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

Let $\mathtt{k}$ be an algebraic closure of a finite field $\mathbb{F}_{q}$ of characteristic $p$. Let $G$ be a connected unipotent group over $\mathtt{k}$ equipped with an $\mathbb{F}_q$-structure given by a Frobenius map $F:G\to G$. We will denote the corresponding algebraic group defined over $\mathbb{F}_q$ by $G_0$. Character sheaves on $G$ are certain objects in the triangulated braided monoidal category $\mathscr{D}_G(G)$ of bounded conjugation equivariant $\bar{\mathbb...

We study the representations of the infinite symmetric group induced from the identity representations of Young subgroups. It turns out that such induced representations can be either of type~I or of type~II. Each Young subgroup corresponds to a partition of the set of positive integers; depending on the sizes of blocks of this partition, we divide Young subgroups into two classes: large and small subgroups. The first class gives representations of type I, in particular, irre...

Consider the pseidounitary group $G=U(p,q)$ and its compact subgroup $K=U(p)$. We construct an explicit unitary intertwining operator from the tensor product of a holomorphic representation and a antiholomorphic representation of $G$ to the space $L^2(G/K)$. This implies the existense of a canonical action of the group $G\times G$ in $L^2(G/K)$. We also give a survey of analysis of Berezin kernels and their relations with special functions.

We study the von Neumann algebra, generated by the unitary representations of infinite-dimensional groups nilpotent group $B_0^{\mathbb N}$. The conditions of the irreducibility of the regular and quasiregular representations of infinite-dimensional groups (associated with some quasi-invariant measures) are given by the so-called Ismagilov conjecture (see [1,2,9-11]). In this case the corresponding von Neumann algebra is type ${\rm I}_\infty$ factor. When the regular represen...

In this paper we give an asymptotic formula for a matrix integral which plays a crucial role in the approach of Diaconis et al. to random matrix eigenvalues. The choice of parameter for the asymptotic analysis is motivated by an invariant theoretic interpretation of this type of integral. For arbitrary regular irreducible representations of arbitrary connected semisimple compact Lie groups, we obtain an asymptotic formula for the trace of certain operators on the space of ten...

In this paper all deformations of the general linear group, subject to certain restrictions which in particular ensure a smooth passage to the Lie group limit, are obtained. Representations are given in terms of certains sets of creation and annihilation operators. These creation and annihilation operators may belong to a generalisation of the $q$-quark type or $q$-hadronic type, of $q$-boson or $q$-fermion type. We are also led to a natural definition of $q$-direct sums of q...

In this paper, we present a vertex operator approach to construct and compute all complex irreducible characters of the general linear group $\mathrm{GL}_n(\mathbb F_q)$. Green's theory of $\mathrm{GL}_n(\mathbb F_q)$ is recovered and enhanced under the realization of the Grothendieck ring of representations $R_G=\bigoplus_{n\geq 0}R(\mathrm{GL}_n(\mathbb F_q))$ as two isomorphic Fock spaces associated to two infinite-dimensional $F$-equivariant Heisenberg Lie algebras $\wide...

Let $\mathbf G$ be a reductive algebraic group over a non-archimedean local field $F$ of characteristic zero and let $G=\mathbf G(F)$ be the group of $F$-rational points. Let $\mathcal H(G)$ be the Hecke algebra and let $\mathcal J(G)$ be the asymptotic Hecke algebra, as defined by Braverman and Kazhdan. We classify irreducible representations of $\mathcal J(G)$. As a consequence, we prove a conjecture of Bezrukavnikov-Braverman-Kazhdan that the inclusion $\mathcal H(G)\subse...

The Lie algebra gl(lambda) dependent on the complex parameter lambda is a continuous version of the Lie algebra gl(inf) of infinite matrices with only finite number of nonzero entries. The gl(lambda) was first introduced by B.L.Feigin in [1] in connection with the Lie algebra cohomologies of the differential operators on the complex line. The paper is devoted to the representation theory of the gl(lambda). The Shapovalov's form determinant is calculated; it depends polynomial...

We study the structure of minimal parabolic subgroups of the classical infinite dimensional real simple Lie groups, corresponding to the classical simple direct limit Lie algebras. This depends on the recently developed structure of parabolic subgroups and subalgebras that are not necessarily direct limits of finite dimensional parabolics. We then discuss the use of that structure theory for the infinite dimensional analog of the classical principal series representations. We...