Harmonic analysis on locally symmetric spaces associated to cocompact discrete subgroups of SL2(R)

Abstract

In his 1956 paper, Selberg proved the famous Trace Formula for a semisimple Lie group G and its discrete subgroup ð€€€. The case when G = SL2(R) is quite well-known. In this thesis, we look at the decomposition of L2(ð€€€nG) into irreducible unitary representations of G. The multiplicities of the spherical representations correspond to the eigenvalues of the Laplacian on the locally symmetric space ð€€€nG=K. Our aim will be to nd a nite threshold on the multiplicity spectrum, or equivalently for the eigenvalue spectrum, which determines the entire spectrum.