Symmetry breaking operators for real reductive groups of rank one
Symmetry breaking operators for real reductive groups of rank one
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DOI：
10.1016/j.jfa.2020.108568
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发表时间：
2018-12
影响因子：
1.7
通讯作者：
Jan Frahm;Clemens Weiske
中科院分区：
文献类型：
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作者：
Jan Frahm;Clemens Weiske
对于一对实还原群G‘⊂G，我们考虑G的球面主级数表示π和G’的τ之间的交织算子的空间Hom G‘(π|G’，τ)，也称为对称破缺算子。限制到其中dimHom G‘(π|G’，τ)<；∞且G和G‘是实一阶的那些对(G，G’)，我们根据它们的分布核对所有对称破缺算子进行了显式分类。这推广了Kobayashi-Speh以前关于(G，G‘)=(O(1，n+1)，O(1，n))的工作，得到了F=C，H，O和F<；U(n−m；F)的约化对(G，G’)=(U(1，n+1；F)，U(1，m+1；F)×F)。在大多数情况下，所有对称破坏算子都可以使用一个亚纯族分布族来构造，我们将详细描述这些分布族的极点和残数。除了这一族外，还可能出现一些我们明确确定的零星对称破缺算符。
For a pair of real reductive groups G′⊂ G we consider the space Hom G′(π| G′, τ) of intertwining operators between spherical principal series representations π of G and τ of G′, also called symmetry breaking operators. Restricting to those pairs (G, G′) where dim Hom G′(π| G′, τ)<∞ and G and G′ are of real rank one, we classify all symmetry breaking operators explicitly in terms of their distribution kernels. This generalizes previous work by Kobayashi–Speh for (G, G′)=(O (1, n+ 1), O (1, n)) to the reductive pairs (G, G′)=(U (1, n+ 1; F), U (1, m+ 1; F)× F) with F= C, H, O and F< U (n− m; F). In most cases, all symmetry breaking operators can be constructed using one meromorphic family of distributions whose poles and residues we describe in detail. In addition to this family, there may occur some sporadic symmetry breaking operators which we determine explicitly.