Langlands classification for real Lie groups with reductive Lie algebra

LetG be a (not necessarily connected) real Lie group with reductive Lie algebra. We consider representations ofG which some call admissible but we call them of Harish-Chandra type. We show that any nontempered irreducible Harish-Chandra type representation ofG is infinitesimally equivalent to the Langlands quotient obtained from an essentially unique triple (M, V, ) of Langlands data; while for tempered irreducible Harish-Chandra type representations we prove they are infinitesimally subrepresentations of some induced representations U^V, with imaginary and withV from the quasi-discrete series of a suitableM (perhapsG=M; we define the quasi-discrete series in Definition 4.5 of this paper.We show that irreducible continuous unitary representations of really reductive groups are of Harish-Chandra type. Then the results above yield the canonical decomposition of the unitary spectrum>G for any really reductiveG. In particular, this holds ifG/G ₀ is finite, so the center of the connected semi-simple subgroup with Lie algebra g, g] may be infinite!Research supported, in part, by the Hungarian National Fund for Scientific Research (grant Nos. 1900 and 2648).