A geometric formula for multiplicities of 𝐾-types of tempered representations
A geometric formula for multiplicities of 𝐾-types of tempered representations
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DOI：
10.1090/tran/7857
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发表时间：
2018-05
影响因子：
1.3
通讯作者：
P. Hochs;Yanli Song;Shilin Yu
中科院分区：
文献类型：
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作者：
P. Hochs;Yanli Song;Shilin Yu
设G G是中心紧的连通线性实约化李群。让K>；G K>；G紧凑。在KK上的一个条件下，特别是如果KK是极大紧的，我们给出了G的任何调和表示(实际上是任何标准表示)的K-K型的重数的几何表达式π\pi。这一表述符合基里洛夫轨道法和约化原理的量子化交换的精神。它是基于早期论文中得到的π|K\pi|_K的几何实现。对于Paradan的离散级数和Duflo和Vergne的具有规则参数的调和表示，都得到了这个表达式。我们得到了支持多重性函数的结果，以及适用于一般可容许表示的多重性自由限制的一个判据。作为例子，我们证明了SU(p，1)、SO0(2，2)、SO0(2，2)、SO0(2，2)的可容许表示将重数限制为极大紧子群。
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.