On Shahidi local coefficients matrix

Szpruch, Dani

doi:10.1007/s00229-018-1052-x

Cited by 7 publications

(2 citation statements)

References 30 publications

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“…[Sha, Chapter 5]), we call it the Shahidi local coefficient matrix in this paper. See also [Bud] and [Szp5]. In the unramified setting, the matrix is computed in [Mc2]; it is also computed for ramified places in [GS].…”

Section: Resultsmentioning

confidence: 99%

Distinguished theta representations for certain covering groups

Gao

2017

Pacific J. Math.

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Abstract. For Brylinski-Deligne covering groups of an arbitrary split reductive group, we consider theta representations attached to certain exceptional genuine characters. The goal of the paper is to study the dimension of the space of Whittaker functionals of a theta representation. In particular, we investigate when the dimension is exactly one, in which case the theta representation is called distinguished. For this purpose, we first give effective lower and upper bounds for the dimension of Whittaker functionals for general theta representations. As a consequence, the dimension in many cases can be reduced to simple combinatorial computations, e.g., the Kazhdan-Patterson covering groups of the general linear groups, or covering groups whose complex dual groups (à la Finkelberg-Lysenko-McNamara-Reich) are of adjoint type. In the second part of the paper, we consider coverings of certain semisimple simply-connected groups and give necessary and sufficient condition for the theta representation to be distinguished. There are subtleties arising from the relation between the rank and the degree of the covering group. However, in each case we will determine the exceptional character such that its associated theta representation is distinguished.

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Section: Resultsmentioning

confidence: 99%

Distinguished theta representations for certain covering groups

Gao

2017

Pacific J. Math.

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“…was obtained by considering the scattering matrix τ ψ * in §8.4 with ψ * = z * ( w G ψ) and z * = −ρ, while the method equally applies to τ w G ψ . In fact, even if χ is ramified, similar results are expect to hold following the work [GS16,Szp19] on the local coefficients matrices and scattering matrices.…”

Section: H(g I) ≃ H(g I)mentioning

confidence: 63%

Genuine pro-$p$ Iwahori--Hecke algebras, Gelfand--Graev representations, and some applications

Gao¹,

Gurevich²,

Karasiewicz³

2022

Preprint

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We study the Iwahori-component of the Gelfand-Graev representation of a central cover of a split linear reductive group and utilize our results for three applications. In fact, it is advantageous to begin at the pro-p level. Thus to begin we study the structure of a genuine pro-p Iwahori-Hecke algebra, establishing Iwahori-Matsumoto and Bernstein presentations. With this structure theory we first describe the pro-p part of the Gelfand-Graev representation and then the more subtle Iwahori part.For the first application we relate the Gelfand-Graev representation to the metaplectic representation of Sahi-Stokman-Venkateswaran, which conceptually realizes the Chinta-Gunnells action from the theory of Weyl group multiple Dirichlet series. For the second we compute the Whittaker dimension of the constituents of regular unramified principal series; for the third we do the same for unitary unramified principal series.

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