit a natural action of ℚ[x_{1}, …, x_{n}]^{sym} on the vertices. Recall that under that action, the monomials act in the following manner: For nonnegative integers a_{i} and indeterminates x_{i}, we say that a vertex w is a x_{1}^{a_{1}}x_{2}^{a_{2}} ⋯ x_{n}^{a_{n}}-translate of v if there exists lattices L_{1} ∈ v and L_{2} ∈ w with L_{2} ⊂ L_{1} and {L_{1} : L_{2}} = {p^{a_{1}}, p^{a_{2}} …, p^{a_{n}}}, where the latter notation represents the elementary divisors of L_{2} relative to L_{1}. Note that since vertices are defined by classes of lattices, any vertex v is an x_{1} ⋯ x_{n} - translate of itself, i.e. the x_{1}x_{2} ⋯ x_{n} - translate of v = [L] is [pL] = v.

Fix a vertex in the apartment, labelled v, and let L = ⊕ℤ_{p}e_{i} be a lattice whose class determines the vertex v. Then the vertices in the apartment can be viewed in any of the following ways: the homothety class of a lattice M, denoted [M] with {L : M} = {p^{a_{1}}, p^{a_{2}} …, p^{a_{n}}}; the class written explicitly in terms of the basis of L: [ℤ_{p}p^{a_{1}}e_{1} ⊕ ⋯ ⊕ ℤ_{p}p^{a_{n}}e_{n}]; the x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}}-translate of v.

Referring to the commutative diagrams of the preceding section, we defined a representation of the polynomial ring (i.e the local Hecke algebra) into the Bruhat-Tits building by

ρ(t_{k}^{n}(p)) = t̃_{k}^{n}(p) = \begin{cases}t_{k}^{n}(p) & if k < n \\ 1 & if k = n \end{cases} .

Notice that this is just the automorphism of ℚ[s_{1}, …, s_{n}] which takes s_{k} ↦ s_{k} (if k < n) and maps s_{n} ↦ 1. Thus any operator (i.e. symmetric polynomial) acts on A by translation (extended linearly) as described above. Thus we can define Hecke operators T_{A}(p^{ℓ}) acting on A by taking T_{A}(p^{ℓ}) = ρ(t_{1}^{n}(p^{ℓ})). Then from Proposition 4.5ff, we have that

∑_{ℓ≥0} T_{A}(p^{ℓ})u^{ℓ} = [∑_{k=0}^{n}(−1)^{k}t̃_{k}^{n}(p)u^{k}]^{−1} .

Moreover, it is clear from the definition that

t_{1}^{n}(p^{ℓ}) = ∑_{\begin{matrix}∑ j_{k}=ℓ \\ j_{k}≥0 \end{matrix}} x_{1}^{j_{1}}x_{2}^{j_{2}} ⋯ x_{n}^{j_{n}}.

We see that the action of Sym_{n}(x_{1}^{j_{1}}x_{2}^{j_{2}} ⋯ x_{n}^{j_{n}}) defines an operator T_{A}(p^{j_{1}}, …, p^{j_{n}}) which acts on the vertex v = [L] by

T_{A}(p^{j_{1}}, …, p^{j_{n}})([L]) =\begin{matrix}∑ & ∗ \\ \{L:M\}=\{p^{j_{1}},... ,p^{jn}\} \end{matrix}[M]

where the sum is over all sublattices M relative to the (unordered) basis {e_{1}, …, e_{n}} of L. Then

T_{A}(p^{ℓ}) = ρ(t_{1}^{n}(p^{ℓ}) = ρ(φ_{A}(T (p^{ℓ}))) = ∑_{\begin{matrix}∑ a_{k}=ℓ \\ a_{k}≥0 \end{matrix}} T_{A}(p^{a_{1}}, …, p^{a_{n}}).

The natural definition of a Hecke operator acting on the Bruhat-Tits building is very similar. With the lattice L fixed as above, we recall [7] that there is a 1 – 1 correspondence between right cosets Γ_{p}ξ_{ν} in Γ_{p} diag(p^{a_{1}}, …, p^{a_{n}})Γ_{p} and lattices M in V = L ⊗ ℚ_{p} with elementary divisors {L : M} = {p^{a_{1}}, …, p^{a_{n}}}.

Thus it is natural to define the Hecke operator T_{B}(p^{a_{1}}, …, p^{a_{n}}) (acting on B) as

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

We also define

T_{B}(p^{ℓ}) = ∑_{i∈P_{n}(ℓ)} T_{B}(p^{i_{1}}, …, p^{i_{n}})

It is important to note that since the ring structure of the Hecke algebra H_{p}^{n} is characterized [7] precisely in terms of elementary divisors of lattices on V, we have for free that the generating series for the operators T_{B}(p^{ℓ}) is expressible as the same rational function as the generating series for the classical Hecke operators T (p^{ℓ}), namely

∑_{ℓ≥0} T_{B}(p^{ℓ})u^{ℓ} = [∑_{k=0}^{n}(−1)^{k}p^{k(k−1)/2}t̃_{k}^{n}(p)u^{k}]^{−1} .

That is,

T_{B}(p^{ℓ}) = ρ(φ_{B}(T (p^{ℓ}))).

References

[1] A. N. Andrianov, Spherical Functions for GL_{n} over Local Fields and Summation of Hecke Series, Math USSR Sbornik 12 (1970) pp 429–452, (Math. Sbornik 83 (125) (1970)).

[2] A. N. Andrianov, V. G. Zhuravlev, Modular Forms and Hecke Operators, Translations of Mathematical Monographs, 145, American Mathematical Society, Providence, 1995.

[3] K. S. Brown, Buildings, Springer-Verlag, New York, 1989.

[4] M. Ronan, Lectures on Buildings, Perspectives in Mathematics, Vol. 7, Academic Press, San Diego, 1989.

[5] A. Schwartz, T. R. Shemanske, Maximal Orders in Central Simple Algebras and Bruhat-Tits Buildings, J. Number Theory, 56, (1996) pp 115–138.

[6] J. P. Serre, Trees, Springer-Verlag, Berlin, 1980.

[7] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, Princeton, 1971.

[8] M-F. Vignéras, Arithmétique des Algèbres de Quaternions, Lecture Notes in Mathematics, 800, Springer-Verlag, New York, 1980.

Department of Mathematics, Bates College, Lewiston, Maine 04240

 E-mail address: jrhodes@bates.edu

Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755

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

Equation (3.1) . . . . . . . . . . . . . . . . . . . .lemeq (Page 11) Theorem 3.4 . . . . . . . . . . . . .thm:globalhecke (Page 12) Equation (3.2) . . . . . . . . . . . . eq:globalhecke (Page 13) Proposition 4.2 . . . . . . . . . . . prop:basiceuler (Page 15) Proposition 4.5 . . . . . . . . . . prop:generaleuler (Page 16) Proposition 4.10 . . . . . . . . . . . prop:sympolys (Page 18)