Spectral transfer morphisms for unipotent affine Hecke algebras

Opdam, Eric

doi:10.1007/s00029-016-0273-7

Cited by 18 publications

(88 citation statements)

References 63 publications

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“…Let G be a connected reductive group over F which is split over an unramified extension of F . Our main theorem is a slight sharpening and extension of the main result of [Opd5].…”

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confidence: 60%

“…One of the main ingredients of the proof of Theorem 4.5.1 is the notion of spectral transfer morphism between normalised affine Hecke algebras [Opd4], [Opd5], which allows us to construct a bijection between the set G temp uni of equivalence classes of tempered irreducible representations of G with unipotent reduction and the set Φ temp nr (G) of G ∨conjugacy classes of unramified bounded enhanced Langlands parameters for G. The construction of such a bijection has some interest in its own right. A key point is the fact that a spectral transfer morphism Ψ : H t (G) H I (G * ) from the Hecke algebra H t (G) of a unipotent type t of G to the Iwahori Hecke algebra H I (G * ) defines Langlands parameters π → ϕ π ∈ Φ temp nr (G) for the tempered representations covered by t such that the conjectures of Hiraga, Ichino and Ikeda hold (up to rational constant factors independent of the cardinality q of the residue field of F ).…”

Section: Affine Hecke Algebras Asmentioning

confidence: 99%

“…(e) G arbitrary, π depth 0 supercuspidal representation (tame regular semisimple case) (DeBacker and Reeder [DeRe], [HII]). The main result we will discuss in these lectures is and extension to general connected reductive G of the following result: 1 Theorem 2.7.1 ([R3], [Opd4], [Opd5], [Fe2]). Let G be absolutely almost simple of adjoint type over a non-archimedean field F such that G splits over an unramified extension of F .…”

Section: We Have Homomorphisms Gmentioning

confidence: 99%

“…(1) Spectral transfer maps between Hecke algebras [Opd4], [Opd5], [FO], [FOS]. We use these tools to deal with the q-rational factors of the formal degree.…”

Section: We Have Homomorphisms Gmentioning

confidence: 99%

“…Let Vol(P) ∈ Λ 0 be the Laurent polynomial defined above, let deg(σ) ∈ Λ 0 be the polynomials such that deg(σ)(v) = deg(σ v ), then the trace τ of H ẽ t is normalised by τ (1) = d t = |Ω P,θ 1 | −1 deg(σ) Vol(P Fru . This turns the summands H ẽ t into normalised affine Hecke algebras in the sense of [Opd5].…”

Section: Unipotent Types and Unipotent Affine Hecke Algebras Letmentioning

confidence: 99%

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Affine Hecke algebras and the conjectures of Hiraga, Ichino and Ikeda on the Plancherel density

Opdam¹

2019

Representations of Reductive Groups

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Hiraga, Ichino and Ikeda have conjectured an explicit expression for the Plancherel density of the group of points of a reductive group defined over a local field F , in terms of local Langlands parameters. In these lectures we shall present a proof of these conjectures for Lusztig's class of representations of unipotent reduction if F is p-adic and G is of adjoint type and splits over an unramified extension of F . This is based on the author's paper [Spectral transfer morphisms for unipotent affine Hecke algebras, Selecta Math. (N.S.) 22 (2016), no. 4, 2143-2207].More generally for G connected reductive (still assumed to be split over an unramified extension of F ), we shall show that the requirement of compatibility with the conjectures of Hiraga, Ichino and Ikeda essentially determines the Langlands parameterisation for tempered representations of unipotent reduction. We shall show that there exist parameterisations for which the conjectures of Hiraga, Ichino and Ikeda hold up to rational constant factors. The main technical tool is that of spectral transfer maps between normalised affine Hecke algebras used in op. cit.Date: 2018-07-30. 2000 Mathematics Subject Classification. Primary 20C08; Secondary 22D25, 43A30. Key words and phrases. Cuspidal unipotent representation, formal degree, discrete unramified Langlands parameter.It is a pleasure to thank Maarten Solleveld for many useful comments. It is also a pleasure to acknowledge the excellent comments and suggestions of the participants of the workshop "Representation Theory of Reductive Groups Over Local Fields and Applications to Automorphic forms" at the Weizmann institute, and of the workshop "Representation theory of p-adic groups" at IISER, Pune.

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“…Let G be a connected reductive group over F which is split over an unramified extension of F . Our main theorem is a slight sharpening and extension of the main result of [Opd5].…”

mentioning

confidence: 60%

Section: Affine Hecke Algebras Asmentioning

confidence: 99%

Section: We Have Homomorphisms Gmentioning

confidence: 99%

“…(1) Spectral transfer maps between Hecke algebras [Opd4], [Opd5], [FO], [FOS]. We use these tools to deal with the q-rational factors of the formal degree.…”

Section: We Have Homomorphisms Gmentioning

confidence: 99%

Section: Unipotent Types and Unipotent Affine Hecke Algebras Letmentioning

confidence: 99%

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Affine Hecke algebras and the conjectures of Hiraga, Ichino and Ikeda on the Plancherel density

Opdam¹

2019

Representations of Reductive Groups

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Spectral correspondences for affine Hecke algebras

Opdam

2016

Advances in Mathematics

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We introduce the notion of spectral transfer morphisms between normalized affine Hecke algebras, and show that such morphisms induce spectral measure preserving correspondences on the level of the tempered spectra of the affine Hecke algebras involved. We define a partial ordering on the set of isomorphism classes of normalized affine Hecke algebras, which plays an important role for the Langlands parameters of Lusztig's unipotent representations.

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On a uniqueness property of supercuspidal unipotent representations

Feng

Opdam

2020

Advances in Mathematics

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The formal degree of a unipotent discrete series character of a simple linear algebraic group over a non-archimedean local field (in the sense of Lusztig [Lus3]), is a rational function of q evaluated at q = q, the cardinality of the residue field. The irreducible factors of this rational function are q and cyclotomic polynomials. We prove that the formal degree of a supercuspidal unipotent representation determines its Lusztig-Langlands parameter, up to twisting by weakly unramified characters. For split exceptional groups this result follows from the work of M. Reeder [R3], and for the remaining exceptional cases this is verified in [Fe2]. In the present paper we treat the classical families.The main result of this article characterizes unramified Lusztig-Langlands parameters which support a cuspidal local system in terms of formal degrees. The result implies the uniqueness of so-called cuspidal spectral transfer morphisms (as introduced in [Opd2]) between unipotent affine Hecke algebras (up to twisting by unramified characters). In [Opd3] the essential uniqueness of arbitrary unipotent spectral transfer morphisms was reduced to the cuspidal case.

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Spectral transfer morphisms for unipotent affine Hecke algebras

Cited by 18 publications

References 63 publications

Affine Hecke algebras and the conjectures of Hiraga, Ichino and Ikeda on the Plancherel density

Affine Hecke algebras and the conjectures of Hiraga, Ichino and Ikeda on the Plancherel density

Spectral correspondences for affine Hecke algebras

On a uniqueness property of supercuspidal unipotent representations

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