Hochschild homology of affine Hecke algebras

Solleveld, Maarten

doi:10.1016/j.jalgebra.2013.03.007

Cited by 21 publications

(40 citation statements)

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“…In particular when k = 1 the varieties are smooth, while the dual case k = n, considered by Solleveld, will always give the most singular case in the tower. In that case each Y µ has cardinality g(µ), recovering Solleveld's formula, [7].…”

Section: Introductionmentioning

confidence: 76%

Stratified Langlands duality in the $A_n$ tower

Niblo

Plymen

Wright

2019

J. Noncommut. Geom.

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Let S k denote a maximal torus in the complex Lie group G = SLn(C)/C k and let T k denote a maximal torus in its compact real form SUn(C)/C k , where k divides n. Let W denote the Weyl group of G, namely the symmetric group Sn. We elucidate the structure of the extended quotient S k //W as an algebraic variety and of T k //W as a topological space, in both cases describing them as bundles over unions of tori. Corresponding to the invariance of K-theory under Langlands duality, this calculation provides a homotopy equivalence between T k //W and its dual T n k //W . Hence there is an isomorphism in cohomology for the extended quotients which is stratified as a direct sum over conjugacy classes of the Weyl group. We use our formula to compute a number of examples.

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Section: Introductionmentioning

confidence: 76%

Stratified Langlands duality in the $A_n$ tower

Niblo

Plymen

Wright

2019

J. Noncommut. Geom.

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“…where W Γ acts on the union by w ′ · (w, t) = (w ′ ww ′−1 , w ′ (t)). This parametrization respects central characters, up to a twists which are constant on connected components of T //W Γ [Sol4,Theorem 2.6]. In this parametrization of Irr(H ⋊ Γ) almost all elements of a piece {w} × T w with w ∈ W (R Q )Γ Q come from representations induced from H Q ⋊ Γ Q , and such w account for all representations induced from proper parabolic subalgebras.…”

Section: The Space Of Irreducible Representationsmentioning

confidence: 99%

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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“…Furthermore we show that the rational isomorphism given above is induced by a homotopy equivalence between the varieties which respects the decomposition. The special case of SU(n) itself was considered by Solleveld in [13].…”

Section: Langlands Duality and K-theorymentioning

confidence: 99%