Periodic cyclic homology of reductive $p$-adic groups

Solleveld, Maarten

doi:10.4171/jncg/45

Cited by 15 publications

(26 citation statements)

References 46 publications

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“…For the groups under consideration the Baum-Connes conjecture can also be formulated and proven more algebraically [HiNi, Schn], with equivariant cosheaf homology (also known as chamber homology) [ABP,§2]. By [Sol1] these two versions of the conjecture are compatible.…”

Section: Equivariant K-theorymentioning

confidence: 82%

Conjectures about 𝑝-adic groups and their noncommutative geometry

Aubert¹,

Baum²,

Plymen

et al. 2017

Contemporary Mathematics

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Abstract. Let G be any reductive p-adic group. We discuss several conjectures, some of them new, that involve the representation theory and the geometry of G.At the heart of these conjectures are statements about the geometric structure of Bernstein components for G, both at the level of the space of irreducible representations and at the level of the associated Hecke algebras. We relate this to two well-known conjectures: the local Langlands correspondence and the BaumConnes conjecture for G. In particular, we present a strategy to reduce the local Langlands correspondence for irreducible G-representations to the local Langlands correspondence for supercuspidal representations of Levi subgroups. Contents

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Section: Equivariant K-theorymentioning

confidence: 82%

Conjectures about 𝑝-adic groups and their noncommutative geometry

Aubert¹,

Baum²,

Plymen

et al. 2017

Contemporary Mathematics

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“…Then [Sol1,Theorem 2.15.a] says that the Langlands constituents of I G P (ω ⊗χ) are precisely its irreducible quotients. Furthermore, by [Sol1,Proposition 2.15.a] I G P (ω ⊗ χ) is a direct sum of representations of the form I G Q (τ ⊗ |χ|), where (Q, τ, log |χ|) is a datum for the Langlands classification of Irr(G). Suppose that π ′ is a constituent of I G Q (τ ⊗ |χ|), but not a quotient.…”

Section: The Space Of Irreducible Representationsmentioning

confidence: 99%

“…By [Sol1, Lemma 2.11.a and Lemma 2.12], π ′ is the Langlands quotient of I G Q ′ (τ ′ ⊗ ν ′ ), for a Langlands datum (Q ′ , τ ′ , log ν ′ ) with Q ′ ⊃ Q and cc(τ ′ ) > cc(τ ) . By [Sol1,Proposition 2.15.a] π ′ is a Langlands…”

Section: The Space Of Irreducible Representationsmentioning

confidence: 99%

“…After that we look at the aforementioned group algebras for a reductive p-adic group G. We recall the Plancherel isomorphism for the Schwartz algebra of G [HC, Wal] and for the reduced C * -algebra of G [Ply]. Like for affine Hecke algebras, we analyse the space of irreducible smooth G-representation in terms of the Langlands classification and parabolic induction of square-integrable representations, following [Sol1].…”

Section: Introductionmentioning

confidence: 99%

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On Completions of Hecke Algebras

Solleveld

2019

Progress in Mathematics

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Let G be a reductive p-adic group and let H(G) s be a Bernstein block of the Hecke algebra of G. We consider two important topological completions of H(G) s : a direct summand S(G) s of the Harish-Chandra-Schwartz algebra of G and a two-sided ideal C * r (G) s of the reduced C * -algebra of G. These are useful for the study of all tempered smooth G-representations.We suppose that H(G) s is Morita equivalent to an affine Hecke algebra H(R, q) -as is known in many cases. The latter algebra also has a Schwartz completion S(R, q) and a C * -completion C * r (R, q), both defined in terms of the underlying root datum R and the parameters q.We prove that, under some mild conditions, a Morita equivalence H(G) s ∼M H(R, q) extends to Morita equivalences S(G) s ∼M S(R, q) and C * r (G) s ∼M C * r (R, q). We also check that our conditions are fulfilled in all known cases of such Morita equivalences between Hecke algebras. This is applied to compute the topological K-theory of the reduced C * -algebra of a classical p-adic group.

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“…Namely, from the work of Kasparov [Kas] it is known that for many groups G the Baum-Connes assembly map is injective, and that its image is a direct summand of K * (C * r (G)). There exist methods [Sol2,§3.4] which enable one to prove that the assembly map becomes an isomorphism after tensoring its domain and range by Q, but which say little about the torsion elements in the K-groups. If one knew in advance that K * (C * r (G)) is torsion-free, then one could prove instances of the Baum-Connes conjecture with such methods.…”

mentioning

confidence: 99%

Topological K-theory of affine Hecke algebras

Solleveld

2018

Ann. K-Th.

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Let H(R, q) be an affine Hecke algebra with a positive parameter function q. We are interested in the topological K-theory of its C * -completion C * r (R, q). We will prove that K * (C * r (R, q)) does not depend on the parameter q, solving a long-standing conjecture of Higson and Plymen. For this we use representation theoretic methods, in particular elliptic representations of Weyl groups and Hecke algebras.Thus, for the computation of these K-groups it suffices to work out the case q = 1. These algebras are considerably simpler than for q = 1, just crossed products of commutative algebras with finite Weyl groups. We explicitly determine K * (C * r (R, q)) for all classical root data R. This will be useful to analyse the K-theory of the reduced C * -algebra of any classical p-adic group.For the computations in the case q = 1 we study the more general situation of a finite group Γ acting on a smooth manifold M . We develop a method to calculate the K-theory of the crossed product C(M ) ⋊ Γ. In contrast to the equivariant Chern character of Baum and Connes, our method can also detect torsion elements in these K-groups.

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Periodic cyclic homology of reductive $p$-adic groups

Cited by 15 publications

References 46 publications

Conjectures about 𝑝-adic groups and their noncommutative geometry

Conjectures about 𝑝-adic groups and their noncommutative geometry

On Completions of Hecke Algebras

Topological K-theory of affine Hecke algebras

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