Kirillov’s orbit method: The case of discrete series representations

Abstract

Let π be a discrete series representation of a real semi-simple Lie group G 1 and let G be a semi-simple subgroup of G 1 . In this paper, we give a geometric expression of the G-multiplicities in π| G when the representation π is G-admissible.

Restriction of irreducible unitary representations of Spin(N,1) to parabolic subgroups

“…If G is compact, then Duflo's conjecture is a special case of the Spin c version of quantization commutes with reduction principle (see [47]). More generally, if G and H are both reductive, then the assertions (ii) and (iii) of the conjecture are consequences of a recent work of Paradan [48]. Note that in this case, Dulfo-Vargas and Paradan [14], [15], [48] proved that π| H is Hadmissible if and only if the moment map q : O π → h * is proper.…”

“…More generally, if G and H are both reductive, then the assertions (ii) and (iii) of the conjecture are consequences of a recent work of Paradan [48]. Note that in this case, Dulfo-Vargas and Paradan [14], [15], [48] proved that π| H is Hadmissible if and only if the moment map q : O π → h * is proper. In order to prove the assertion (i) of the conjecture in this case, one needs to prove the equivalence between properness and weak properness of the moment map.…”