On the unramified Eisenstein spectrum

Simons Symposia

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. Contents 4. The twisted graded Hecke algebra of a cuspidal quasi-support Appendix A. Compatibility with parabolic induction References

2019

It is well-known that affine Hecke algebras are very useful to describe the smooth representations of any connected reductive p-adic group G, in terms of the supercuspidal representations of its Levi subgroups. The goal of this paper is to create a similar role for affine Hecke algebras on the Galois side of the local Langlands correspondence. To every Bernstein component of enhanced Langlands parameters for G we canonically associate an affine Hecke algebra (possibly extended with a finite R-group). We prove that the irreducible representations of this algebra are naturally in bijection with the members of the Bernstein component, and that the set of central characters of the algebra is naturally in bijection with the collection of cuspidal supports of these enhanced Langlands parameters. These bijections send tempered or (essentially) square-integrable representations to the expected kind of Langlands parameters. Furthermore we check that for many reductive p-adic groups, if a Ber...

Journal of Algebra, 2013

We show that every automorphism of a thick twin building interchanging the halves of the building maps some residue to an opposite one. Furthermore we show that no automorphism of a locally finite 2-spherical twin building of rank at least 3 maps every residue of one fixed type to an opposite (a key step in the proof is showing that every duality of a thick finite projective plane admits an absolute point). Our results also hold for all finite irreducible spherical buildings of rank at least 3, and imply that every involution of a thick irreducible finite spherical building of rank at least 3 has a fixed residue.

arXiv (Cornell University), 2022

We consider four classes of classical groups over a non-archimedean local field F : symplectic, (special) orthogonal, general (s)pin and unitary. These groups need not be quasi-split over F. The main goal of the paper is to obtain a local Langlands correspondence for any group G of this kind, via Hecke algebras. To each Bernstein block Rep(G) s in the category of smooth complex G-representations, an (extended) affine Hecke algebra H(s) can be associated with the method of Heiermann. On the other hand, to each Bernstein component Φe(G) s ∨ of the space Φe(G) of enhanced L-parameters for G, one can also associate an (extended) affine Hecke algebra, say H(s ∨). For the supercuspidal representations underlying Rep(G) s , a local Langlands correspondence is available via endoscopy, due to Moeglin and Arthur. Using that we assign to each Rep(G) s a unique Φe(G) s ∨. Our main new result is an algebra isomorphism H(s) op ∼ = H(s ∨), canonical up to inner automorphisms. In combination with earlier work, that provides an injective local Langlands correspondence Irr(G) → Φe(G) which satisfies Borel's desiderata. This parametrization map is probably surjective as well, but we could not show that in all cases. Our framework is suitable to (re)prove many results about smooth G-representations (not necessarily reducible), and to relate them to the geometry of a space of L-parameters. In particular our Langlands parametrization yields an independent way to classify discrete series G-representations in terms of Jordan blocks and supercuspidal representations of Levi subgroups. We show that it coincides with the classification of the discrete series obtained twenty years ago by Moeglin and Tadić.

Annals of Mathematics, 2011

We define the spherical Hecke algebra for an (untwisted) affine Kac-Moody group over a local non-archimedian field. We prove a generalization of the Satake isomorphism for this algebra, relating it to integrable representations of the Langlands dual affine Kac-Moody group. In the next publication we shall use these results to define and study the notion of Hecke eigenfunction for the group G aff .

2005

Let K be a local non-archimedian field, F=K((t)) and let G be a split semi-simple group. The purpose of this paper is to study certain analogs of spherical (and Iwahori) Hecke algebras for representations of the group G(F) and its central extension by means of K*. For instance our spherical Hecke algebra corresponds to the subgroup G(A) where A is

Science China-mathematics, 2017

This paper aims at developing a "local-global" approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the cross-characteristic representation theory of finite groups of Lie type. The authors first review the notions of quasi-hereditary and stratified algebras over a Noetherian commutative ring. They prove that many global properties of these algebras hold if and only if they hold locally at every prime ideal. When the commutative ring is sufficiently good, it is often sufficient to check just the prime ideals of height at most one. These methods are applied to construct certain generalized q-Schur algebras, proving they are often quasi-hereditary (the "good" prime case) but always stratified. Finally, these results are used to prove a triangular decomposition matrix theorem for the modular representations of Hecke algebras at good primes. In the bad prime case, the generalized q-Schur algebras are at least stratified, and a block triangular analogue of the good prime case is proved, where the blocks correspond to Kazhdan-Lusztig cells. Contents 1. Introduction. 1 2. Localization of integral quasi-hereditary algebras (QHAs). 3 3. Stratified algebras and their localizations. 10 4. Some Morita equivalences. 15 5. The Hecke algebras at good primes. 19 6. Bad primes and standardly stratified algebras.

Nagoya Mathematical Journal, 2006

Let K be a local non-archimedian field, F = K((t)) and let G be a split semi-simple group. The purpose of this paper is to study certain analogs of spherical and Iwahori Hecke algebras for representations of the group G = G(F) and its central extension Ĝ. For instance our spherical Hecke algebra corresponds to the subgroup G (A) ⊂ G(F) where A ⊂ F is the subring OK((t)) where OK ⊂ K is the ring of integers. It turns out that for generic level (cf. [4]) the spherical Hecke algebra is trivial; however, on the critical level it is quite large. On the other hand we expect that the size of the corresponding Iwahori-Hecke algebra does not depend on a choice of a level (details will be considered in another publication).

2022

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 subgroups of the split exceptional group of type $G_2$, 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 $G_2$.

Journal de l’École polytechnique — Mathématiques, 2019

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