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 $W^{\ast }((\pi \otimes \pi )(G\times G))$ 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 \[ \lim_{n\to\infty}\frac{1}{\int_{B_n}Ξ(g)^2dg}\int_{B_n}π(g)\otimesπ(g^{-1})dg=F, \] 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 $G$.

24 pages with an appendix, this version contains an additional reference