Complex Structure on the Smooth Dual of GL(n)

Let G denote the p-adic group GL(n). With the aid of
Langlands parameters, we equip each Bernstein component Z in the smooth dual of G with the structure of complex
algebraic variety. We prove that the periodic cyclic homology of the corresponding ideal in the Hecke algebra H(G) is isomorphic to the de Rham cohomology of Z. We show how the structure of the variety Z is related to Xi's affirmation of a conjecture of Lusztig for GL(n,C). The smooth dual of G admits a deformation retraction onto the tempered dual of G.


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): 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 (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 Then Π(Ω) is a smooth complex algebraic variety with finitely many irreducible components.We have the following Bernstein decomposition of Π(G): 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]: 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): 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]: 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 where G is a profinite group, Z is the profinite completion of Z, and v is the valuation.
Let F be a separable algebraic closure of F .Then the absolute Galois group G(F |F ) is the projective limit of the finite Galois groups G(E|F ) taken over the finite extensions E of F in F .Let F be the maximal unramified extension of F .The map d is in this case the projection map The group G( F |F ) is procyclic.It has a single topological generator: the Frobenius automorphism φ F of F |F .The Weil group W F is by definition the pre-image of < φ F > in G(F | F ).We thus have the surjective map 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(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( F |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] now create an isomorphism [30, p.69]: 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.
The naturality of the bijection involves compatibility of the L-factors andfactors 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.
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 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 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 = 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 where φ 1 is repeated l 1 times, . .., φ k is repeated l k times, and the representations φ j | IF ×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 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 and so is an affine complex algebraic variety.
. 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 , 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 . ., n be the elementary symmetric polynomials in n variables.Then from the last isomorphism we have 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 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 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 -factor (s, π v ) are functions of several complex variables: Example 1.8.Unramified representations.Let ψ 1 , . . ., ψ n be unramified quasicharacters of the Weil group W F .Then we have 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 This exhibits the local L-factor as a function on the complex manifold C × Sym n C × .

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.
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 Documenta Mathematica 7 (2002) 91-112 number of left cells contained in c.The Lusztig conjecture says that there is a ring isomorphism where R Fc 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 Fc is isomorphic to the tensor product over Z of the representation rings R GL(ni,C) , 1 ≤ i ≤ p.For the ring R GL(n,C) we have where the elements X 1 , X 2 , . . ., X n , X −1 We recall the extended quotient.Let the finite group Γ act on the space X.Let X = {(x, γ) : γx = x}, let Γ act on X by γ 1 (x, γ) = (γ 1 x, γ 1 γγ −1 1 ).Then X/Γ is the extended quotient of X by Γ, and we have where one γ is chosen in each Γ-conjugacy class.
There is a canonical projection X/Γ → X/Γ.
and so Then 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 X/S n .

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 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 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): 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]: 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).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 Documenta Mathematica 7 (2002) 91-112 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 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 )) for x ∈ K and 0 otherwise.Then e ρ is an idempotent in the Hecke algebra H(G).Then we have 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 : 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 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 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: Now the extended quotient is multiplicative, i.e.
and by [8, p. 217] we have Π(Ω) Ω + .It now follows that 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 where 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.

Documenta Mathematica 7 (2002) 91-112
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: Remark 3.4.We now consider further the disjoint union If we apply the local Langlands correspondence π F then we obtain 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 In fact this is a special stratification of Irr(Ω) in the precise sense of their article [34, p.198]. Let 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 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 . ., µ r = 2j r + 1 and define partitions as follows The natural partial order on partitions is: λ ≤ µ if and only if 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].
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.

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 D/Γ = D γ /Z γ , where D is the complex torus C ×m .Let γ be a permutation of n letters with cycle type where 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 (α1−1)/2 z 1 , . . ., q (1−α1)/2 z 1 , . . ., q (αr−1)/2 z r , . . ., q (1−αr)/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 D/Γ to the ordinary quotient D/Γ.This creates a map π q from the extended Bernstein variety to the Bernstein variety: Definition 4.1.The map π q is called the q-projection.
The q-projection π q occurs in the following commutative diagram [8]: 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 , 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 , 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: where St(2) is the Steinberg representation of GL (2).The orbit of (T, 1) is D = (C × ) 2 , and The q−projection works as follows: 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.
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 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 The map π q works as follows: If ψ(w) = z d(w) then we will write ψ = z.With this understood, let

Documenta Mathematica 7 (2002) 91-112
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 Then the distinct L-parameters φ 1 , φ 2 , φ 3 , φ 4 all have the same image under the q-projection π q .

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: The Baum-Connes assembly map µ is an isomorphism [1,24].The map 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 [7] and is an isomorphism [7,Theorem 7].The map 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 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.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 wr m (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 wr m (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 wr m (F ) → Π wr m (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 even c (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)).

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 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 top 0 (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.
The group of unramified quasicharacters of M is denoted Ψ(M ).We have Ψ(M ) ∼ = (C × ) where 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.

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[ (C × ) e /S e ]) H * ( (C × ) e /S e ; C).
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: (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.