Transfer of plancherel measures between p-adic inner forms

Kwangho Choiy, Purdue University

Abstract

The primary aims of this thesis are to understand the behavior of Plancherel measures between p-adic inner forms, and to support the conjecture by Shahidi that Plancherel measures are preserved by inner forms. While Plancherel measures are well understood for many cases of quasi-split groups over a p-adic field F of characteristic 0, little is known in any generality for F-inner forms. We also study Plancherel measures for a non quasi-split group, by transferring Plancherel measures for its quasi-split F-inner form. This work gives a conditional proof of the following: all the known facts involving the Plancherel measure for a quasi-split group can be transferred to its inner forms. In Chapter 3, we prove that the Plancherel measure attached to any unitary supercuspidal representation of any F-Levi subgroup M(F) with [special characters omitted] is identically transferred under the local Jacquet-Langlands type correspondence. Our strategy of the proof is a local to global argument, assuming the conjectural fact that Plancherel measures are invariant on L-packets. This work extends the result of Muić and Savin (2000) for Siegel Levi subgroups of SO4 n or Sp4n under the local Jacquet-Langlands correspondence, to any F-Levi subgroup of any connected reductive group under the local Jacquet-Langlands type correspondence. It can be also applied to simply connected groups of type E6 or E7, and connected reductive groups of type An, Bn, Cn, or D n. In Chapter 4, we first verify that the Plancherel measures attached to Steinberg representations of F-Levi subgroups are preserved by p-adic inner forms. To prove this, we construct a purely local method which is inspired by the ideas of Arthur and Clozel (1989) for GLn. Then, we develop this approach with help of the conjectural character identity, to identically transfer Plancherel measures for discrete series representations with unramified central characters between p-adic inner forms. This work extends the result of Arthur and Clozel, for discrete series representations of Levi subgroups in GLn , to discrete series representations with unramified central characters of Levi subgroups in any connected reductive group where the character identities are known. As a corollary, we also prove that Plancherel measures are invariant on L-packets for these cases.

Degree

Ph.D.

Advisors

Shahidi, Purdue University.

Subject Area

Applied Mathematics|Mathematics

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