AI 中文总结
本文研究局部域上Chevalley群的一般球面酉对偶，通过结合多种代数工具证明其与域无关，且具有统一的简单描述。
AI 中文摘要
我们证明，具有等参数的分次仿射Hecke代数的一般球面参数是酉的，当且仅当其正规化交织形式在反射表示及其第二对称幂的不可约分量上为正。证明结合了反射表示的符号公式、Yu的统一两壁操作、单链Jantzen递推以及根偏序分析。在Bₙ型和Cₙ型中，我们更精确地证明，仅需反射表示V与Sym²(V)的无迹对角分量即可。当允许任意Weyl群型时，我们还给出了一个更简短的第二证明。第一个证明的推论（通过Barbasch-Vogan小K型）是，实数域或非阿基米德局部域上Chevalley群的一般球面酉对偶，对于单链群和复辛群也在复数域上，与域无关。此外，该结果具有简单统一的描述：即Weyl共轭的、由仿射Weyl群排列在基本Weyl腔中的2个alcove的幂的不交并，其中指数是短余根子系统的Dynkin图的匹配数。
英文摘要
We prove that a generic spherical parameter for the graded affine Hecke algebra with equal parameters is unitary if and only if its normalized intertwining forms are positive on the reflection representation and on the irreducible constituents of its second symmetric power. The proof combines a signature formula for the reflection representation, Yu's uniform two-wall operation, a simply-laced Jantzen recursion, and a root-poset analysis. In types B_n and C_n, we prove more sharply that the reflection representation V and the traceless-diagonal constituent of Sym^2(V) suffice. We also give a shorter second proof when arbitrary Weyl-group types are allowed. The consequence of the first proof (via Barbasch-Vogan petite K-types) is that the generic spherical unitary dual of a Chevalley group over the real numbers or a nonarchimedean local field, and for simply-laced groups and complex symplectic groups also over the complex numbers, is independent of the field. Moreover, the answer has a simple uniform description as the Weyl conjugates of a disjoint union of a power of 2 alcoves in the affine Weyl group arrangement in the fundamental Weyl chamber, where the exponent is the matching number of the Dynkin diagram of the subsystem of short coroots.
Comments34 pages, 1 figure, use of AI (see acknowledgements)
