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arXiv · 2604.09138

The K_+-fixed vectors of Iwahori-spherical GL_n-representations: connections with Zelevinsky's segments

Runze Wang
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Abstract

We study the space of K_+-fixed vectors of Iwahori-spherical representations of GL_n over a non-archimedean local field. For a generic Iwahori-spherical representation, we show that its decomposition into irreducible modules of the finite Lie group K/K_+ is controlled by a partition determined by the representation: an irreducible module occurs only if its partition is dominated by that partition, and when it occurs the multiplicity is a Kostka number. For an arbitrary irreducible Iwahori-spherical representation, we attach a partition from its data and prove a necessary condition: any occurring module must correspond to a partition dominated by this one, and the module attached to the partition itself occurs exactly once. We also give a combinatorial algorithm which, by further computation, determines precisely which modules actually occur and with what multiplicities. This answers a question of Prasad.

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BibTeXRIS

Runze Wang. 2026-04-10. The K_+-fixed vectors of Iwahori-spherical GL_n-representations: connections with Zelevinsky's segments. https://arxiv.org/abs/2604.09138

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