SU(2,2) 的酉 (g,K)-模的系数演算

中文摘要 AI 辅助

我们为 SU(2,2) 的可容许 (g,K)-模发展了一套显式的算子与系数演算，其中 g = sl(4,C)，K = S(U(2) x U(2))。K-型由三元组 (n,k,m) 索引。我们引入了与非紧根相关的算子 A_delta 和 B_delta，以及辅助算子 Q 和 R。我们建立了它们的交换子关系、中心系数关系，以及在酉情形下相应的伴随与范数因子恒等式。这将 (g,K)-模分析的一部分归结为研究由这些算子复合产生的标量系数。该系数公式适用于重数为一的权，包括所有边界权。我们推导了必要的符号限制和约束系数空间的显式关系。这些系数也决定了不可约性。根据 n+m 的最小值，我们分析了若干族酉 (g,K)-模。对于 N=0，该构造产生双参数族的酉模，并描述了当某些系数为零时的可约性。对于 N>0，该方法产生更大的 K-型族和无重数阶梯型族。所得模式与 Knapp 和 Speh 对 SU(2,2) 酉对偶的已知描述相容。本文提供了一个从 K-型结构恢复酉 (g,K)-模的显式框架，并提出了对其他实约化群的可能方法。

英文摘要

We develop an explicit operator and coefficient calculus for admissible (g,K)-modules of SU(2,2), where g = sl(4,C) and K = S(U(2) x U(2)). The K-types are indexed by triples (n,k,m). We introduce operators A_delta and B_delta associated with the noncompact roots, together with auxiliary operators Q and R. We establish their commutator relations, the central coefficient relations, and, in the unitary setting, the corresponding adjoint and norm-factor identities. This reduces part of the analysis of (g,K)-modules to the study of scalar coefficients arising from compositions of these operators. The coefficient formula applies to weights of multiplicity one, including all boundary weights. We derive necessary sign restrictions and explicit relations constraining the coefficient space. These coefficients also determine irreducibility. Several families of unitary (g,K)-modules are analyzed according to the minimal value of n+m. For N=0, the construction yields two-parameter families of unitary modules and describes reducibility when certain coefficients vanish. For N>0, the method produces both larger families of K-types and multiplicity-free ladder-type families. The resulting patterns are compatible with the known description of the unitary dual of SU(2,2) due to Knapp and Speh. The paper provides an explicit framework for recovering unitary (g,K)-modules from their K-type structure and suggests a possible approach to other real reductive groups.

发表机构

• University of Zagreb, Faculty of Electrical Engineering and Computing（萨格勒布大学电气工程和计算学院）

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