Irreducibility of the Koopman representations for the group ${\rm GL}_0(2\infty,{\mathbb R})$ acting on three infinite rows

Let $O(\infty)$ and $U(\infty)$ be the inductively compact infinite orthogonal group and infinite unitary group respectively. The classifications of ergodic probability measures with respect to the natural group action of $O(\infty)\times O(m)$ on $\mathrm{Mat}(\mathbb{N}\times m, \mathbb{R})$ and that of $U(\infty)\times U(m)$ on $\mathrm{Mat}(\mathbb{N}\times m, \mathbb{C})$ are due to Olshanski. The original proofs for these results are based on the asymptotic representati...

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...

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.

Spherical representations and functions are the building blocks for harmonic analysis on riemannian symmetric spaces. In this paper we consider spherical functions and spherical representations related to certain infinite dimensional symmetric spaces $G_\infty/K_\infty = \varinjlim G_n/K_n$. We use the representation theoretic construction $\phi (x) = <e, \pi(x)e>$ where $e$ is a $K_\infty$--fixed unit vector for $\pi$. Specifically, we look at representations $\pi_\infty = \...

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...

The Cohn-Umans (FOCS '03) group-theoretic framework for matrix multiplication produces fast matrix multiplication algorithms from three subsets of a finite group $G$ satisfying a simple combinatorial condition (the Triple Product Property). The complexity of such an algorithm then depends on the representation theory of $G$. In this paper we extend the group-theoretic framework to the setting of infinite groups. In particular, this allows us to obtain constructions in Lie gro...

We construct the so-called quasiregular representations of the group $B_0^{\mathbb N}({\mathbb F}_p)$ of infinite upper triangular matrices with coefficients in a finite field and give the criteria of theirs irreducibility in terms of the initial measure. These representations are particular case of the Koopman representation hence, we find new conditions of its irreducibility. Since the field ${\mathbb F}_p$ is compact some new operators in the commutant emerges. Therefore, ...

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...

It is well-known that characters classify linear representations of finite groups, that is if characters of two representations of a finite group are the same, these representations are equivalent. It is also well-known that, in general, this is not true for representations of infinite groups, even if they are finitely generated. The goal of this paper is to establish a characterization of representations of finitely generated groups in terms of projective joint spectra. This...