Decay estimates for matrix coefficients of unitary representations of semisimple Lie groups

Abstract

Let $G$ be a connected semisimple Lie group with finite centre and $K$ be a maximal compact subgroup thereof. Given a function $u$ on $G$, we define $\mathcal{A} u$ to be the root mean square average over $K$, acting both on the left and the right, of $u$. We show that for all unitary representations $π$ of $G$, there exists a unique minimal positive-real-valued spherical function $ϕ_λ$ on $G$ such that $\mathcal{A} \langle π(\cdot) ξ, η\rangle \leq \Vert ξ\Vert_{\mathcal{H}_π} \Vert η\Vert_{\mathcal{H}_π} ϕ_λ$. This estimate has nice features of both asymptotic pointwise estimates and Lebesgue space estimates; indeed it is equivalent to pointwise estimates $\vert \langle π(\cdot) ξ, η\rangle \vert \leq C(ξ, η) \,ϕ_λ$ for $K$-finite or smooth vectors $ξ$ and $η$, and it exhibits different decay rates in different directions at infinity in $G$. Further, if we assume the latter inequality with arbitrary $C( ξ, η)$, we can prove the former inequality and then return to the latter inequality with explicit knowledge of $C( ξ, η)$. On the other hand, it holds everywhere in $G$, in contrast to asymptotic estimates which are not global. We also provide some applications.