Irreducible unitary representations with non-zero relative Lie algebra cohomology of the Lie group $SO_0(2,m)$

Abstract

By a theorem of D. Wigner, an irreducible unitary representation with non-zero $(\frak{g},K)$-cohomology has trivial infinitesimal character, and hence up to unitary equivalence, these are finite in number. We have determined the number of equivalence classes of these representations and the Poincar\'{e} polynomial of cohomologies of these representations for the Lie group $SO_0(2,m)$ for any positive integer $m.$ We have also determined, among these, which are discrete series representations and holomorphic discrete series representations.