Finite traces and representations of the group of infinite matrices over a finite field

The aim of the present survey paper is to provide an accessible introduction to a new chapter of representation theory - harmonic analysis for noncommutative groups with infinite-dimensional dual space. I omitted detailed proofs but tried to explain the main ideas of the theory and its connections with other fields. The fact that irreducible representations of the groups in question depend on infinitely many parameters leads to a number of new effects which never occurred i...

We review the recent development in the representation theory of the $W_{1+\infty}$ algebra. The topics that we concern are, Quasifinite representation, Free field realizations, (Super) Matrix Generalization, Structure of subalgebras such as $W_\infty$ algebra, Determinant formula, Character formula. (Invited talk at ``Quantum Field Theory, Integrable Models and Beyond", YITP, 14-17 February 1994. To appear in Progress of Theoretical Physics Proceedings Supplement.)

The orthogonal groups are a series of simple Lie groups associated to symmetric bilinear forms. There is no analogous series associated to symmetric trilinear forms. We introduce an infinite dimensional group-like object that can be viewed as the limit of this non-existent series, were it to exist. We show that the representation theory of this object is well-behaved, and similar to the stable representation theory of orthogonal groups. Our theory is not specific to symmetric...

The classification of irreducible, spherical characters of the infinite-dimensional unitary/orthogonal/symplectic groups can be obtained by finding all possible limits of normalized, irreducible characters of the corresponding finite-dimensional groups, as the rank tends to infinity. We solve a q-deformed version of the latter problem for orthogonal and symplectic groups, extending previously known results for the unitary group. The proof is based on novel determinantal and d...

Let $\mathfrak{S}_\infty$ be the infinity permutation group and $\Gamma$ an arbitrary group. Then $\mathfrak{S}_\infty$ admits a natural action on $\Gamma^\infty$ by automorphisms, so one can form a semidirect product $\Gamma^\infty\rtimes \mathfrak{S}_\infty$, known as the {\it wreath} product $\Gamma\wr\mathfrak{S}_\infty$ of $\Gamma$ by $\mathfrak{S}_{\infty}$. We obtain a full description of unitary $II_1-$factor-representations of $\Gamma\wr\mathfrak{S}_\infty$ in terms ...

A full set of (higher order) Casimir invariants for the Lie algebra $gl(\infty )$ is constructed and shown to be well defined in the category $O_{FS}$ generated by the highest weight (unitarizable) irreducible representations with only a finite number of non-zero weight components. Moreover the eigenvalues of these Casimir invariants are determined explicitly in terms of the highest weight. Characteristic identities satisfied by certain (infinite) matrices with entries from $...

In this note we consider the algebra $U_q(\hat{sl}_\infty)$ and we study the category O of its integrable representations. The main motivations are applications to quantum toroidal algebras, more precisely predictions of character formulae for representations of quantum toroidal algebras. In this context, we state a general positivity conjecture for representations of $U_q(\hat{sl}_\infty)$ viewed as representations of quantum toroidal algebras, that we prove for Kirillov-Res...

Let $H$ be the space of all Hermitian matrices of infinite order and $U(\infty)$ be the inductive limit of the chain $U(1)\subset U(2)\subset...$ of compact unitary groups. The group $U(\infty)$ operates on the space $H$ by conjugations, and our aim is to classify the ergodic $U(\infty)$-invariant probability measures on $H$ by making use of a general asymptotic approach proposed in Vershik's note \cite{V}. The problem is reduced to studying the limit behavior of orbital inte...

In this paper we study (asymptotic) properties of the $*$-distribution of irreducible characters of finite quantum groups. We proceed in two steps, first examining the representation theory to determine irreducible representations and their powers, then we study the $*$-distribution of their trace with respect to the Haar measure. For the Sekine family we look at the asymptotic distribution (as the dimension of the algebra goes to infinity).

Asymptotic representation theory of general linear groups GL(n,q) over a finite field leads to studying probability measures \rho on the group U of all infinite uni-uppertriangular matrices over F_q, with the condition that \rho is invariant under conjugations by arbitrary infinite matrices. Such probability measures form an infinite-dimensional simplex, and the description of its extreme points (in other words, ergodic measures \rho) was conjectured by Kerov in connection wi...