## Open Access Dissertations

January 2015

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Jiu-Kang Yu

Freydoon Shahidi

Tong Liu

David Goldberg

#### Abstract

For a reductive group $G$ over a $p$-adic field $k$, one may grade the associated Lie algebra $\g$ by an automorphism of order $m$. It has been shown that stable vectors $v\in\g_{a}$ arise only when $a$ is coprime to $m$. Given a stable vector $v\in\g_{a}$, we construct packets of supercuspidal representations $\{\pi_{v,\rho}\}$ as well as discrete Langlands parameters $\varphi_{v}$. Both the parameter and representations are of depth $a/m$. We further show that for a fixed vector, $\pi_{v,\rho}$ and $\varphi_{v}$ satisfy both sides of the formal degree conjecture.

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