Representation growth and representation zeta functions of groups

This is the first of two papers in which we introduce and study two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. One of these zeta functions encodes the numbers of isomorphism classes of irreducible complex representations of finite dimensions of congruence quotients of the associated group and the other one encodes the numbers of conjugacy classes of each size of such quotients. In this paper, we show that these zeta...

This is my talk delivered at the workshop 'Automorphic L-Functions and related prpblems' (March 10--13, 2012, Tokyo University). We showed an instance of applications of the theory of automorphic representations to a genuinely traditional problem in the theory of the zeta and allied functions. We restricted ourselves to very basic issues and results, because of the purpose of the workshop.

We introduce new methods from p-adic integration into the study of representation zeta functions associated to compact p-adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of generic members of families of p-adic analytic pro-p groups obtained from a global, `perfect' Lie lattice satisfy functional equations. In the case of `semisimple' compact p-adic analytic groups, we exhibit a link between the relevant p-adic inte...

These are the notes of some lectures given by the author for a workshop held at TIFR, Mumbai in December, 2011, giving an exposition of the Deligne-Lusztig theory.

We adapt methods from quiver representation theory and Hall algebra techniques to the counting of representations of virtually free groups over finite fields. This gives rise to the computation of the E-polynomials of $\mathbf{GL}_d(\mathbb{C})$-character varieties of virtually free groups. As examples we discuss the representation theory of $\mathbb{D}_\infty$ , $\mathbf{PSL}_2(\mathbb{Z})$ , $\mathbf{SL}_2(\mathbb{Z})$ , $\mathbf{GL}_2(\mathbb{Z})$ and $\mathbf{PGL}_2(\math...

This is a short introduction (in Spanish) to the study of growth in finite groups, with SL_2 as an example. Emphasis is put on developments of the decade 2005--2015, originating partly in combinatorics. Little algebraic-geometrical background is assumed, but the proofs are nevertheless presented in a relatively abstract and generalizable way.

The growth rate function $r_N$ counts the number of irreducible representations of simple complex Lie groups of dimension $N$. While no explicit formula is known for this function, previous works have found bounds for $R_N=\sum_{i=1}^Nr_i$. In this paper we improve on previous bounds and show that $R_N=O(N)$.

This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups defined over the fields which admit arbitrary cyclic extensions.

In this paper we study the asymptotic behavior of the number of summands in tensor products of finite dimensional representations of affine (semi)group (super)schemes and related objects.

There is both theoretical and numerical evidence that the set of irreducible representations of a reductive group over local or finite fields is naturally partitioned into families according to analytic properties of representations. Examples of such properties are the rate of decay at infinity of "matrix coefficients" in the local field setting, and the order of magnitude of "character ratios" in the finite field situation. In these notes we describe known results, new resul...