Representation growth of linear groups over local rings

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Zeta functions play a major role in number theory, starting with the Riemann zeta function, and continuing with Dedekind zeta functions and more general Dirichlet series. More recently zeta functions have started to play a role also in group theory and representation theory. In these cases the zeta function encodes an infinite sequence of numbers a_1, a_2,..., where a_n is, for instance, the number of irreducible n-dimensional representations of a group. The zeta function defined using this sequence is called the representation zeta function of the group. Among other things, the zeta functions help us to say something about the growth rate of the sequence a_1, a_2,.... For large classes of interesting groups the precise rate of growth is governed by a rational number called the abscissa of convergence of the zeta function. The first part of this project will study the abscissa of convergence associated to the groups SL_N(o), that is, the group of N by N matrices with determinant 1 and with entries in a local principal ideal ring with finite residue field, such as the p-adic integers Z_p. The precise value of the abscissa of SL_N(o) is a mysterious object and there is currently not even a guess as to its precise value. We will develop and use several constructions for representations of the related group GL_N(o) of invertible matrices over o to pin down the value of the abscissa more precisely than has been done previously. A prominent role here is played by the so-called regular representations. The first goal is to find a precise conjectural value of the abscissa and to prove it for as many groups as possible. Since our methods work for all rings o, the second goal will be to address the problem of whether or not the abscissa of SL_N(o) is independent of o, where N and the residue field of o is fixed but o varies.The second part of the project will generalise certain results from classical Deligne-Lusztig theory to the generalised unramified Deligne-Lusztig construction. These results will then be used to prove a generalisation of an asymptotic formula of Liebeck and Shalev for Chevalley groups over finite local rings G(o_r). There is some evidence that this formula can be viewed as a finite group analogue of some formulae for the abscissa of G(o), but the precise explanation for this is still not known, and our work aims to shed some light on this.Finally, we aim to tie together the above topics by showing that the representations given by the generalised unramified Deligne-Lusztig construction for GL_n(o_r) are regular, and have therefore been included in the counting in the first part of the project.