Braiding and asymptotic Schur's orthogonality

TL;DR

This paper investigates the properties of the braiding operator in unitary representations of locally compact groups, establishing conditions for its inclusion in von Neumann algebras and deriving an asymptotic orthogonality relation for semisimple groups over local fields.

Contribution

It characterizes when the braiding operator belongs to the von Neumann algebra generated by a representation and proves an asymptotic Schur's orthogonality relation for certain semisimple groups.

Findings

The braiding operator is in the von Neumann algebra if and only if the representation is irreducible.

Established an asymptotic limit for the quasi-regular representation involving the Harish-Chandra function.

Derived conditions under which the limit of averaged tensor products converges to the braiding operator.

Abstract

Let $\pi:G\to U(\mathcal H)$ be a unitary representation of a locally compact group. The braiding operator $F:\mathcal H\otimes\mathcal H\to \mathcal H\otimes\mathcal H$, which flips the components of the Hilbert tensor product $F(v\otimes w)=w\otimes v$, belongs to the von Neumann algebra $W^*((\pi\otimes\pi)(G\times G))$ if and only if $\pi$ is irreducible. Suppose $G$ is semisimple over a local field. If $G$ is non-compact with finite center, $P<G$ is a minimal parabolic, $\pi:G\to U(L^2(G/P))$ is the quasi-regular representation, then \[ \lim_{n\to\infty}\frac{1}{\int_{B_n}\Xi(g)^2dg}\int_{B_n}\pi(g)\otimes\pi(g^{-1})dg=F, \] in the weak operator topology, where $\Xi$ is the Harish-Chandra function of $G$ and $B_n$ is the ball of radius $n$ around the identity defined by a natural length function on $G$.