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 $\pi$ of $G$, there exists a unique minimal positive-real-valued spherical function ${\varphi }_{\lambda }$ on $G$ such that $\mathcal{A}⟨\pi (\cdot )\xi ,\eta ⟩\le \Vert \xi {\Vert }_{{\mathcal{H}}_{\pi }}\Vert \eta {\Vert }_{{\mathcal{H}}_{\pi }}{\varphi }_{\lambda }$. This estimate has nice features of both asymptotic pointwise estimates and Lebesgue space estimates; indeed it is equivalent to pointwise estimates $|⟨\pi (\cdot )\xi ,\eta ⟩|\le C(\xi ,\eta ){\varphi }_{\lambda }$ for $K$-finite or smooth vectors $\xi$ and $\eta$, and it exhibits different decay rates in different directions at infinity in $G$. Further, if we assume the latter inequality with arbitrary $C(\xi ,\eta )$, we can prove the former inequality and then return to the latter inequality with explicit knowledge of $C(\xi ,\eta )$. On the other hand, it holds everywhere in $G$, in contrast to asymptotic estimates which are not global. We also provide some applications.

arXiv comment20 pages, revised version to improve readability, one new result added