Let $(\tau , V_{\tau })$ be a finite dimensional representation of a maximal compact subgroup K of a connected non-compact semisimple Lie group G, and let $\Gamma $ be a uniform torsion-free lattice in G. We obtain an infinitesimal version of the celebrated Matsushima–Murakami formula, which relates the dimension of the space of automorphic forms associated to $\tau $ and multiplicities of irreducible $\tau ^\vee $-spherical spectra in $L^2(\Gamma \backslash G)$. This result gives a promising tool to study the joint spectra of all central operators on the homogenous bundle associated to the locally symmetric space and hence its infinitesimal $\tau $-isospectrality. Along with this, we prove that the almost equality of $\tau $-spherical spectra of two lattices assures the equality of their $\tau $-spherical spectra.

The results of Orihara [10] and Malliavin2 [7] on the asymptotic behaviour in polar coordinates of Brownian motion on a symmetric space of non-compact type are obtained by means of a skew product representation on K/M x A+of the Brownian motion on the set of regular points of X. Results of Norris, Rogers, and Williams [9] are interpreted in this context.