L spectral theory and heat dynamics of locally symmetric spaces

In this paper we first derive several results concerning the L spectrum of arithmetic locally symmetric spaces whose Q-rank equals one. In particular, we show that there is an open subset of C consisting of eigenvalues of the L Laplacian if p < 2 and that corresponding eigenfunctions are given by certain Eisenstein series. On the other hand, if p > 2 there is at most a discrete set of real eigenvalues of the L Laplacian. These results are used in the second part of this paper in order to show that the dynamics of the L heat semigroups for p < 2 is very different from the dynamics of the L heat semigroups if p ≥ 2.