Zelevinsky Segments and Hyperspecial Branching Laws for All Depth-Zero Representations of GL_N(F)

Abstract

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.