Conjectures about 𝑝-adic groups and their
                    noncommutative geometry

Aubert, Anne-Marie; Baum, Paul; Plymen, Roger; Solleveld, Maarten

doi:10.1090/conm/691/13892

Cited by 30 publications

(32 citation statements)

References 36 publications

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“…An enhancement of φ is defined to be an irreducible complex representation ρ of S φ . We refer to [Art2,ABPS6] for a motivation of this particular kind of enhancements. h · (φ, ρ) = (hφh −1 , h · ρ) where (h · ρ)(g) = ρ(h −1 gh).…”

Section: Cuspidal Langlands Parametersmentioning

confidence: 99%

“…It is conjectured [Art2,ABPS6] that the local Langlands correspondence for H can be enhanced to a bijection Irr(H) ←→ Φ e,ζ H (H).…”

Section: In This Way Every Inner Twist Of H Is Associated To a Uniquementioning

confidence: 99%

“…According to the original setup [Bor, Lan], it should be possible to associate to every L-parameter a finite packet of irreducible admissible representations. Later this was improved by enhancing L-parameters [Lus1,KaLu], and the modern interpretation [ABPS6,Vog] says that the LLC should be a bijection (when formulated appropriately).…”

Section: Introductionmentioning

confidence: 99%

“…Let L ⊂ H be a Levi subgroup, and let W (H, L) = N H (L)/L be its "Weyl" group. In [ABPS6] it was shown to be naturally isomorphic to N H ∨ (L ∨ ⋊ W F )/L ∨ , so it acts on both Irr cusp (L) and Φ cusp (L).…”

Section: Introductionmentioning

confidence: 99%

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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: Cuspidal Langlands Parametersmentioning

confidence: 99%

“…It is conjectured [Art2,ABPS6] that the local Langlands correspondence for H can be enhanced to a bijection Irr(H) ←→ Φ e,ζ H (H).…”

Section: In This Way Every Inner Twist Of H Is Associated To a Uniquementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalizations of the Springer correspondence and cuspidal Langlands parameters

2018

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“…It is expected that all other summands H(G) s are also Morita equivalent to AHAs, or to closely related algebras. Indeed, this has been proven in many cases, see [ABPS3,§2.4] for an overview.…”

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confidence: 92%

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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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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Conjectures about 𝑝-adic groups and their noncommutative geometry

Cited by 30 publications

References 36 publications

Generalizations of the Springer correspondence and cuspidal Langlands parameters

Generalizations of the Springer correspondence and cuspidal Langlands parameters

Topological K-theory of affine Hecke algebras

Graded Hecke Algebras for Disconnected Reductive Groups

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