Braiding and asymptotic Schur's orthogonality

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 $G$.