Regular representations of GL _n(O) and the inertial Langlands correspondence

This thesis is divided into two parts. The first one comes from the representation theory of reductive -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 be a non-Archimedean local field and let be its ring of integers. We give an explicit description of cuspidal types on , with 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 of is regular if and only if the normalised level of is equal to or for . The second part of the thesis comes from the theory of integer-valued polynomials and simultaneous -orderings. This is a joint work with Mikołaj Frączyk. The notion of simultaneous -ordering was introduced by Bhargava in his early work on integer-valued polynomials. Let be a number field and let be its ring of integers. Roughly speaking a simultaneous -ordering is a sequence of elements from which is equidistributed modulo every power of every prime ideal in as well as possible. Bhargava asked which subsets of Dedekind domains admit simultaneous -ordering. Together with Mikołaj Frączyk we proved that the only number field with admitting a simultaneous -ordering is .