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 Au to be the root-mean-square average over K, acting both on the left and the right, of u. We take a positive-real-valued spherical function ϕλ on G, and study the Banach convolution algebra of Cc(G)-functions u with the norm ‖u‖(λ):=∫GAu(x)ϕλ(x)dx. The C⁎ completion of this algebra is an exotic C⁎-algebra on G, in the sense that it lies “between” the reduced C⁎-algebra of G and the full C⁎-algebra of G, and in the sense that it arise as the completion of a star-algebra that does not contain an approximate identity.Using functional analysis and representation theory, we show that for all unitary representations π of G, there exists a unique minimal positive-real-valued spherical function ϕλ on G such that A〈π(⋅)ξ,η〉≤‖ξ‖Hπ‖η‖Hπϕλ. This estimate has nice features of both asymptotic pointwise estimates and Lebesgue space estimates; indeed it is equivalent to pointwise estimates |〈π(⋅)ξ,η〉|≤C(ξ,η)ϕλ 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). Finally, it holds everywhere in G, in contrast to asymptotic estimates which are not global and to Lp estimates which carry no pointwise information.