Four Drafts of The Representation Theory of the Group of Infinite Matrices over Finite Fields

There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio trace(\pi(g)) / dim(\pi), for an irreducible representation \pi of G and an element g of G. It turns out [Gurevich-Howe15, Gurevich-Howe17] that for classical groups G over finite fields there are several (compatible) invariants of representations that provide ...

This is a survey article written for a workshop on L-functions and random matrix theory at the Newton Institute in July, 2004. The goal is to give some insight into how well-distributed sets of matrices in classical groups arise from families of $L$-functions in the context of function fields of curves over finite fields. The exposition is informal and no proofs are given; rather, our aim is to illustrate what is true by considering key examples.

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 paper describes a family of Hecke algebras H_\mu=End_G(Ind_U^G(\psi_\mu)), where U is the subgroup of unipotent upper-triangular matrices of G=GL_n(F_q) and \psi_\mu is a linear character of U. The main results combinatorially index a basis of H_\mu, provide a large commutative subalgebra of H_\mu, and after describing the combinatorics associated with the representation theory of H_\mu, generalize the RSK correspondence that is typically found in the representation theo...

Let ${\bf G}$ be a connected reductive group defined over $\mathbb{F}_q$, the finite field with $q$ elements. Let ${\bf B}$ be an Borel subgroup defined over $\mathbb{F}_q$. In this paper, we completely determine the composition factors of the induced module $\mathbb{M}(\op{tr})=\Bbbk{\bf G}\otimes_{\Bbbk{\bf B}}\op{tr}$ ($\op{tr}$ is the trivial ${\bf B}$-module) for any field $\Bbbk$.

In the first chapters, this paper contains a survey on the theory of ordinary characters of finite reductive groups with non-connected centre. The last chapters are devoted to the proof of Lusztig's conjecture on characteristic functions of character sheaves for all finite reductive groups of type A, split or not.

The dominant theme of this thesis is the construction of matrix representations of finite solvable groups using a suitable system of generators. For a finite solvable group $G$ of order $N = p_{1}p_{2}\dots p_{n}$, where $p_{i}$'s are primes, there always exists a subnormal series: $\langle {e} \rangle = G_{o} < G_{1} < \dots < G_{n} = G$ such that $G_{i}/G_{i-1}$ is isomorphic to a cyclic group of order $p_{i}$, $i = 1,2,\dots,n$. Associated with this series, there exists a ...

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.

Let $U_n(q)$ denote the upper triangular group of degree $n$ over the finite field $\F_q$ with $q$ elements. It is known that irreducible constituents of supercharacters partition the set of all irreducible characters $\Irr(U_n(q)).$ In this paper we present a correspondence between supercharacters and pattern subgroups of the form $U_k(q)\cap {}^wU_k(q)$ where $w$ is a monomial matrix in $GL_k(q)$ for some $k<n.$

The infinite-dimensional Iwahori--Hecke algebras $\mathcal{H}_\infty(q)$ are direct limits of the usual finite-dimensional Iwahori--Hecke algebras. They arise in a natural way as convolution algebras of bi-invariant functions on groups $\mathrm{GLB}(\mathbb{F}_q)$ of infinite-dimensional matrices over finite-fields having only finite number of non-zero matrix elements under the diagonal. In 1988 Vershik and Kerov classified all indecomposable positive traces on $\mathcal{H}_\...