Parameters of Hecke algebras for Bernstein components of p-adic groups

Solleveld, Maarten

doi:10.48550/arxiv.2103.13113

Cited by 3 publications

(3 citation statements)

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“…Independently, the parameters of the Hecke algebras H s and H τ can be investigated. In many cases they can be determined [Sol6], and in all those instances they agree with the parameters of some Hecke algebras on the Galois side (or equivalently, with the parameters of a Hecke algebra for a Bernstein block of unipotent representations. )…”

Section: Local Langlands Correspondencementioning

confidence: 92%

Graded Hecke algebras, constructible sheaves and the p-adic Kazhdan--Lusztig conjecture

Solleveld¹

2021

Preprint

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The local Langlands correspondence matches irreducible representations of a reductive p-adic group G(F ) with enhanced L-parameters. It is conjectured by Hellmann and Zhu that it can be categorified. Then it should become a fully faithful functor from a derived category of representations to a derived category of equivariant sheaves on some variety of L-parameters.We approach this conjecture in the case of finite length G(F )-representations. Then it runs via graded Hecke algebras associated to Bernstein components or to enhanced L-parameters. Here we work with graded Hecke algebras H on the Galois side, those can be constructed entirely in terms of a complex reductive group, endowed with data from L-parameters.We fix an arbitrary central character (σ, r) of H (which encodes the image of Frobenius by an L-parameter). That leads to a variety g σ,rN of nilpotent elements in the Lie algebra of G (possibilities for the monodromy operator from an Lparameter) and to a complex of equivariant constructible sheaves KN,σ,r on g σ,r N . We relate the (derived) endomorphism algebra of (g σ,r N , KN,σ,r) to a localization of H, which yields an equivalence between the appropriate categories of finite length modules of these algebras. From there we construct a fully faithful functor between:• the bounded derived category of finite length H-modules specified by the central character (σ, r), • the equivariant bounded derived category of constructible sheaves on g σ,rN . Also, we explicitly determine the images of standard modules under this functor. We expect that these results pave the way for more general instances of the aforementioned conjectural extension of the local Langlands correspondence. ContentsGRADED HECKE ALGEBRAS AND EQUIVARIANT CONSTRUCTIBLE SHEAVES 6. Structure of the localized complexes K N,σ,r 31 7. A functor from sheaves to Hecke algebra modules 34 8. A functor from Hecke algebra modules to sheaves 39 9. Twisted graded Hecke algebras with a fixed r 42 Appendix A. Compatibility with parabolic induction 45 Appendix B. Localization in equivariant cohomology 52 References

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Section: Local Langlands Correspondencementioning

confidence: 92%

Graded Hecke algebras, constructible sheaves and the p-adic Kazhdan--Lusztig conjecture

Solleveld¹

2021

Preprint

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“…Let α be the element of a * M defined as α := ρ P , α ∨ −1 ρ P , where ρ P is half the sum of the roots of A M in Lie N , with P = M N . Then s α ∈ a * M ⊗ R C. We recall the description of the Plancherel measure from [Sil79] (see also [Sol21b] or [Hei11] for the notations used here): for α ∈ Σ O,µ , where is Σ O,µ is the root system defined in (2.1.10), there exists q α , q…”

Section: General Backgroundmentioning

confidence: 99%

Hecke algebras for p-adic reductive groups and Local Langlands Correspondences for Bernstein blocks

Aubert,

2024

Advances in Mathematics

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Section: General Backgroundmentioning

confidence: 99%

Hecke algebras for $p$-adic reductive groups and Local Langlands Correspondence for Bernstein blocks

Aubert¹,

Xu²

2022

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We study the endomorphism algebras attached to Bernstein components of reductive p-adic groups. By using recent results of Solleveld, we prove a reduction to depth zero case result for the components attached to regular supercuspidal representations of Levi subgroups, and construct a correspondence with the appropriate set of enhanced L-parameters.In particular, for Levi subgroups of maximal parabolic of the split exceptional group of type G2, we compute the explicit parameters for the corresponding Hecke algebras, and show that they satisfy a conjecture of Lusztig's. We also give examples for a generalized version of Yu's conjecture using type theory for G2. G ∼

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Parameters of Hecke algebras for Bernstein components of p-adic groups

Cited by 3 publications

References 32 publications

Graded Hecke algebras, constructible sheaves and the p-adic Kazhdan--Lusztig conjecture

Graded Hecke algebras, constructible sheaves and the p-adic Kazhdan--Lusztig conjecture

Hecke algebras for p-adic reductive groups and Local Langlands Correspondences for Bernstein blocks

Hecke algebras for $p$-adic reductive groups and Local Langlands Correspondence for Bernstein blocks

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