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A geometric formula for multiplicities of š¾-types of tempered representations
- Source :
- Transactions of the American Mathematical Society. 372:8553-8586
- Publication Year :
- 2019
- Publisher :
- American Mathematical Society (AMS), 2019.
-
Abstract
- 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) Ļ \pi of G G . This expression is in the spirit of Kirillovās orbit method and the quantisation commutes with reduction principle. It is based on the geometric realisation of Ļ | 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.
Details
- ISSN :
- 10886850 and 00029947
- Volume :
- 372
- Database :
- OpenAIRE
- Journal :
- Transactions of the American Mathematical Society
- Accession number :
- edsair.doi...........20ae9f64cf0f69043f4276b5867b5ce9