## Departmental PhD Thesis Exam – Jerrod Smith

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, February 1, 2017
2:10 p.m.
BA6183

PhD Candidate:  Jerrod Smith
Supervisor:  Fiona Murnaghan
Thesis title:  Construction of relative discrete series representations for $p$-adic $\mathbf{GL}_n$

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Abstract:

Let $F$ be a nonarchimedean local field of characteristic zero and odd residual characteristic.  Let $G$ be the $F$-points of a connected reductive group defined over $F$ and let $\theta$ be an $F$-rational involution of $G$.  Define $H$ to be the closed subgroup of $\theta$-fixed points in $G$.
The quotient variety $H \backslash G$ is a $p$-adic symmetric space.  A fundamental problem in the harmonic analysis on $H \backslash G$ is to understand the irreducible subrepresentations of the right-regular representation of $G$ acting on the space $L^2(H\backslash G)$ of complex-valued square integrable functions on $H\backslash G$.  The irreducible subrepresentations of $L^2(H\backslash G)$ are called relative discrete series representations.

In this thesis, we give an explicit construction of relative discrete series representations for three $p$-adic symmetric spaces, all of which are quotients of the general linear group.  We consider $\mathbf{GL}_n(F) \times \mathbf{GL}_n(F) \backslash \mathbf{GL}_{2n}(F)$, $\mathbf{GL}_n(F) \backslash \mathbf{GL}_n(E)$, and $\mathbf {U}_{E/F} \backslash \mathbf{GL}_{2n}(E)$, where $E$ is a quadratic Galois extension of $F$ and $\mathbf {U}_{E/F}$ is a quasi-split unitary group.  All of the representations that we construct are parabolically induced from $\theta$-stable parabolic subgroups admitting a certain type of Levi subgroup.  In particular, we give a sufficient condition for the relative discrete series representations that we construct to be non-relatively supercuspidal.  Finally, in an appendix, we describe all of the relative discrete series of $\mathbf{GL}_{n-1}(F)\times \mathbf{GL}_{1}(F) \backslash \mathbf{GL}_{n}(F)$.

A copy of the thesis can be found here: thesis_jmsmith-27-01-2017