Complex Structure on the Smooth Dual of GL(n)

Jacek Brodzki^{1} and Roger Plymen

Received: December 10, 2001 Revised: April 24, 2002

Communicated by Peter Schneider

Abstract. Let G denote the p-adic group GL(n), let Π(G) denote the smooth dual of G, let Π(Ω) denote a Bernstein component of Π(G) and let ℋ(Ω) denote a Bernstein ideal in the Hecke algebra ℋ(G). With the aid of Langlands parameters, we equip Π(Ω) with the structure of complex algebraic variety, and prove that the periodic cyclic homology of ℋ(Ω) is isomorphic to the de Rham cohomology of Π(Ω). We show how the structure of the variety Π(Ω) is related to Xi’s affirmation of a conjecture of Lusztig for GL(n, ℂ). The smooth dual Π(G) admits a deformation retraction onto the tempered dual Π^{t}(G).

2000 Mathematics Subject Classification: 46L80 22E50 46L87 11S37 Keywords and Phrases: Langlands correspondence, p-adic GL(n), Baum-Connes map, smooth dual, tempered dual.

Introduction

The use of unramified quasicharacters to create a complex structure is well established in number theory. The group of unramified quasicharacters of the idele class group of a global field admits a complex structure: this complex structure provides the background for the functional equation of the zeta integral Z(ω, Φ), see [39, Theorem 2, p. 121]. Let now G be a reductive p-adic group and let M be a Levi subgroup of G. Let Π^{sc}(M) denote the set of equivalence classes of irreducible supercuspidal representations of M. Harish-Chandra creates a complex structure on the set Π^{sc}(M) by using unramified quasicharacters of M [16, p.84]. This complex structure provides the background for the Harish-Chandra functional equations [16, p. 91]. Bernstein considered the set Ω(G) of all conjugacy classes of pairs (M, σ) where M is a Levi subgroup of G and σ is an irreducible supercuspidal representation of M. Making use of unramified quasicharacters of M, Bernstein gave the set Ω(G) the structure of a complex algebraic variety. Each irreducible component Ω of Ω(G) has the structure of a complex affine algebraic variety [5]. Let Π(G) denote the set of equivalence classes of irreducible smooth representations of G. We will call Π(G) the smooth dual of G. Bernstein defines the infinitesimal character from Π(G) to Ω(G):

1Partially supported by a Leverhulme Trust Fellowship

inf.ch. : Π(G) → Ω(G).

The infinitesimal character is a finite-to-one map from the set Π(G) to the variety Ω(G). Let F be a nonarchimedean local field and from now on let G = GL(n) = GL(n, F). Let now W_{F} be the Weil group of the local field F, then W_{F} admits unramified quasicharacters, namely those which are trivial on the inertia subgroup I_{F}. Making use of the unramified quasicharacters of W_{F}, we introduced in [8] a complex structure on the set of Langlands parameters for GL(n). In view of the local Langlands correspondence for GL(n) this creates, by transport of structure, a complex structure on the smooth dual of GL(n). In Section 1 of this article, we describe in detail the complex structure on the set of L-parameters for GL(n). We prove that the smooth dual Π(GL(n)) has the structure of complex manifold. The local L-factors L(s, π) then appear as complex valued functions of several complex variables. We illustrate this with the local L-factors attached to the unramified principal series of GL(n). The complex structure on Π(GL(n)) is well adapted to the periodic cyclic homology of the Hecke algebra ℋ(GL(n)). The identical structure arises in the work of Xi on Lusztig’s conjecture [40]. Let W be the extended affine Weyl group associated to GL(n, ℂ), and let J be the associated based ring (asymptotic algebra) [27, 40]. Xi confirms Lusztig’s conjecture and proves that J ⊗_{ℤ} ℂ is Morita equivalent to the coordinate ring of the complex algebraic variety { \atop (ℂ^{×})^{n}}/S_{n}, the extended quotient by the symmetric group S_{n} of the complex n-dimensional torus (ℂ^{×})^{n}. In Section 2 we describe the theorem of Xi on the structure of the based ring J. So the structure of extended quotient, which runs through our work, occurs in the work of Xi at the level of algebras. The link with our work is now provided by the theorem of Baum and Nistor [3, 4]

HP_{∗}(ℋ(n, q)) ≃ HP_{∗}(J)

where ℋ(n, q) is the associated extended affine Hecke algebra. Let Ω be a component in the Bernstein variety Ω(GL(n)), and let ℋ(G) = ⨁ ℋ(Ω) be the Bernstein decomposition of the Hecke algebra. Let

Π(Ω) = (inf.ch.)^{−1}Ω.

Then Π(Ω) is a smooth complex algebraic variety with finitely many irreducible components. We have the following Bernstein decomposition of Π(G):

Π(G) = ⨆ Π(Ω).

Let M be a compact C^{∞} manifold. Then C^{∞}(M) is a Fréchet algebra, and we have Connes’ fundamental theorem [14, Theorem 2, p. 208]:

HP_{∗}(C^{∞}(M)) ≅ H^{∗}(M; ℂ).

Now the ideal ℋ(Ω) is a purely algebraic object, and, in computing its periodic cyclic homology, we would hope to find an algebraic variety to play the role of the manifold M. This algebraic variety is Π(Ω).

Theorem 0.1. Let Ω be a component in the Bernstein variety Ω(G). Then the periodic cyclic homology of ℋ(Ω) is isomorphic to the periodised de Rham cohomology of Π(Ω):

HP_{∗}(ℋ(Ω)) ≅ H^{∗}(Π(Ω); ℂ).

This theorem constitutes the main result of Section 3, which is then used to show that the periodic cyclic homology of the Hecke algebra of GL(n) is isomorphic to the periodic cyclic homology of the Schwartz algebra of GL(n). We also provide an explicit numerical formula for the dimension of the periodic cyclic homology of ℋ(Ω) in terms of certain natural number invariants attached to Ω. The smooth dual Π(GL(n)) has a natural stratification-by-dimension. We compare this stratification with the Schneider-Zink stratification [34]. Stratification-by-dimension is finer than the Schneider-Zink stratification, see Section 3. A scheme X is a topological space, called the support of X and denoted |X|, together with a sheaf 𝒪_{X} of rings on X, such that the pair (|X|, 𝒪_{X}) is locally affine, see [15, p. 21]. The smooth dual Π(G) determines a reduced scheme, see [18, Prop. 2.6]. If S is the reduced scheme determined by the Bernstein variety Ω(G), then Π(G) is a scheme over S, i.e. a scheme together with a morphism Π(G) → S. This morphism is the q-projection introduced in [8]:

π_{q} : Π(G) → S.

In Section 4 we give a detailed description of the q-projection and prove that the q-projection is a finite morphism. From the point of view of noncommutative geometry it is natural to seek the spaces which underlie the noncommutative algebras ℋ(G) and 𝒮(G). The space which underlies the Hecke algebra ℋ(G) is the complex manifold Π(G). The space which underlies the Schwartz algebra is the Harish-Chandra parameter space, which is a disjoint union of compact orbifolds. In Section 5 we construct a deformation retraction of the smooth dual onto the tempered dual. We view this deformation retraction as a geometric counterpart of the Baum-Connes assembly map for GL(n). In Section 6 we track the fate of supercuspidal representations of G through the diagram which appears in Section 5. In particular, the index map μ manifests itself as an example of Ahn reciprocity. We would like to thank Paul Baum, Alain Connes, Jean-Francois Dat and Nigel Higson for many valuable conversations. Jacek Brodzki was supported in part by a Leverhulme Trust Fellowship. This article was completed while Roger Plymen was at IHES, France.

1. The complex structure on the smooth dual of GL(n)

The field F is a nonarchimedean local field, so that F is a finite extension of ℚ_{p}, for some prime p or F is a finite extension of the function field 𝔽_{p}((x)). The residue field k_{F} of F is the quotient o_{F} /m_{F} of the ring of integers o_{F} by its unique maximal ideal m_{F}. Let q be the cardinality of k_{F}. The essence of local class field theory, see [29, p.300], is a pair of maps

(d : G ⟶ ℤ̂, v : F^{×} ⟶ ℤ)

where G is a profinite group, ℤ̂ is the profinite completion of ℤ, and v is the valuation. Let \overline{F} be a separable algebraic closure of F. Then the absolute Galois group G(\overline{F} |F) is the projective limit of the finite Galois groups G(E|F) taken over the finite extensions E of F in \overline{F}. Let F̃ be the maximal unramified extension of F. The map d is in this case the projection map

d : G(\overline{F} |F) ⟶ G(F̃|F) ≅ ℤ̂

The group G(F̃|F) is procyclic. It has a single topological generator: the Frobenius automorphism ϕ_{F} of F̃|F. The Weil group W_{F} is by definition the pre-image of < ϕ_{F} > in G(\overline{F} | F). We thus have the surjective map

d : W_{F} ⟶ ℤ

The pre-image of 0 is the inertia group I_{F}. In other words we have the following short exact sequence

1 → I_{F} → W_{F} → ℤ → 0

The group I_{F} is given the profinite topology induced by G(\overline{F} |F). The topology on the Weil group W_{F} is dictated by the above short exact sequence. The Weil group W_{F} is a locally compact group with maximal compact subgroup I_{F}. The map

W_{F} ⟶ G(F̃|F)

is a continuous homomorphism with dense image. A detailed account of the Weil group for local fields may be found in [37]. For a topological group G we denote by G^{ab} the quotient G^{ab} = G/G^{c} of G by the closure G^{c} of the commutator subgroup of G. Thus G^{ab} is the maximal abelian Hausdorff quotient of G. The local reciprocity laws [29, p.320]

r_{E|F} : G(E|F)^{ab} ≅ F^{×}/N_{E|F} E^{×}

now create an isomorphism [30, p.69]:

r_{F} : W_{F}^{ab} ≅ F^{×}

We have W_{F} = ⊔Φ^{n}I_{F} , n ∈ ℤ. The Weil group is a locally compact, totally disconnected group, whose maximal compact subgroup is I_{F}. This subgroup is also open. There are three models for the Weil-Deligne group.

One model is the crossed product W_{F} ⋉ ℂ, where the Weil group acts on ℂ by w · x = ∥w∥x, for all w ∈ W_{F} and x ∈ ℂ. The action of W_{F} on ℂ extends to an action of W_{F} on SL(2, ℂ). The semidirect product W_{F} ⋉SL(2, ℂ) is then isomorphic to the direct product W_{F} ×SL(2, ℂ), see [22, p.278]. Then a complex representation of W_{F} × SL(2, ℂ) is determined by its restriction to W_{F} × SU(2), where SU(2) is the standard compact Lie group. From now on, we shall use this model for the Weil-Deligne group:

ℒ_{F} = W_{F} × SU(2).

Definition 1.1. An L-parameter is a continuous homomorphism

ϕ : ℒ_{F} → GL(n, ℂ)

such that ϕ(w) is semisimple for all w ∈ W_{F}. Two L-parameters are equivalent if they are conjugate under GL(n, ℂ). The set of equivalence classes of Lparameters is denoted Φ(G).

Definition 1.2. A representation of G on a complex vector space V is smooth if the stabilizer of each vector in V is an open subgroup of G. The set of equivalence classes of irreducible smooth representations of G is the smooth dual Π(G) of G.

Theorem 1.3. Local Langlands Correspondence for GL(n). There is a natural bijection between Φ(GL(n)) and Π(GL(n)).

The naturality of the bijection involves compatibility of the L-factors and ϵfactors attached to the two types of objects. The local Langlands conjecture for GL(n) was proved by Laumon, Rapoport and Stuhler [25] when F has positive characteristic and by Harris-Taylor [17] and Henniart [19] when F has characteristic zero. We recall that a matrix coefficient of a representation ρ of a group G on a vector space V is a function on G of the form f(g) = ⟨ρ(g)v, w⟩, where v ∈ V, w ∈ V^{∗}, and V^{∗} denotes the dual space of V. The inner product is given by the duality between V and V^{∗}. A representation ρ of G is called supercuspidal if and only if the support of every matrix coefficient is compact modulo the centre of G. Let τ_{j} = spin(j) denote the (2j + 1)-dimensional complex irreducible representation of the compact Lie group SU(2), j = 0, 1/2, 1, 3/2, 2, . . .. For GL(n) the local Langlands correspondence works in the following way.

• Let ρ be an irreducible representation of the Weil group W_{F}. Then π_{F} (ρ ⊗ 1) is an irreducible supercuspidal representation of GL(n), and every irreducible supercuspidal representation of GL(n) arises in this way. If det(ρ) is a unitary character, then π_{F} (ρ⊗1) has unitary central character, and so is pre-unitary.

• We have π_{F} (ρ⊗spin(j)) = Q(Δ), the Langlands quotient associated to the segment {| |^{−(j−1)/2}π_{F} (ρ), …, | |^{(j−1)/2}π_{F} (ρ)}. If det(ρ) is unitary, then Q(Δ) is in the discrete series. In particular, if ρ = 1 then π_{F} (1 ⊗ spin(j)) is the Steinberg representation St(2j + 1) of GL(2j + 1).

• If ϕ is an L-parameter for GL(n) then ϕ = ϕ_{1} ⊕ … ⊕ ϕ_{m} where ϕ_{j} = ρ_{j} ⊗ spin(j). Then π_{F} (ρ) is the Langlands quotient Q(Δ_{1}, …, Δ_{m}). If det(ρ_{j}) is a unitary character for each j, then π_{F} (ϕ) is a tempered representation of GL(n).

This correspondence creates, as in [23, p. 381], a natural bijection

π_{F} : Φ(GL(n)) → Π(GL(n)).

A quasi-character ψ : W_{F} → ℂ^{×} is unramified if ψ is trivial on the inertia group I_{F}. Recall the short exact sequence

0 → I_{F} → W_{F} →^{d} ℤ → 0

Then ψ(w) = z^{d(w)} for some z ∈ ℂ^{×}. Note that ψ is not a Galois representation unless z has finite order in the complex torus ℂ^{×}, see [37]. Let Ψ(W_{F}) denote the group of all unramified quasi-characters of W_{F}. Then

\begin{matrix}Ψ(W_{F}) & ≃ & ℂ^{×} \\ ψ & ↦ & z \end{matrix}

Each L-parameter ϕ : ℒ_{F} → GL(n, ℂ) is of the form ϕ_{1} ⊕ ⋯ ⊕ ϕ_{m} with each ϕ_{j} irreducible. Each irreducible L-parameter is of the form ρ ⊗ spin(j) with ρ an irreducible representation of the Weil group W_{F}.

Definition 1.4. The orbit 𝒪(ϕ) ⊂ Φ_{F} (G) is defined as follows

𝒪(ϕ) = {⨁_{r=1}^{m} ψ_{r}ϕ_{r} | ψ_{r} ∈ Ψ(W_{F}), 1 ≤ r ≤ m}

where each ψ_{r} is an unramified quasi-character of W_{F}.

Definition 1.5. Let det ϕ_{r} be a unitary character, 1 ≤ r ≤ m and let ϕ = ϕ_{1} ⊕ … ⊕ ϕ_{m}. The compact orbit 𝒪^{t}(ϕ) ⊂ Φ^{t}(G) is defined as follows:

𝒪^{t}(ϕ) = {⨁_{r=1}^{m} ψ_{r}ϕ_{r} | ψ_{r} ∈ Ψ^{t}(W_{F}), 1 ≤ r ≤ m}

where each ψ_{r} is an unramified unitary character of W_{F}.

We note that I_{F} ×SU(2) ⊂ W_{F} ×SU(2) and in fact I_{F} ×SU(2) is the maximal compact subgroup of ℒ_{F}. Now let ϕ be an L-parameter. Moving (if necessary) to another point in the orbit 𝒪(ϕ) we can write ϕ in the canonical form

ϕ = ϕ_{1} ⊕ … ⊕ ϕ_{1} ⊕ … ⊕ ϕ_{k} ⊕ … ⊕ ϕ_{k}

where ϕ_{1} is repeated l_{1} times, …, ϕ_{k} is repeated l_{k} times, and the representations

ϕ_{j}|_{I_{F} ×SU(2)}

are irreducible and pairwise inequivalent, 1 ≤ j ≤ k. We will now write k = k(ϕ). This natural number is an invariant of the orbit 𝒪(ϕ). We have

𝒪(ϕ) = Sym^{l_{1}}ℂ^{×} × … × Sym^{l_{k}}ℂ^{×} the product of symmetric products of ℂ^{×}.

Theorem 1.6. The set Φ(GL(n)) has the structure of complex algebraic variety. Each irreducible component 𝒪(ϕ) is isomorphic to the product of a complex affine space and a complex torus

𝒪(ϕ) = 𝔸^{l} × (ℂ^{×})^{k}

where k = k(ϕ).

Proof. Let Y = 𝕍(x_{1}y_{1} − 1, …, x_{n}y_{n} − 1) ⊂ ℂ^{2n}. Then Y is a Zariski-closed set in ℂ^{2n}, and so is an affine complex algebraic variety. Let X = (ℂ^{×})^{n}. Set α : Y → X, α(x_{1}, y_{1}, …, x_{n}, y_{n}) = (x_{1}, …, x_{n}) and β : X → Y, β(x_{1}, …, x_{n}) = (x_{1}, x_{1}^{−1}, …, x_{n}, x_{n}^{−1}). So X can be embedded in affine space ℂ^{2n} as a Zariskiclosed subset. Therefore X is an affine algebraic variety, as in [36, p.50]. Let A = ℂ[X] be the coordinate ring of X. This is the restriction to X of polynomials on ℂ^{2n}, and so A = ℂ[X] = ℂ[x_{1}, x_{1}^{−1}, …, x_{n}, x_{n}^{−1}], the ring of Laurent polynomials in n variables x_{1}, …, x_{n}. Let S_{n} be the symmetric group, and let Z denote the quotient variety X/S_{n}. The variety Z is an affine complex algebraic variety. The coordinate ring of Z is

ℂ[Z] ≃ ℂ[x_{1}, …, x_{n}, x_{1}^{−1}, …, x_{n}^{−1}]^{S_{n}}.

Let σ_{i}, i = 1, …, n be the elementary symmetric polynomials in n variables. Then from the last isomorphism we have

ℂ[Z] ≃ ℂ[x_{1}, …, x_{n}]^{S_{n}} ⊗ ℂ[σ_{n}^{−1}]
≃ ℂ[σ_{1}, …, σ_{n}] ⊗ ℂ[σ_{n}^{−1}]
≃ ℂ[σ_{1}, …, σ_{n−1}] ⊗ ℂ[σ_{n}, σ_{n}^{−1}]
≃ ℂ[𝔸^{n−1}] ⊗ ℂ[𝔸 − {0}]
≃ ℂ[𝔸^{n−1} × (𝔸 − {0})]

where 𝔸^{n} denotes complex affine n-space. The coordinate ring of the quotient variety ℂ^{×n}/S_{n} is isomorphic to the coordinate ring of 𝔸^{n−1} × (𝔸 − {0}). Now the categories of affine algebraic varieties and of finitely generated reduced ℂ-algebras are equivalent, see [36, p.26]. Therefore the variety ℂ^{×n}/S_{n} is isomorphic to the variety 𝔸^{n−1} × (𝔸 − {0}). Consider 𝔸 − {0} = 𝕍(f) where f(x) = x_{1}x_{2} − 1. Then ∂f/∂x_{1} = x_{2} ≠ 0 and ∂f/∂x_{2} = x_{1} ≠ 0 on the variety 𝕍(f). So 𝔸 − {0} is smooth. Then 𝔸^{n−1} × (𝔸 − {0}) is smooth. Therefore the quotient variety ℂ^{×n}/S_{n} is a smooth complex affine algebraic variety of dimension n. Now each orbit 𝒪(ϕ) is a product of symmetric products of ℂ^{×}. Therefore each orbit 𝒪(ϕ) is a smooth complex affine algebraic variety. We have

𝒪(ϕ) = Sym^{l_{1}}ℂ^{×} × … × Sym^{l_{k}}ℂ^{×} = 𝔸^{l} × (ℂ^{×})^{k}

where l = l_{1} + … + l_{k} − k and k = k(ϕ). □

We now transport the complex structure from Φ(GL(n)) to Π(GL(n)) via the local Langlands correspondence. This leads to the next result.

Theorem 1.7. The smooth dual Π(GL(n)) has a natural complex structure. Each irreducible component is a smooth complex affine algebraic variety.

The smooth dual Π(GL(n)) has countably many irreducible components of each dimension d with 1 ≤ d ≤ n. The irreducible supercuspidal representations of GL(n) arrange themselves into the 1-dimensional tori. It follows from Theorems 1.6 and 1.7 that the smooth dual Π(GL(n)) is a complex manifold. Then ℂ × Π(GL(n)) is a complex manifold. So the local L-factor L(s, π_{v}) and the local ϵ-factor ϵ(s, π_{v}) are functions of several complex variables:

L : ℂ × Π(GL(n)) ⟶ ℂ
ϵ : ℂ × Π(GL(n)) ⟶ ℂ.

Example 1.8. Unramified representations. Let ψ_{1}, …, ψ_{n} be unramified quasicharacters of the Weil group W_{F}. Then we have

ψ_{j}(w) = z_{j}^{d(w)}

with z_{j} ∈ ℂ^{×} for all 1 ≤ j ≤ n. Let ϕ be the L-parameter given by ψ_{1}⊕…⊕ψ_{n}. Then the image π_{F} (ϕ) of ϕ under the local Langlands correspondence π_{F} is an unramified principal series representation. For the local L-factors L(s, π) see [23, p. 377]. The local L-factor attached to such an unramified representation of GL(n) is given by

L(s, π_{F} (ϕ)) = ∏_{j=1}^{n}(1 − z_{j}q^{−s})^{−1}.

This exhibits the local L-factor as a function on the complex manifold ℂ × Sym^{n}ℂ^{×}.

2. The structure of the based ring J

Let W be the extended affine Weyl group associated to GL(n, ℂ). For each two-sided cell c of W we have a corresponding partition λ of n. Let μ be the dual partition of λ. Let u be a unipotent element in GL(n, ℂ) whose Jordan blocks are determined by the partition μ. Let the distinct parts of the dual partition μ be μ_{1}, …, μ_{p} with μ_{r} repeated n_{r} times, 1 ≤ r ≤ p. Let C_{G}(u) be the centralizer of u in G = GL(n, ℂ). Then the maximal reductive subgroup F_{c} of C_{G}(u) is isomorphic to GL(n_{1}, ℂ)×GL(n_{2}, ℂ)×⋯×GL(n_{p}, ℂ). Following Lusztig [27] and Xi [40, 1.5] let J be the free ℤ-module with basis {t_{w} | w ∈ W }. The multiplication t_{w}t_{u} = ∑_{v∈W} γ_{w,u,v}t_{v} defines an associative ring structure on J. The ring J is the based ring of W. For each two-sided cell c of W the ℤ-submodule J_{c} of J, spanned by all t_{w}, w ∈ c, is a two-sided ideal of J. The ring J_{c} is the based ring of the two-sided cell c. Let |Y | be the number of left cells contained in c. The Lusztig conjecture says that there is a ring isomorphism

J_{c} ≃ M_{|Y |}(R_{F_{c}}), t_{w} ↦ π(w)

where R_{F_{c}} is the rational representation ring of F_{c}. This conjecture for GL(n, ℂ) has been proved by Xi [40, 1.5, 4.1, 8.2]. Since F_{c} is isomorphic to a direct product of the general linear groups GL(n_{i}, ℂ) (1 ≤ i ≤ p) we see that R_{F_{c}} is isomorphic to the tensor product over ℤ of the representation rings R_{GL(n_{i},ℂ)}, 1 ≤ i ≤ p. For the ring R_{GL(n,ℂ)} we have

R_{GL(n,ℂ)} ≃ ℤ[X_{1}, X_{2}, …, X_{n}][X_{n}^{−1}]

where the elements X_{1}, X_{2}, …, X_{n}, X_{n}^{−1} are described in [40, 4.2][6, IX.125]. Then

\begin{aligned}R_{GL(n,ℂ)} & ≃ & ℤ[σ_{1}, …, σ_{n}, σ_{n}^{−1}] \\ & ≃ & ℤ[x_{1}, …, x_{n}, x_{1}^{−1}, …, x_{n}^{−1}]^{S_{n}} \end{aligned}

We have

R_{GL(n,ℂ)} ⊗_{ℤ} ℂ ≃ ℂ[Sym^{n}ℂ^{×}]

and

R_{F_{c}} ⊗_{ℤ} ℂ ≃ ℂ[Sym^{n_{1}}ℂ^{×} × ⋯ × Sym^{n_{p}}ℂ^{×}]

We recall the extended quotient. Let the finite group Γ act on the space X. Let X̃ = {(x, γ) : γx = x}, let Γ act on X̂ by γ_{1}(x, γ) = (γ_{1}x, γ_{1}γγ_{1}^{−1}). Then X̃/Γ is the extended quotient of X by Γ, and we have

X̃/Γ = ⨆ X^{γ}/Z(γ)

where one γ is chosen in each Γ-conjugacy class. There is a canonical projection X̃/Γ → X/Γ. Let γ ∈ S_{n} have cycle type μ, let X = (ℂ^{×})^{n}. Then

\begin{aligned}X^{γ} & ≃ & (C^{×})^{n_{1}} × ⋯ × (C^{×})^{n_{p}} \\ Z(γ) & ≃ & (ℤ/μ_{1}ℤ) ≀ S_{n_{1}} × ⋯ × (ℤ/μ_{p}ℤ) ≀ S_{n_{p}} \\ X^{γ}/Z(γ) & ≃ & Sym^{n_{1}}ℂ^{×} × ⋯ × Sym^{n_{p}}ℂ^{×} \end{aligned}

and so

R_{F_{c}} ⊗_{ℤ} ℂ ≃ ℂ[X^{γ}/Z(γ)]

Then

J ⊗_{ℤ} ℂ = ⊕_{c}(J_{c} ⊗_{ℤ} ℂ) ∼ ⊕_{c}(R_{F_{c}} ⊗_{ℤ} ℂ) ≃ ℂ[X̃/S_{n}]

The algebra J ⊗_{ℤ} ℂ is Morita equivalent to a reduced, finitely generated, commutative unital ℂ-algebra, namely the coordinate ring of the extended quotient X̃/S_{n}.

3. Periodic cyclic homology of the Hecke algebra

The Bernstein variety Ω(G) of G is the set of G-conjugacy classes of pairs (M, σ), where M is a Levi (i.e. block-diagonal) subgroup of G, and σ is an irreducible supercuspidal representation of M. Each irreducible smooth representation of G is a subquotient of an induced representation i_{GM}σ. The pair (M, σ) is unique up to conjugacy. This creates a finite-to-one map, the infinitesimal character, from Π(G) onto Ω(G). Let Ω(G) be the Bernstein variety of G. Each point in Ω(G) is a conjugacy class of cuspidal pairs (M, σ). A quasicharacter ψ : M ⟶ ℂ^{×} is unramified if ψ is trivial on M^{∘}. The group of unramified quasicharacters of M is denoted Ψ(M). We have Ψ(M) ≅ (ℂ^{×})^{ℓ} where ℓ is the parabolic rank of M. The group Ψ(M) now creates orbits: the orbit of (M, σ) is {(M, ψ ⊗ σ) : ψ ∈ Ψ(M)}. Denote this orbit by D, and set Ω = D/W (M, D), where W (M) is the Weyl group of M and W (M, D) is the subgroup of W (M) which leaves D globally invariant. The orbit D has the structure of a complex torus, and so Ω is a complex algebraic variety. We view Ω as a component in the algebraic variety Ω(G). The Bernstein variety Ω(G) is the disjoint union of ordinary quotients. We now replace the ordinary quotient by the extended quotient to create a new variety Ω^{+}(G). So we have

Ω(G) = ⨆ D/W (M, D) and Ω^{+}(G) = ⨆ D̃/W (M, D)

Let Ω be a component in the Bernstein variety Ω(GL(n)), and let ℋ(G) = ⨁ ℋ(Ω) be the Bernstein decomposition of the Hecke algebra. Let

Π(Ω) = (inf.ch.)^{−1}Ω.

Then Π(Ω) is a smooth complex algebraic variety with finitely many irreducible components. We have the following Bernstein decomposition of Π(G):

Π(G) = ⨆ Π(Ω).

Let M be a compact C^{∞} manifold. Then C^{∞}(M) is a Fréchet algebra, and we have Connes’ fundamental theorem [14, Theorem 2, p. 208]:

HP_{∗}(C^{∞}(M)) ≅ H^{∗}(M; ℂ).

Now the ideal ℋ(Ω) is a purely algebraic object, and, in computing its periodic cyclic homology, we would hope to find an algebraic variety to play the role of the manifold M. This algebraic variety is Π(Ω).

Theorem 3.1. Let Ω be a component in the Bernstein variety Ω(G). Then the periodic cyclic homology of ℋ(Ω) is isomorphic to the periodised de Rham cohomology of Π(Ω):

HP_{∗}(ℋ(Ω)) ≅ H^{∗}(Π(Ω); ℂ).

Proof. We can think of Ω as a vector (τ_{1}, …, τ_{r}) of irreducible supercuspidal representations of smaller general linear groups, the entries of this vector being only determined up to tensoring with unramified quasicharacters and permutation. If the vector is equivalent to (σ_{1}, …, σ_{1}, …, σ_{r}, …, σ_{r}) with σ_{j} repeated e_{j} times, 1 ≤ j ≤ r, and σ_{1}, …, σ_{r} are pairwise distinct, then we say that Ω has exponents e_{1}, …, e_{r}. Then there is a Morita equivalence

ℋ(Ω) ∼ ℋ(e_{1}, q_{1}) ⊗ … ⊗ ℋ(e_{r}, q_{r})

where q_{1}, …, q_{r} are natural number invariants attached to Ω. This result is due to Bushnell-Kutzko [11, 12, 13]. We describe the steps in the proof. Let (ρ, W) be an irreducible smooth representation of the compact open subgroup K of G. As in [12, 4.2], the pair (K, ρ) is an Ω-type in G if and only if, for (π, V) ∈ Π(G), we have inf.ch.(π) ∈ Ω if and only if π contains ρ. The existence of an Ω-type in GL(n), for each component Ω in Ω(GL(n)), is established in [13, 1.1]. So let (K, ρ) be an Ω-type in GL(n). As in [12, 2.9], let

e_{ρ}(x) = (volK)^{−1}(dim ρ) Trace_{W} (ρ(x^{−1}))

for x ∈ K and 0 otherwise. Then e_{ρ} is an idempotent in the Hecke algebra ℋ(G). Then we have

ℋ(Ω) ≅ ℋ(G) ∗ e_{ρ} ∗ ℋ(G)

as in [12, 4.3] and the two-sided ideal ℋ(G) ∗ e_{ρ} ∗ ℋ(G) is Morita equivalent to e_{ρ} ∗ ℋ(G) ∗ e_{ρ}. Now let ℋ(K, ρ) be the endomorphism-valued Hecke algebra attached to the semisimple type (K, ρ). By [12, 2.12] we have a canonical isomorphism of unital ℂ-algebras :

ℋ(G, ρ) ⊗_{ℂ} End_{ℂ} W ≅ e_{ρ} ∗ ℋ(G) ∗ e_{ρ}

so that e_{ρ} ∗ ℋ(G) ∗ e_{ρ} is Morita equivalent to ℋ(G, ρ). Now we quote the main theorem for semisimple types in GL(n) [13, 1.5]: there is an isomorphism of unital ℂ-algebras

ℋ(G, ρ) ≅ ℋ(G_{1}, ρ_{1}) ⊗ … ⊗ ℋ(G_{r}, ρ_{r})

The factors ℋ(G_{i}, ρ_{i}) are (extended) affine Hecke algebras whose structure is given explicitly in [11, 5.6.6]. This structure is in terms of generators and relations [11, 5.4.6]. So let ℋ(e, q) denote the affine Hecke algebra associated to the affine Weyl group ℤ^{e} ⋊ S_{e}. Putting all this together we obtain a Morita equivalence

ℋ(Ω) ∼ ℋ(e_{1}, q_{1}) ⊗ … ⊗ ℋ(e_{r}, q_{r})

The natural numbers q_{1}, …, q_{r} are specified in [11, 5.6.6]. They are the cardinalities of the residue fields of certain extension fields E_{1}/F, …, E_{r}/F. Using the Künneth formula the calculation of HP_{∗}(ℋ(Ω)) is reduced to that of the affine Hecke algebra ℋ(e, q). Baum and Nistor demonstrate the spectral invariance of periodic cyclic homology in the class of finite type algebras [3, 4]. Now ℋ(e, q) is the Iwahori-Hecke algebra associated to the extended affine Weyl group ℤ^{e} ⋊ S_{e}, and let J be the asymptotic Hecke algebra (based ring) associated to ℤ^{e}⋊S_{e}. According to [3, 4], Lusztig’s morphisms ϕ_{q} : ℋ(e, q) → J induce isomorphisms

(ϕ_{q})_{∗} : HP_{∗}(ℋ(e, q)) → HP_{∗}(J)

for all q ∈ ℂ^{×} that are not proper roots of unity. At this point we can back track and deduce that

HP_{∗}(ℋ(e, q)) ≃ HP_{∗}(J) ≃ HP_{∗}(ℋ_{1})

and use the fact that ℋ(e, 1) ≃ ℂ[ℤ^{e} ⋊ S_{e}]. It is much more illuminating to quote Xi’s proof of the Lusztig conjecture for the based ring J, see Section 2. Then we have

HP_{∗}(ℋ(e, q)) ≃ HP_{∗}(J) ≃ HP_{∗}(ℂ[{ \atop (ℂ^{×})^{e}}/S_{e}]) ≃ H^{∗}({ \atop (ℂ^{×})^{e}}/S_{e}; ℂ).

If Ω has exponents e_{1}, …, e_{r} then e_{1} + … + e_{r} = d(Ω) = dim_{ℂ} Ω, and W (Ω) is a product of symmetric groups:

W (Ω) = S_{e_{1}} × … × S_{e_{r}}

We have

HP_{∗}(ℋ(Ω)) ≃ HP_{∗}(ℋ(e_{1}, q_{1}) ⊗ ⋯ ⊗ ℋ(e_{r}, q_{r}))
≃ HP_{∗}(ℋ(e_{1}, q_{1})) ⊗ ⋯ ⊗ HP_{∗}(ℋ(e_{r}, q_{r}))
≃ H^{∗}({ \atop (ℂ^{×})^{e_{1}}}/S_{e_{1}}; ℂ) ⊗ ⋯ ⊗ H^{∗}({ \atop (ℂ^{×})^{e_{r}}}/S_{e_{r}}; ℂ)

Now the extended quotient is multiplicative, i.e.

{ \atop (ℂ^{×})^{d(Ω)}}/W (Ω) = { \atop (ℂ^{×})^{e_{1}}}/S_{e_{1}} × ⋯ × { \atop (ℂ^{×})^{e_{r}}}/S_{e_{r}}

which implies that

HP_{∗}(ℋ(Ω)) = H^{∗}({ \atop (ℂ^{×})^{d(Ω)}}/W (Ω); ℂ)

Recall that

\begin{aligned}Ω & = & (ℂ^{×})^{d(Ω)}/W (Ω) \\ Ω^{+} & = & { \atop (ℂ^{×})^{d(Ω)}}/W (Ω) \end{aligned}

and by [8, p. 217] we have Π(Ω) ≃ Ω^{+}. It now follows that

HP_{∗}(ℋ(Ω)) ≃ H^{∗}(Π(Ω); ℂ)
□

Lemma 3.2. Let Ω be a component in the variety Ω(G) and let Ω have exponents {e_{1}, …, e_{r}}. Then for j = 0, 1 we have

dim_{ℂ} HP_{j} ℋ(Ω) = 2^{r−1}β(e_{1}) ⋯ β(e_{r})

where

β(e) = ∑_{|λ|=e} 2^{α(λ)−1}

and where α(λ) is the number of unequal parts of λ. Here |λ| is the weight of λ, i.e. the sum of the parts of λ so that λ is a partition of e.

Proof. Suppose first that Ω has the single exponent e. By Theorem 3.1 the periodic cyclic homology of ℋ(Ω) is isomorphic to the periodised de Rham cohomology of the extended quotient of (ℂ^{×})^{e} by the symmetric group S_{e}. The components in this extended quotient correspond to the partitions of e. In fact, if α(λ) is the number of unequal parts in the partition λ then the corresponding component is homotopy equivalent to the compact torus of dimension α(λ). We now proceed by induction, using the fact that the extended quotient is multiplicative and the Künneth formula. □

Theorem 3.1, combined with the calculation in [7], now leads to the next result.

Theorem 3.3. The inclusion ℋ(G) ⟶ 𝒮(G) induces an isomorphism at the level of periodic cyclic homology:

HP_{∗}(ℋ(G)) ≃ HP_{∗}(𝒮(G)).

Remark 3.4. We now consider further the disjoint union

Φ(Ω) = 𝒪(ϕ_{1}) ⊔ ⋯ ⊔ 𝒪(ϕ_{r}) ≃ Ω^{+}

If we apply the local Langlands correspondence π_{F} then we obtain

Π(Ω) = π_{F} (𝒪(ϕ_{1})) ⊔ ⋯ ⊔ π_{F} (𝒪(ϕ_{r})) ≃ Ω^{+}

This partition of Π(Ω) is identical to that in Schneider-Zink [34, p. 198], modulo notational differences. In their notation, for each 𝒫 ∈ ℬ there is a natural map

Q_{𝒫} : X_{nr}(N_{𝒫} ) → Irr(Ω)

such that

Irr(Ω) = ⨆_{𝒫∈ℬ} im(Q_{𝒫} ).

In fact this is a special stratification of Irr(Ω) in the precise sense of their article [34, p.198]. Let

Z_{𝒫} = ⋃_{𝒫′≤𝒫} im(Q_{𝒫} )

Then Z_{𝒫} is a Jacobson closed set, in fact Z_{𝒫} = V (J_{𝒫} ), where J_{𝒫} is a certain 2-sided ideal [34, p.198]. We note that the set Z_{𝒫} is also closed in the topology of the present article: each component in Ω^{+} is equipped with the classical (analytic) topology. Issues of stratification play a dominant role in [34]. The stratification of the tempered dual Π^{t}(GL(n)) arises from their construction of tempered K-types, see [34, p. 162, p. 189]. In the context of the present article, there is a natural stratification-by-dimension as follows. Let 1 ≤ k ≤ n and define

k-stratum = {𝒪(ϕ) | dim_{ℂ} 𝒪(ϕ) ≤ k}

If π_{F} (𝒪(ϕ)) is the complexification of the component Θ ⊂ Π^{t}(G) then we have

dim_{ℝ} Θ = dim_{ℂ} 𝒪(ϕ).

The partial order in [34] on the components Θ transfers to a partial order on complex orbits 𝒪(ϕ). This partial order originates in the opposite of the natural partial order on partitions, and the partitions manifest themselves in terms of Langlands parameters. For example, let

ϕ = ρ ⊗ spin(j_{1}) ⊕ ⋯ ⊕ ρ ⊗ spin(j_{r})
ϕ′ = ρ ⊗ spin(j′_{1}) ⊕ ⋯ ⊕ ρ ⊗ spin(j′_{r})

Let λ_{1} = 2j_{1} + 1, …, λ_{r} = 2j_{r} + 1, μ_{1} = 2j′_{1} + 1, …, μ_{r} = 2j′_{r} + 1 and define partitions as follows

\begin{aligned}λ & = & (λ_{1}, …, λ_{r}), λ_{1} ≥ λ_{2} ≥ … \\ μ & = & (μ_{1}, …, μ_{r}), μ_{1} ≥ μ_{2} ≥ … \end{aligned}

The natural partial order on partitions is: λ ≤ μ if and only if

λ_{1} + ⋯ + λ_{i} ≤ μ_{1} + ⋯ + μ_{i}

for all i ≥ 1, see [28, p.6]. Let l(λ) be the length of λ, that is the number of parts in λ. Then dim_{ℂ} 𝒪(ϕ) = l(λ). Let λ′, μ′ be the dual partitions as in [28]. Then we have [28, 1.11] λ ≥ μ if and only if μ′ ≥ λ′. Note that l(λ) = λ′_{1}, l(μ) = μ′_{1}. Then

Θ_{λ} ≤ Θ_{μ} ⇔ λ ≥ μ ⇔ μ′ ≥ λ′ ⇒ λ′_{1} ≤ μ′_{1}

So if Θ_{λ} ≤ Θ_{μ} then dim_{ℝ} Θ_{λ} ≤ dim_{ℝ} Θ_{μ}, similarly 𝒪(ϕ) ≤ 𝒪(ϕ′) implies dim_{ℂ} 𝒪(ϕ) ≤ dim_{ℂ} 𝒪(ϕ′). Stratification-by-dimension is finer than the Schneider-Zink stratification [34].

Let now R denote the ring of all regular functions on Π(G). The ring R is a commutative, reduced, unital ring over ℂ which is not finitely generated. We will call R the extended centre of G. It is natural to believe that the extended centre R of G is the centre of an ‘extended category’ made from smooth Gmodules. The work of Schneider-Zink [34, p. 201] contains various results in this direction.

4. The q-projection

Let Ω be a component in the Bernstein variety. This component is an ordinary quotient D/Γ. We now consider the extended quotient D̃/Γ = ⨆ D^{γ}/Z_{γ}, where D is the complex torus ℂ^{×m}. Let γ be a permutation of n letters with cycle type

γ = (1 … α_{1}) ⋯ (1 … α_{r})

where α_{1} + ⋯ + α_{r} = m. On the fixed set D^{γ} the map π_{q}, by definition, sends the element (z_{1}, …, z_{1}, …, z_{r}, …, z_{r}) where z_{j} is repeated α_{j} times, 1 ≤ j ≤ r, to the element

(q^{(α_{1}−1)/2}z_{1}, …, q^{(1−α_{1})/2}z_{1}, …, q^{(α_{r}−1)/2}z_{r}, …, q^{(1−α_{r})/2}z_{r})

The map π_{q} induces a map from D^{γ}/Z_{γ} to D/Γ, and so a map, still denoted π_{q}, from the extended quotient D̃/Γ to the ordinary quotient D/Γ. This creates a map π_{q} from the extended Bernstein variety to the Bernstein variety:

π_{q} : Ω^{+}(G) ⟶ Ω(G).

Definition 4.1. The map π_{q} is called the q-projection.

The q-projection π_{q} occurs in the following commutative diagram [8]:

Φ(G) ⟶ Π(G)
α↓  ↓^{inf. ch.}
Ω^{+}(G) ⟶^{π_{q}} Ω(G)

Let A, B be commutative rings with A ⊂ B, 1 ∈ A. Then the element x ∈ B is integral over A if there exist a_{1}, …, a_{n} ∈ A such that

x^{n} + a_{1}x^{n−1} + … + a_{n} = 0.

Then B is integral over A if each x ∈ B is integral over A. Let X, Y be affine varieties, f : X ⟶ Y a regular map such that f(X) is dense in Y. Then the pull-back f^{#} defines an isomorphic inclusion ℂ[Y] ⟶ ℂ[X]. We view ℂ[Y] as a subring of ℂ[X] by means of f^{#}. Then f is a finite map if ℂ[X] is integral over ℂ[Y], see [35]. This implies that the pre-image F^{−1}(y) of each point y ∈ Y is a finite set, and that, as y moves in Y, the points in F^{−1}(y) may merge together but not disappear. The map 𝔸^{1} −{0} ⟶ 𝔸^{1} is the classic example of a map which is not finite.

Lemma 4.2. Let X be a component in the extended variety Ω^{+}(G). Then the q-projection π_{q} is a finite map from X onto its image π_{q}(X).

Proof. Note that the fixed-point set D^{γ} is a complex torus of dimension r, that π_{q}(D^{γ}) is a torus of dimension r and that we have an isomorphism of affine varieties D^{γ} ≅ π_{q}(D^{γ}). Let X = D^{γ}/Z_{γ}, Y = π_{q}(D^{γ})/Γ where Z_{γ} is the Γcentralizer of γ. Now each of X and Y is a quotient of the variety D^{γ} by a finite group, hence X, Y are affine varieties [35, p.31]. We have D^{γ} ⟶ X ⟶ Y and ℂ[Y] ⟶ ℂ[X] ⟶ ℂ[D^{γ}]. According to [35, p.61], ℂ[D^{γ}] is integral over ℂ[Y] since Y = D^{γ}/Γ. Therefore the subring ℂ[X] is integral over ℂ[Y]. So the map π_{q} : X ⟶ Y is finite. □

Example 4.3. GL(2). Let T be the diagonal subgroup of G = GL(2) and let Ω be the component in Ω(G) containing the cuspidal pair (T, 1). Then σ ∈ Π(GL(2)) is arithmetically unramified if inf.ch.σ ∈ Ω. If π_{F} (ϕ) = σ then ϕ is a 2-dimensional representation of ℒ_{F} and there are two possibilities: ϕ is reducible, ϕ = ψ_{1} ⊕ ψ_{2} with ψ_{1}, ψ_{2} unramified quasicharacters of W_{F}. So ψ_{j}(w) = z_{j}^{d(w)}, z_{j} ∈ ℂ^{×}, j = 1, 2. We have π_{F} (ϕ) = Q(ψ_{1}, ψ_{2}) where ψ_{1} does not precede ψ_{2}. In particular we obtain the 1-dimensional representations of G as follows:

π_{F} (| |^{1/2}ψ ⊕ | |^{−1/2}ψ) = Q(| |^{1/2}ψ, | |^{−1/2}ψ) = ψ ∘ det.

ϕ is irreducible, ϕ = ψ ⊗ spin(1/2). Then π_{F} (ϕ) = Q(Δ) with Δ = {| |^{−1/2}ψ, | |^{1/2}ψ} so π_{F} (ϕ) = ψ ⊗ St(2) where St(2) is the Steinberg representation of GL(2). The orbit of (T, 1) is D = (ℂ^{×})^{2}, and W (T, D) = ℤ/2ℤ. Then Ω ≅ (ℂ^{×})^{2}/ℤ/2ℤ ≅ Sym^{2} ℂ^{×}. The extended quotient is Ω^{+} = Sym^{2} ℂ^{×} ⊔ ℂ^{×}. The q−projection works as follows:

π_{q} : {z_{1}, z_{2}} ↦ {z_{1}, z_{2}}
π_{q} : z ↦ {q^{1/2}z, q^{−1/2}z}

where q is the cardinality of the residue field of F.

Let A = ℋ(GL(2)//I) be the Iwahori-Hecke algebra of GL(2). This is a finite type algebra. Following [21, p. 327], denote by Prim_{n}(A) ⊂ Prim(A) the set of primitive ideals B ⊂ A which are kernels of irreducible representations of A of dimension n. Set X_{1} = Prim_{1}(A), X_{2} = Prim_{1}(A) ⊔ Prim_{2}(A) = Prim(A). Then X_{1} and X_{2} are closed sets in Prim(A) defining an increasing filtration of Prim(A). Now A is Morita equivalent to the Bernstein ideal ℋ(Ω), and Π(Ω) ≃ Prim(A). Let ϕ_{1} = 1 ⊗ spin(1/2), ϕ_{2} = 1 ⊗ 1 ⊕ 1 ⊗ 1. The 1-dimensional representations of GL(2) determine 1-dimensional representations of ℋ(G//I) and so lie in X_{1}. The L-parameters of the 1-dimensional representations of GL(2) do not lie in the 1-dimensional orbit 𝒪(ϕ_{1}): they lie in the 2-dimensional orbit 𝒪(ϕ_{2}). The Kazhdan-Nistor-Schneider stratification [21] does not coincide with stratification-by-dimension.

Example 4.4. GL(3). In the above example, the q-projection is stratifiedinjective, i.e. injective on each orbit type. This is not so in general, as shown by the next example. Let T be the diagonal subgroup of GL(3) and let Ω be the component containing the cuspidal pair (T, 1). Then Ω = Sym^{3} ℂ^{×} and

Ω^{+} = Sym^{3} ℂ^{×} ⊔ (ℂ^{×})^{2} ⊔ ℂ^{×}

. The map π_{q} works as follows:

{z_{1}, z_{2}, z_{3}} ↦ {z_{1}, z_{2}, z_{3}}
(z, w, w) ↦ {z, q^{1/2}w, q^{−1/2}w}
(z, z, z) ↦ {qz, z, q^{−1}z}.

Consider the L-parameter

ϕ = ψ_{1} ⊗ 1 ⊕ ψ_{2} ⊗ spin(1/2) ∈ Φ(GL(3)).

If ψ(w) = z^{d(w)} then we will write ψ = z. With this understood, let

ϕ_{1} = q ⊗ 1 ⊕ q^{−1/2} ⊗ spin(1/2)
ϕ_{2} = q^{−1} ⊗ 1 ⊕ q^{1/2} ⊗ spin(1/2).

Then α(ϕ_{1}), α(ϕ_{2}) are distinct points in the same stratum of the extended quotient, but their image under the q-projection π_{q} is the single point {q^{−1}, 1, q} ∈ Sym^{3} ℂ^{×}. Let

ϕ_{3} = 1 ⊗ spin(3/2)
ϕ_{4} = q^{−1} ⊗ 1 ⊕ 1 ⊗ 1 ⊕ q ⊗ 1.

Then the distinct L-parameters ϕ_{1}, ϕ_{2}, ϕ_{3}, ϕ_{4} all have the same image under the q-projection π_{q}.

5. The diagram

In this section we create a diagram which incorporates several major results. The following diagram serves as a framework for the whole article:

[figure: K_{∗}^{top}(G); K_{∗}(C_{r}^{∗}(G)); H_{∗}(G; βG); HP_{∗}(ℋ(G)); HP_{∗}(𝒮(G)); H_{c}^{∗}(Φ(G); ℂ); H_{c}^{∗}(Π(G); ℂ); H_{c}^{∗}(Π^{t}(G); ℂ); ch; μ; ch; ı_{∗}]

The Baum-Connes assembly map μ is an isomorphism [1, 24]. The map

H_{∗}(G; βG) → HP_{∗}(ℋ(G))

is an isomorphism [20, 33]. The map ı_{∗} is an isomorphism by Theorem 3.3. The right hand Chern character is constructed in [9] and is an isomorphism after tensoring over ℤ with ℂ [9, Theorem 3]. The Chern character on the left hand side of the diagram is the unique map for which the top half of the diagram is commutative. In the diagram, H_{c}^{∗}(Π^{t}(G); ℂ) denotes the (periodised) compactly supported de Rham cohomology of the tempered dual Π^{t}(G), and H_{c}^{∗}(Π(G); ℂ) denotes the (periodised) de Rham cohomology supported on finitely many components of the smooth dual Π(G). The map

HP_{∗}(𝒮(G)) → H_{c}^{∗}(Π^{t}(G); ℂ)

is constructed in [7] and is an isomorphism [7, Theorem 7]. The map

H_{c}^{∗}(Π(G); ℂ) → H_{c}^{∗}(Π^{t}(G); ℂ)

is constructed in the following way. Given an L-parameter ϕ : ℒ_{F} → GL(n, ℂ) we have

ϕ = ϕ_{1} ⊕ … ⊕ ϕ_{m}

with each ϕ_{j} an irreducible representation. We have ϕ_{j} = ρ_{j} ⊗ spin(j) where each ρ_{j} is an irreducible representation of the Weil group W_{F}. We shall assume that det ρ_{j} is a unitary character. Let 𝒪(ϕ) be the orbit of ϕ as in Definition 1.4. The map 𝒪(ϕ) → 𝒪^{t}(ϕ) is now defined as follows

ψ_{1}ϕ_{1} ⊕ … ⊕ ψ_{m}ϕ_{m} ↦ |ψ_{1}|^{−1} · ψ_{1}ϕ_{1} ⊕ … ⊕ |ψ_{m}|^{−1} · ψ_{m}ϕ_{m}.

This map is a deformation retraction of the complex orbit 𝒪(ϕ) onto the compact orbit 𝒪^{t}(ϕ). Since Φ(G) is a disjoint union of such complex orbits this formula determines, via the local Langlands correspondence for GL(n), a deformation retraction of Π(G) onto the tempered dual Π^{t}(GL(n)), which implies that the induced map on cohomology is an isomorphism. The map

H_{c}^{∗}(Φ(G); ℂ) → H_{c}^{∗}(Π(G); ℂ)

is an isomorphism, induced by the local Langlands correspondence π_{F}. The map

HP_{∗}(ℋ(G)) → H_{c}^{∗}(Π(G); ℂ)

is an isomorphism by Theorem 3.1. There is at present no direct definition of the map

H_{∗}(G; βG) → H_{c}^{∗}(Φ(G); ℂ).

Suppose for the moment that F has characteristic 0 and has residue field of characteristic p. An irreducible representation ρ of the Weil group W_{F} is called wildly ramified if dim ρ is a power of p and ρ ≄ ρ ⊗ ψ for any unramified quasicharacter ψ ≠ 1 of W_{F}. We write Φ_{m}^{wr}(F) for the set of equivalence classes of such representations of dimension p^{m}. An irreducible supercuspidal representation π of GL(n) is wildly ramified if n is a power of p and π ≄ π ⊗ (ψ ∘ det) for any unramified quasicharacter ψ ≠ 1 of F^{×}. We write Π_{m}^{wr}(F) for the set of equivalence classes of such representations of GL(p^{m}, F). In this case Bushnell-Henniart [10] construct, for each m, a canonical bijection

π_{F,m} : Φ_{m}^{wr}(F) → Π_{m}^{wr}(F).

Now the maximal simple type (J, λ) of an irreducible supercuspidal representation determines an element in the chamber homology of the affine building [2, 6.7]. The construction of Bushnell-Henniart therefore determines a map from a subspace of H_{c}^{even}(Φ(G); ℂ) to a subspace of H_{0}(G; βG). In the context of the above diagram the Baum-Connes map has a geometric counterpart: it is induced by the deformation retraction of Π(GL(n)) onto the tempered dual Π^{t}(GL(n)).

6. Supercuspidal representations of GL(n)

In this section we track the fate of supercuspidal representations of GL(n) through the diagram constructed in the previous Section. Let ρ be an irreducible n-dimensional complex representation of the Weil group W_{F} such that det ρ is a unitary character and let ϕ = ρ ⊗ 1. Then ϕ is the L-parameter for a pre-unitary supercuspidal representation ω of GL(n). Let 𝒪(ϕ) be the orbit of ϕ and 𝒪^{t}(ϕ) be the compact orbit of ϕ. Then 𝒪(ϕ) is a component in the Bernstein variety isomorphic to ℂ^{×} and 𝒪^{t}(ϕ) is a component in the tempered dual, isomorphic to 𝕋. The L-parameter ϕ now determines the following data.

6.1. Let (J, λ) be a maximal simple type for ω in the sense of Bushnell and Kutzko [11, chapter 6]. Then J is a compact open subgroup of G and λ is a smooth irreducible complex representation of J. We will write

𝕋 = {ψ ⊗ ω : ψ ∈ Ψ^{t}(G)}

where Ψ^{t}(G) denotes the group of unramified unitary characters of G.

Theorem 6.1. Let K be a maximal compact subgroup of G containing J and form the induced representation W = Ind_{J}^{K}(λ). We then have

ℓ^{2}(G ×_{K} W) ≃ Ind_{K}^{G}(W) ≃ Ind_{J}^{G}(λ) ≃ ∫_{𝕋} πdπ.

Proof. The supercuspidal representation ω contains λ and, modulo unramified unitary twist, is the only irreducible unitary representation with this property [11, 6.2.3]. Now the Ahn reciprocity theorem expresses Ind_{J}^{G} as a direct integral [26, p.58]:

Ind_{J}^{G}(λ) = ∫ n(π, λ)πdπ

where dπ is Plancherel measure and n(π, λ) is the multiplicity of λ in π|_{J}. But the Hecke algebra of a maximal simple type is commutative (a Laurent polynomial ring). Therefore ω|_{J} contains λ with multiplicity 1 (thanks to C. Bushnell for this remark). We then have n(ψ ⊗ ω, λ) = 1 for all ψ ∈ Ψ^{t}(G). We note that Plancherel measure induces Haar measure on 𝕋, see [31]. The affine building of G is defined as follows [38, p. 49]:

βG = ℝ × βSL(n)

where g ∈ G acts on the affine line ℝ via t ↦ t + val(det(g)). Let G^{∘} = {g ∈ G : val(det(g)) = 0}. We use the standard model for βSL(n) in terms of equivalence classes of o_{F} - lattices in the n-dimensional F -vector space V. Then the vertices of βSL(n) are in bijection with the maximal compact subgroups of G^{∘}, see [32, 9.3]. Let P ∈ βG be the vertex for which the isotropy subgroup is K = GL(n, o_{F}). Then the G-orbit of P is the set of all vertices in βG and the discrete space G/K can be identified with the set of vertices in the affine building βG. Now the base space of the associated vector bundle G ×_{K} W is the discrete coset space G/K, and the Hilbert space of ℓ^{2}-sections of this homogeneous vector bundle is a realization of the induced representation Ind_{K}^{G}(W). □

The C_{0}(βG)-module structure is defined as follows. Let f ∈ C_{0}(βG), s ∈ ℓ^{2}(G ×_{K} W) and define

(fs)(v) = f(v)s(v)

for each vertex v ∈ βG. We proceed to construct a K-cycle in degree 0. This K-cycle is

(C_{0}(βG), ℓ^{2}(G ×_{K} W) ⊕ 0, 0)

interpreted as a ℤ/2ℤ-graded module. This triple satisfies the properties of a (pre)-Fredholm module [14, IV] and so creates an element in K_{0}^{top}(G). By Theorem 5.1 this generator creates a free C(𝕋)-module of rank 1, and so provides a generator in K_{0}(C_{r}^{∗}(G)).

6.2. The Hecke algebra of the maximal simple type (J, λ) is commutative (the Laurent polynomials in one complex variable). The periodic cyclic homology of this algebra is generated by 1 in degree zero and dz/z in degree 1. The corresponding summand of the Schwartz algebra 𝒮(G) is Morita equivalent to the Fréchet algebra C^{∞}(𝕋). By an elementary application of Connes’ theorem [14, Theorem 2, p. 208], the periodic cyclic homology of this Fréchet algebra is generated by 1 in degree 0 and dθ in degree 1.

6.3. The corresponding component in the Bernstein variety is a copy of ℂ^{×}. The cohomology of ℂ^{×} is generated by 1 in degree 0 and dθ in degree 1. The corresponding component in the tempered dual is the circle 𝕋. The cohomology of 𝕋 is generated by 1 in degree 0 and dθ in degree 1.

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Jacek Brodzki Faculty of Mathematical Studies University of Southampton Southampton SO17 1BJ, U.K. j.brodzki@maths.soton.ac.uk Roger Plymen Department of Mathematics University of Manchester Manchester M13 9PL, U.K. roger@maths.man.ac.uk