Positivity of Principal Matrix Coefficients of Principal Series Representations of GL(n)(R)

Let G = GL(n)(R), with the usual Cartan decomposition G = K exp(p(0)) and the usual Iwasawa decomposition G = NAK. We determine the image of exp(p(0)) under the projection of G to K through the Iwasawa decomposition. As an application, we prove a positivity result about the matrix coefficients of principal series representations of G.