Structure of the unramified L-packet

Manish Kumar Mishra, Purdue University

Abstract

Let G be an unramified connected reductive group defined over a non-archemedian local field k and let T be a maximal torus in G. Let λ be an unramified character of T. Then the conjugacy classes of hyperspecial subgroups of G(k ) is a principal homogenous space for a certain finite abelian group [special characters omitted]. Also, the L-packet Π(ϕλ) associated to λ is parametrized by an abelian group Rˆ. In the first part of the thesis, we show that Rˆ is naturally a homogenous space for [special characters omitted]. Further, let πρ ∈ Π(ϕλ ), where ρ ∈ Rˆ and let [K] denote the conjugacy class of hyperspecial subgroup K. Then we show that [special characters omitted] ≠ 0 if and only if [special characters omitted] ≠ 0 where ω ∈ [special characters omitted] and Kω is any hyperspecial subgroup in the conjugacy class ω · [K]. In the second part, we describe the image, under the local Langlands correspondece for tori, of the characters of a torus which are trivial on its Iwahori subgroup. To a representation of a connected reductive quasi-split group G , having a non-zero vector fixed under a special maximal parahoric subgroup, we associate in a natural way, a twisted semi-simple conjugacy class in a certain subgroup of the dual group Ĝ. These results generalize well known classical results to the ramified case.

Degree

Ph.D.

Advisors

Yu, Purdue University.

Subject Area

Mathematics|Theoretical Mathematics

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