Character identities and asymptotic behavior of matrix coefficients of discrete series (Q1072660)

Let G be a connected semisimple Lie group with finite center, K its maximal compact subgroup, and \(G=KAN\) the Iwasawa decomposition of G. \textit{P. C. Trombi} and \textit{V. S. Varadarajan} [Acta Math. 129, 237-280 (1972; Zbl 0244.43006)] have derived a growth estimate for K-finite matrix coefficients c of the discrete series representations \(\pi_{\lambda}\) of G: \[ | c(a)| \leq C \delta^{(1+k)/2}(a) (1+\| \log a\|)^ r,\quad C,r\geq 0,\quad k>0,\quad a\in A^- \] \((A^-\) is a subgroup of A), \(\delta\) be the modular function of AN. The argument of Trombi and Varadarajan is based on a long analysis of differential equations for the matrix coefficients. In the paper under review it is shown that the growth estimate is a consequence of \textit{W. Schmid}'s character identities [Lect. Notes Math. 587, 196-225 (1977; Zbl 0362.22015)]. The authors prove that the growth estimate given by Trombi and Varadarajan is the best possible estimate which does not contradict the character identities.