Representation growth of linear groups over local rings

Zeta函数在数论中起着重要的作用，从Riemann zeta函数开始，继续到Dedekind zeta函数和更一般的Dirichlet级数。最近zeta函数也开始在群论和表示论中发挥作用。在这些情况下，zeta函数编码一个无限序列的数字a_1，a_2，.，其中a_n是例如群的不可约n维表示的数目。使用这个序列定义的zeta函数称为群的表示zeta函数。除此之外，zeta函数帮助我们说明序列a_1，a_2，.的增长率。对于大类有趣的群体，精确的增长率由一个有理数决定，这个有理数称为zeta函数收敛的横坐标。本项目的第一部分将研究与群SL_N（o）相关的收敛横坐标，SL_N（o）是行列式为1的N × N矩阵群，其元素在具有有限剩余域的局部主理想环中，如p-adic整数Z_p。SL_N（o）的横坐标的精确值是一个神秘的对象，目前甚至没有关于它的精确值的猜测。我们将开发和使用几个结构表示的相关群GL_N（o）的可逆矩阵在o钉下的横坐标的值比以前做的更精确。在这里，所谓的正则表示起着重要的作用。第一个目标是找到一个精确的横坐标值，并证明它为尽可能多的群体。由于我们的方法适用于所有环o，第二个目标将是解决SL_N（o）的横坐标是否独立于o的问题，其中N和o的剩余域是固定的，但o是变化的。这些结果将被用来证明有限局部环G（o_r）上Chevalley群的Liebeck和Shalev渐近公式的推广。有一些证据表明，这个公式可以看作是G（o）横坐标的某些公式的有限群类似物，但对此的精确解释还不清楚，我们的工作旨在阐明这一点。最后，我们的目的是通过证明GL_n（o_r）的广义非分歧Deligne-Lusztig构造给出的表示是正则的，因此已包括在第一阶段的点算内。

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.