AI 中文总结
该研究针对非阿基米德局部域上GL_N(F)的不可约零深度表示，结合Bushnell-Kutzko型等工具，确定了超特殊极大紧子群幂幺根基固定向量空间的分解，建立了p进与有限侧对偶的交织关系，得到了完整有效的超特殊分支律。
AI 中文摘要
对于非阿基米德局部域上一般线性群的每个不可约零深度表示，该表示由Zelevinsky意义下的段多重集参数化，我们确定了超特殊极大紧子群的幂幺根基固定向量空间的完全分解。该分解根据Zelevinsky参数明确描述：不可约成分由Kostka数和Zelevinsky分解数控制，且自然有界于两个分划之间，这两个分划源于该表示及其Aubert对偶的Zelevinsky段。在一般情形下，结果完全由Kostka数给出。对于一般的零深度Bernstein块，重数是对应单块重数的乘积。我们首先对任意非分歧连通约化群证明，取幂幺固定向量的函子将p进侧的Zelevinsky-Aubert对偶与有限侧的Alvis-Curtis对偶交织起来。一般线性群的证明结合了Bushnell-Kutzko型、有限Harish-Chandra系列、Iwahori-Hecke代数和Zelevinsky分类，为所有不可约零深度表示提供了完整且有效的超特殊分支律。
英文摘要
For every irreducible depth-zero representation of the general linear group over a non-Archimedean local field, parameterized by multisets of segments in the sense of Zelevinsky, we determine the complete decomposition of the space of vectors fixed by the pro-unipotent radical of a hyperspecial maximal compact subgroup. The decomposition is described explicitly in terms of the Zelevinsky parameters: the irreducible constituents are governed by Kostka numbers and Zelevinsky decomposition numbers, and they are naturally bounded between two partitions arising from the Zelevinsky segments of the representation and of its Aubert dual. In the generic case the result is given entirely by Kostka numbers. For general depth-zero Bernstein blocks, the multiplicities are products of the corresponding single-block multiplicities. We first prove, for an arbitrary unramified connected reductive group, that the functor of taking pro-unipotent fixed vectors intertwines Zelevinsky--Aubert duality on the p-adic side with Alvis--Curtis duality on the finite side. The proof for general linear groups combines Bushnell--Kutzko types, finite Harish--Chandra series, Iwahori--Hecke algebras, and the Zelevinsky classification to provide a complete and effective hyperspecial branching law for all irreducible depth-zero representations.
