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Abstract We show, under an orientation hypothesis, that a log symplectic manifold with simple normal crossing singularities has a stable almost complex structure, and hence is Spin$$_c$$. In the compact Hamiltonian case we prove that the index of the Spin$$_c$$ Dirac operator twisted by a prequantum line bundle satisfies a $[Q,R]=0$ theorem. 

Let G G be a connected, linear, real reductive Lie group with compact centre. Let K > G K>G be compact. Under a condition on K K , which holds in particular if K K is maximal compact, we give a geometric expression for the multiplicities of the K K -types of any tempered representation (in fact, any standard representation) π \pi of G G . This expression is in the spirit of Kirillov’s orbit method and the quantisation commutes with reduction principle. It is based on the geometric realisation of π | K \pi |_K obtained in an earlier paper. This expression was obtained for the discrete series by Paradan, and for tempered representations with regular parameters by Duflo and Vergne. We obtain consequences for the support of the multiplicity function, and a criterion for multiplicity-free restrictions that applies to general admissible representations. As examples, we show that admissible representations of SU ( p , 1 ) \textrm {SU}(p,1) , SO 0 ( p , 1 ) \textrm {SO}_0(p,1) , and SO 0 ( 2 , 2 ) \textrm {SO}_0(2,2) restrict multiplicity freely to maximal compact subgroups.