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The Hecke algebra of a reductive padic group: a geometric conjecture
 in: C. Consani, M. Marcolli (Eds.), Noncommutative geometry and number theory, Aspects of Mathematics E37, Vieweg Verlag
, 2006
"... Let H(G) be the Hecke algebra of a reductive padic group G. We formulate a conjecture for the ideals in the Bernstein decomposition of H(G). The conjecture says that each ideal is geometrically equivalent to an algebraic variety. Our conjecture is closely related to Lusztig’s conjecture on the asym ..."
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Let H(G) be the Hecke algebra of a reductive padic group G. We formulate a conjecture for the ideals in the Bernstein decomposition of H(G). The conjecture says that each ideal is geometrically equivalent to an algebraic variety. Our conjecture is closely related to Lusztig’s conjecture on the asymptotic Hecke algebra. We prove our conjecture for SL(2) and GL(n). We also prove part (1) of the conjecture for the Iwahori ideals of the groups PGL(n) and SO(5). 1
Complex Structure on the Smooth Dual of GL(n)
 DOCUMENTA MATH.
, 2002
"... Let G denote the padic 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 tha ..."
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Cited by 7 (3 self)
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Let G denote the padic 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).
A NONCOMMUTATIVE GEOMETRY APPROACH TO THE REPRESENTATION THEORY OF REDUCTIVE pADIC GROUPS: HOMOLOGY OF HECKE ALGEBRAS, A SURVEY AND SOME NEW RESULTS
, 2004
"... ABSTRACT. We survey some of the known results on the relation between the homology of the full Hecke algebra of a reductive padic group G, and the representation theory of G. Let us denote by C ∞ c (G) the full Hecke algebra of G and by HP∗(C ∞ c (G)) its periodic cyclic homology groups. Let ˆ G de ..."
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Cited by 3 (0 self)
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ABSTRACT. We survey some of the known results on the relation between the homology of the full Hecke algebra of a reductive padic group G, and the representation theory of G. Let us denote by C ∞ c (G) the full Hecke algebra of G and by HP∗(C ∞ c (G)) its periodic cyclic homology groups. Let ˆ G denote the admissible dual of G. One of the main points of this paper is that the groups HP∗(C ∞ c (G)) are, on the one hand, directly related to the topology of ˆ G and, on the other hand, the groups HP∗(C ∞ c (G)) are explicitly computable in terms of G (essentially, in terms of the conjugacy classes of G and the cohomology of their stabilizers). The relation between HP∗(C ∞ c (G)) and the topology of ˆG is established as part of a more general principle relating HP∗(A) to the topology of Prim(A), the primitive ideal spectrum of A, for any finite typee algebra A. We provide several new examples illustrating in detail this principle. We also prove in this paper a few new results, mostly in order to better explain and tie together the results that are presented here. For example, we compute the Hochschild homology of O(X) ⋊Γ, the crossed product of the ring of regular functions on a smooth, complex algebraic variety X by a finite group Γ. We also outline a very tentative program to use these results to construct and classify the cuspidal representations of G. At the end of the paper, we also recall the definitions of Hochschild and cyclic homology. CONTENTS
Homology of graded Hecke algebras
"... Abstract. Let H be a graded Hecke algebra with real parameters and Weyl group W. We show that the Hochschild, cyclic and periodic cyclic homologies of H are all independent of the parameters, and compute them explicitly. We use this to prove that the collection of irreducible tempered Hmodules with ..."
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Abstract. Let H be a graded Hecke algebra with real parameters and Weyl group W. We show that the Hochschild, cyclic and periodic cyclic homologies of H are all independent of the parameters, and compute them explicitly. We use this to prove that the collection of irreducible tempered Hmodules with real central character forms a Qbasis of the representation ring of W. Our method involves a new interpretation of the periodic cyclic homology of finite type algebras, in terms of the cohomology of a sheaf over the underlying complex
Geometric structure in the principal series of the padic group G2
 Journal of Representation Theory
"... Abstract. In the representation theory of reductive padic groups G, the issue of reducibility of induced representations is an issue of great intricacy. It is our contention, expressed as a conjecture in (2007), that there exists a simple geometric structure underlying this intricate theory. We wil ..."
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Abstract. In the representation theory of reductive padic groups G, the issue of reducibility of induced representations is an issue of great intricacy. It is our contention, expressed as a conjecture in (2007), that there exists a simple geometric structure underlying this intricate theory. We will illustrate here the conjecture with some detailed computations in the principal series of G2. A feature of this article is the role played by cocharacters hc attached to twosided cells c in certain extended affine Weyl groups. The quotient varieties which occur in the Bernstein programme are replaced by extended quotients. We form the disjoint union A(G) of all these extended quotient varieties. We conjecture that, after a simple algebraic deformation, the space A(G) is a model of the smooth dual Irr(G). In this respect, our programme is a conjectural refinement of the Bernstein programme. The algebraic deformation is controlled by the cocharacters hc. Thecocharacters themselves appear to be closely related to Langlands parameters. 1.
Periodic cyclic homology of Hecke algebras and their
, 2006
"... Abstract. We show that the inclusion of an affine Hecke algebra in its Schwartz completion induces an isomorphism on periodic cyclic homology. Mathematics Subject Classification (2000) 16E40, 19D55, 20C08 Let O(V) and C ∞ (X) be the algebras of regular functions on a nonsingular affine complex varie ..."
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Abstract. We show that the inclusion of an affine Hecke algebra in its Schwartz completion induces an isomorphism on periodic cyclic homology. Mathematics Subject Classification (2000) 16E40, 19D55, 20C08 Let O(V) and C ∞ (X) be the algebras of regular functions on a nonsingular affine complex variety V and of smooth (complex valued) functions on a differentiable manifold X. The HochschildKostantRosenberg theorem [HKR] states that there is a natural isomorphism HH∗(O(V)) ∼ = Ω ∗ (V) (1) between the Hochschild homology of O(V) and the algebra of differential forms on V, both in the algebraic sense. The smooth analogue of this theorem, due to [Con, §II.6], is HH∗(C ∞ (X)) ∼ = Ω ∗ (X; C) (2) but now both sides must be interpreted in the topological sense. 1 Moreover the exterior differential d on Ω ∗ corresponds to the map B on HH∗, which implies that HP∗(O(V)) ∼ = H ∗ DR (V) (3)
Geometric counterpart of the BaumConnes map for GL(n)
"... We describe a geometric counterpart of the BaumConnes map for the padic group GL(n) ..."
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We describe a geometric counterpart of the BaumConnes map for the padic group GL(n)