Positivity of matrix coefficients of representations with real infinitesimal characters

Let G be a semisimple Lie group with a Cartan decomposition G = K exp(). For every finite-dimensional representation of G, there is an inner product on the representation space such that with respect to it, K acts as unitary operators, and exp() acts as positive definite Hermitian operators. In this paper, we observe that similar phenomena appear for infinite-dimensional representations with real infinitesimal characters. Supported by Tianyuan Mathematics Foundation of NSFC (Grant No. 10626050).