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Durham e-Theses
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Regular representations of GL _n(O) and the inertial Langlands correspondence

SZUMOWICZ, ANNA,MARIA (2019) Regular representations of GL _n(O) and the inertial Langlands correspondence. Doctoral thesis, Durham University.

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This thesis is divided into two parts. The first one comes from the representation theory of reductive $\mathfrak{p} $-adic groups. The main motivation behind this part of the thesis is to find new explicit information and invariants of the types in general linear groups. Let $F$ be a non-Archimedean local field and let $\mathcal{O}_{F}$ be its ring of integers. We give an explicit description of cuspidal types on $\GL _{p}(\mathcal{O}_{F})$, with $p$ prime, in terms of orbits. We determine which of them are regular representations and we provide an example which shows that an orbit of a representation does not always determine whether it is a cuspidal type or not. At the same time we prove that a cuspidal type for a representation $\pi$ of $GL_{p}(F)$ is regular if and only if the normalised level of $\pi$ is equal to $m$ or $m -\frac{1}{p}$ for $m \in \mathbb{Z}$.

The second part of the thesis comes from the theory of integer-valued polynomials and simultaneous $\mathfrak{p}$-orderings. This is a joint work with Mikołaj Frączyk. The notion of simultaneous $\mathfrak{p}$-ordering was introduced by Bhargava in his early work on integer-valued polynomials. Let $k$ be a number field and let $\mathcal{O}_{k}$ be its ring of integers. Roughly speaking a simultaneous $\mathfrak{p}$-ordering is a sequence of elements from $\mathcal{O}_{k}$ which is equidistributed modulo every power of every prime ideal in $\mathcal{O}_{k}$ as well as possible. Bhargava asked which subsets of Dedekind domains admit simultaneous $\mathfrak{p}$-ordering. Together with Mikołaj Frączyk we proved that the only number field $k$ with $\mathcal{O}_{k}$ admitting a simultaneous $\mathfrak{p}$-ordering is $\mathbb{Q}$.

Item Type:Thesis (Doctoral)
Award:Doctor of Philosophy
Keywords:Representation theory of p-adic reductive group; cuspidal types; number theory; integer-valued polynomials; simultaneous p-orderings
Faculty and Department:Faculty of Science > Mathematical Sciences, Department of
Thesis Date:2019
Copyright:Copyright of this thesis is held by the author
Deposited On:13 Jan 2020 10:45

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