Criteria of irreducibility of the Koopman representations for the group ${\rm GL}_0(2\infty,{\mathbb R})$

Let $\mathbb F$ be a finite field. Consider a direct sum $V$ of an infinite number of copies of $\mathbb F$, consider the dual space $V^\diamond$, i.~e., the direct product of an infinite number of copies of $\mathbb F$. Consider the direct sum ${\mathbb V}=V\oplus V^\diamond$. The object of the paper is the group $\mathbf{GL}$ of continuous linear operators in $\mathbb V$. We reduce the theory of unitary representations of $\mathbf{GL}$ to projective representations of a cer...

We study the representations of the quantum Galilei group by a suitable generalization of the Kirillov method on spaces of non commutative functions. On these spaces we determine a quasi-invariant measure with respect to the action of the quantum group by which we discuss unitary and irreducible representations. The latter are equivalent to representations on \ell^2, i.e. on the space of square summable functions on a one dimensional lattice.

Necessary and sufficient conditions for a group to possess a faithful irreducible representation are investigated

We introduce a notion of measure contracting actions and show that Koopman representations corresponding to ergodic measure contracting actions are irreducible. As a corollary we obtain that Koopman representations associated to canonical actions of Higman-Thompson groups are irreducible. We also show that the actions of weakly branch groups on the boundaries of rooted trees are measure contracting. This gives a new point of view on irreducibility of the corresponding Koopman...

We study direct limits $(G,K) = \varinjlim (G_n,K_n)$ of compact Gelfand pairs. First, we develop a criterion for a direct limit representation to be a multiplicity--free discrete direct sum of irreducible representations. Then we look at direct limits $G/K = \varinjlim G_n/K_n$ of compact riemannian symmetric spaces, where we combine our criterion with the Cartan--Helgason Theorem to show in general that the regular representation of $G = \varinjlim G_n$ on a certain functio...

The goal of harmonic analysis on a (noncommutative) group is to decompose the most `natural' unitary representations of this group (like the regular representation) on irreducible ones. The infinite-dimensional unitary group U(infinity) is one of the basic examples of `big' groups whose irreducible representations depend on infinitely many parameters. Our aim is to explain what the harmonic analysis on U(infinity) consists of. We deal with unitary representations of a reaso...

We formalize the notion of matrix coefficients for distributional vectors in a representation of a real reductive group, which consist of generalized functions on the group. As an application, we state and prove a Gelfand-Kazhdan criterion for a real reductive group in very general settings.

A "2-group" is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2-groups have representations on "2-vector spaces", which are categories analogous to vector spaces. Unfortunately, Lie 2-groups typically have few representations on the finite-dimensional 2-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional 2...

Let $F$ be a non-discrete non-Archimedean locally compact field and $\mathcal{O}_F$ the ring of integers in $F$. The main results of this paper are Theorem 1.2 that classifies ergodic probability measures on the space $\mathrm{Mat}(\mathbb{N}, F)$ of infinite matrices with enties in $F$ with respect to the natural action of the group $\mathrm{GL}(\infty,\mathcal{O}_F) \times \mathrm{GL}(\infty,\mathcal{O}_F)$ and Theorem 1.6 that, for non-dyadic $F$, classifies ergodic probab...

The groups mentioned in the title are certain matrix groups of infinite size over a finite field $\mathbb F_q$. They are built from finite classical groups and at the same time they are similar to reductive $p$-adic Lie groups. In the present paper, we initiate the study of invariant measures for the coadjoint action of these infinite-dimensional groups. We examine first the group $\mathbb{GLB}$, a topological completion of the inductive limit group $\varinjlim GL(n, \mathb...