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

TL;DR

This paper classifies and counts irreducible unitary representations with non-zero Lie algebra cohomology for the group SO_0(2,m), including discrete and holomorphic discrete series, and computes their cohomology Poincaré polynomial.

Contribution

It determines the number of such representations, their cohomology structure, and identifies discrete and holomorphic discrete series within this class for SO_0(2,m).

Findings

Number of equivalence classes of these representations is established.

Poincaré polynomial of their cohomologies is computed.

Identification of discrete and holomorphic discrete series among these representations.

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.