#### Document Type

Dissertation

#### Date of Degree

Summer 2015

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Mathematics

#### First Advisor

Muthu Krishnamurthy

#### Abstract

Let *R(G)* denote the category of smooth representations of a *p*-adic group. Bernstein has constructed an indexing set *B(G)* such that *R(G)* decomposes into a direct sum over s ∈ *B(G)* of full subcategories *R ^{s}(G)* known as Bernstein subcategories. Bushnell and Kutzko have developed a method to study the representations contained in a given subcategory. One attempts to associate to that subcategory a smooth irreducible representation (τ,W) of a compact open subgroup

*J < G*. If the functor V ↦ Hom

_{J}(W,V) is an equivalence of categories from

*R*→

^{s}(G)*H(G,τ)*mod we call

*(J,τ)*a type.

Given a Levi subgroup *L < G* and a type *(J _{L}, τ_{L})* for a subcategory of representations on L, Bushnell and Kutzko further show that one can construct a type on G that “lies over”

*(J*by constructing an object known as a cover. In particular, a cover implements induction of

_{L}, τ_{L})*H(L,τ*-modules in a manner compatible with parabolic induction of

_{L})*L*-representations.

In this thesis I construct a cover for certain representations of the Siegel Levi subgroup of *Sp(2k)* over an archimedean local field of characteristic zero. In partic- ular, the representations I consider are twisted by highly ramified characters. This compliments work of Bushnell, Goldberg, and Stevens on covers in the self-dual case. My construction is quite concrete, and I also show that the cover I construct has a useful property known as splitness. In fact, I prove a fairly general theorem characterizing when covers are split.

#### Public Abstract

Number theory studies the integers (i.e., 1, 2, 3, . . .) and their properties. The integers have been studied for over three thousand years, so most of the interesting questions that remain are quite difficult. Thus, to find answers, mathematicians have been forced to invent sophisticated new mathematical tools. One fruitful strategy has been to notice that the integers live inside bigger sets of numbers, like the rational numbers (i.e. fractions), and the real numbers (i.e. the whole number line), and use the extra structure of these bigger sets to better understand the integers.

The real numbers can be built from the rationals via a process that can be thought of as “filling in the holes between the numbers.” In the late 19th century mathematicians began to investigate other ways to fill in these holes. In fact, there is precisely one way for each prime number *p*. For each *p*, this “filling in” produces a set of numbers called the *p*-adic numbers. The *p*-adic numbers have proved a useful tool in number theory. A key motivation in modern research is the idea that if one can get information about the real numbers and the *p*-adic numbers for all primes *p*, one should be able to translate this into information about the integers. This goal has inspired much of the recent progress in modern number theory.

In my thesis I study the structure of certain sets of matrices with entries in the *p*-adic numbers. I construct tools that can be used to analyze their structure, and I prove a general result about the nature of these tools. It is my goal that these constructions be used to calculate further number theoretic data.

#### Keywords

publicabstract, Langlands Program, Math, p-adic numbers, Representation Theory

#### Pages

v, 58 pages

#### Bibliography

Includes bibliographical references (pages 57-58).

#### Copyright

Copyright 2015 Luke Samuel Wassink