The K_+-fixed vectors of Iwahori-spherical GL_n-representations: connections with Zelevinsky's segments
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918451690340352 |
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| author | Wang, Runze |
| author_facet | Wang, Runze |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_09138 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The K_+-fixed vectors of Iwahori-spherical GL_n-representations: connections with Zelevinsky's segments Wang, Runze Representation Theory 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. |
| title | The K_+-fixed vectors of Iwahori-spherical GL_n-representations: connections with Zelevinsky's segments |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2604.09138 |