Branching rules of singular unitary representations with respect to symmetric pairs (A_{2n-1}, D_n)
Branching rules of singular unitary representations with respect to symmetric pairs (A_{2n-1}, D_n)
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对称对 (A_{2n-1}, D_n) 的奇异酉表示的分支规则
DOI：
10.1142/s0129167x13500110
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
Hideko Sekiguchi
中科院分区：
文献类型：
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作者：
M.Hamilton;H.Konno;Hideko Sekiguchi
当限制到半单对称对（G，H）时，标量全纯离散级数表示的不可约分解由Schmid [Die Randwerte holomorphe funktionen auf hermetisch symmetrischen Raumen，Invent. Math.9（1969-1970）61-80]和小林[Multiplicity-Free Theorems of the Restrictions of Unitary Highest Weight Modules with Respect to Reductive Symmetric Pairs，Progress in Mathematics，Vol. 255（Birhäuser，2007），pp. 45-109]为H非紧凑型。本文讨论了对称对（U（n，n），SO ~*（2n）），并将Kobayashi-Schmid公式推广到具有不定度量的开Grassmannian流形上的Dolbeault上同调群中的某些非调和酉表示.得到的分支规则是无重性和离散可分解的，这适合于小林的离散可分解限制的一般理论的框架[Aλ（λ）相对于约化子群的限制的离散可分解性II -微局部分析和渐近K-支撑，Ann.Math.147（1998），709-729]。
The irreducible decomposition of scalar holomorphic discrete series representations when restricted to semisimple symmetric pairs (G, H) is explicitly known by Schmid [Die Randwerte holomorphe funktionen auf hermetisch symmetrischen Raumen,Invent. Math.9(1969–1970) 61–80] for H compact and by Kobayashi [Multiplicity-Free Theorems of the Restrictions of Unitary Highest Weight Modules with Respect to Reductive Symmetric Pairs, Progress in Mathematics, Vol. 255 (Birhäuser, 2007), pp. 45–109] for H non-compact. In this paper, we deal with the symmetric pair (U(n, n), SO*(2n)), and extend the Kobayashi–Schmid formula to certain non-tempered unitary representations which are realized in Dolbeault cohomology groups over open Grassmannian manifolds with indefinite metric. The resulting branching rule is multiplicity-free and discretely decomposable, which fits in the framework of the general theory of discrete decomposable restrictions by Kobayashi [Discrete decomposability of the restriction of A𝔮(λ) with respect to reductive subgroups II — micro-local analysis and asymptotic K-support,Ann. Math.147(1998), 709–729].