Parabolically Induced Representations of Graded Hecke Algebras

Solleveld, Maarten

doi:10.1007/s10468-010-9240-8

Cited by 17 publications

(23 citation statements)

References 29 publications

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“…With Proposition 1.1, [Sol2,Appendix] becomes valid in our situation. The theorem is a reformulation of parts (d) and (e) of [Sol2,Theorem 11.2]. For completeness we note that the connecting homomorphism…”

Section: Twisted Group Algebras and Normal Subgroupsmentioning

confidence: 80%

Generalizations of the Springer correspondence and cuspidal Langlands parameters

2018

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Let H be any reductive p-adic group. We introduce a notion of cuspidality for enhanced Langlands parameters for H, which conjecturally puts supercuspidal H-representations in bijection with such L-parameters. We also define a cuspidal support map and Bernstein components for enhanced L-parameters, in analogy with Bernstein's theory of representations of p-adic groups. We check that for several well-known reductive groups these analogies are actually precise.Furthermore we reveal a new structure in the space of enhanced L-parameters for H, that of a disjoint union of twisted extended quotients. This is an analogue of the ABPS conjecture (about irreducible H-representations) on the Galois side of the local Langlands correspondence. Only, on the Galois side it is no longer conjectural. These results will be useful to reduce the problem of finding a local Langlands correspondence for H-representations to the corresponding problem for supercuspidal representations of Levi subgroups of H.The main machinery behind this comes from perverse sheaves on algebraic groups. We extend Lusztig's generalized Springer correspondence to disconnected complex reductive groups G. It provides a bijection between, on the one hand, pairs consisting of a unipotent element u in G and an irreducible representation of the component group of the centralizer of u in G, and, on the other hand, irreducible representations of a set of twisted group algebras of certain finite groups. Each of these twisted group algebras contains the group algebra of a Weyl group, which comes from the neutral component of G.

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Section: Twisted Group Algebras and Normal Subgroupsmentioning

confidence: 80%

Generalizations of the Springer correspondence and cuspidal Langlands parameters

2018

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“…Notice that the intertwining operators π(gu, Q, σ, t) depend algebraically on t ∈ T Q . This implies that, for every gu ∈ G, π(Q, σ, t) and π(gu(Q, σ, t)) have the same irreducible subquotients (counted with multiplicity), see [Sol3,Lemma 3.1.7] or [Sol2,Lemma 3.4]. Since every G-orbit in Ξ contains an element in positive position, we may assume that (Q, δ, t) is positive.…”

Section: B]mentioning

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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“…Let the central extension R + L → R L and p ♮ L be as in the proof of Proposition 2.2, and let R + be the inverse image of R L,y,σ,ρ • in R + L . As in (4), H(G • R L,y,σ,ρ • , L, L) is the direct summand [Sol1,Theorem 1.2] or [RaRa,p. 24]) there is a bijection…”

Section: It Induces Admentioning

confidence: 99%

Graded Hecke Algebras for Disconnected Reductive Groups

2018

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We introduce graded Hecke algebras H based on a (possibly disconnected) complex reductive group G and a cuspidal local system L on a unipotent orbit of a Levi subgroup M of G. These generalize the graded Hecke algebras defined and investigated by Lusztig for connected G.We develop the representation theory of the algebras H. obtaining complete and canonical parametrizations of the irreducible, the irreducible tempered and the discrete series representations. All the modules are constructed in terms of perverse sheaves and equivariant homology, relying on work of Lusztig. The parameters come directly from the data (G, M, L) and they are closely related to Langlands parameters.Our main motivation for considering these graded Hecke algebras is that the space of irreducible H-representations is canonically in bijection with a certain set of "logarithms" of enhanced L-parameters. Therefore we expect these algebras to play a role in the local Langlands program. We will make their relation with the local Langlands correspondence, which goes via affine Hecke algebras, precise in a sequel to this paper. Erratum. Theorem 3.4 and Proposition 3.22 were not entirely correct as stated. This is repaired in a new appendix.

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Parabolically Induced Representations of Graded Hecke Algebras

Cited by 17 publications

References 29 publications

Generalizations of the Springer correspondence and cuspidal Langlands parameters

Generalizations of the Springer correspondence and cuspidal Langlands parameters

On Completions of Hecke Algebras

Graded Hecke Algebras for Disconnected Reductive Groups

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