复对称对的正则性与Gelfand性质
Regularity and the Gelfand Property for Complex Symmetric Pairs
AI总结：
本文证明连通复约化群的所有对称对均满足Aizenbud-Gourevitch正则性，解决了复域上的相关正则性猜想，并结合广义Harish-Chandra下降法等工具，将该结果推广至Gelfand-Kazhdan性质，同时处理了Rubio归约的四类不可约正则性问题。
AI中文摘要：
我们证明，连通复约化群的每个对称对都满足艾曾布德-古列维奇（Aizenbud--Gourevitch）意义下的正则性，这解决了复域上的艾曾布德-古列维奇正则性猜想。随后，广义哈里什-钱德拉（Harish--Chandra）下降法将每个复对称对的典范中心覆盖转化为一个盖尔范德-卡兹丹（Gelfand--Kazhdan）对。由相容谢瓦莱（Chevalley）对合导出的反自同构将覆盖上得到的GP2界升级为GP1，而有限中心下降法又将GP1传递至原对称对。特别地，这推出了范·戴克（van Dijk）关于复对称对的猜想。鲁比奥（Rubio）将未解决的不可约正则性问题归约为四类：DIII族$(D_r,A_{r-1}+\boldsymbol{C})$、平衡CII族$(C_{2r},C_r+C_r)$、部分剩余旋量块对以及EVII对$(E_7,E_6+\boldsymbol{C})$。我们通过四种不同的机制处理这些情形：对于DIII，我们在正则集上构造一个带符号等变的施瓦茨分布，并利用陈-孙（Chen--Sun）定理将其延拓至公共轨道边界；对于平衡CII，我们结合齐性性、特殊幂零轨道以及中心化子表示的稳定密度定理；对于旋量块，我们证明不等奇偶-奇偶块的优良性，在奇小秩情形下使用普热宾达（Przebinda）的正交分布定理，并在偶小秩情形下构造一个具有自动延拓性质的有限闭轨道；对于EVII，我们计算所有22个幂零轨道的分次$\boldsymbol{\frak{sl}}_2$数据，并利用中心环面特征消除剩余共振，包括两个残留的三重中心化子情形。最后，有限分量组装定理处理任意连通中心商及单因子间的对角耦合。
英文摘要：
We prove that every symmetric pair of a connected complex reductive group is regular in the sense of Aizenbud--Gourevitch. This settles the Aizenbud--Gourevitch regularity conjecture over the complex numbers. Generalized Harish--Chandra descent then makes the canonical central cover of every complex symmetric pair a Gelfand--Kazhdan pair. An anti-automorphism arising from a compatible Chevalley involution upgrades the resulting GP2 bound to GP1 on the cover, and finite central descent transfers GP1 to the original pair. In particular, van Dijk's conjecture on complex symmetric pairs follows. Rubio reduced the unresolved irreducible regularity problem to four families: the DIII family $(D_r,A_{r-1}+\mathbb{C})$, the balanced CII family $(C_{2r},C_r+C_r)$, some remaining Spin block pairs, and the EVII pair $(E_7,E_6+\mathbb{C})$. We treat these cases by four different mechanisms. For DIII we construct a sign-equivariant Schwartz distribution on the regular set and extend it across a common orbit boundary by the Chen--Sun theorem. For balanced CII we combine homogeneity, distinguished nilpotent orbits, and a stable-density theorem for the centralizer representation. For Spin blocks we prove pleasantness for unequal odd--odd blocks, use Przebinda's orthogonal-distribution theorem in odd smaller rank, and construct a finite orbit closure with automatic extension in even smaller rank. For EVII we compute the graded-$\mathfrak{sl}_2$ data for all twenty-two nilpotent orbits and use central-torus characters to eliminate the remaining resonances, including the two residual triple-centralizer cases. A finite-component assembly theorem then handles arbitrary connected central quotients and diagonal couplings among simple factors.
