Parabolically induced representations of $GL_n(O_l)$

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arXiv · 2610.15762

Parabolically induced representations of $GL_n(O_l)$

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Abstract

Let $\mathcal O$ be the ring of integers of a non-Archimedean local field with uniformizer $\varpi$, and let $\mathcal O_\ell=\mathcal O/\varpi^\ell\mathcal O$ for $\ell\geq1$. In this article, we study representations of $\mathrm{GL}_n(\mathcal O_\ell)$ parabolically induced from characters of the Borel subgroup $\mathrm{B}_n(\mathcal O_\ell)$. For $n=3$, we establish a criterion for the irreducibility of these induced representations in terms of the relative conductors of the inducing characters. Motivated by this result, we formulate a conjecture characterizing the irreducibility of such representations for arbitrary $n$. We further show that when all inducing characters are equal, the resulting induced representation is not multiplicity-free, implying that the corresponding triple $(\mathrm{GL}_n(\mathcal O_\ell),\mathrm{B}_n(\mathcal O_\ell),χ)$ is not a Gelfand pair. Finally, we study the irreducible decomposition of these representations for $\mathrm{GL}_3(\mathcal O_\ell)$ when two relative conductors attain their maximal values. Using the decomposition theory for $\mathrm{GL}_2(\mathcal O_\ell)$, we describe the irreducible constituents and determine their dimensions.

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Prem Dagar. 2026-10-11. Parabolically induced representations of $GL_n(O_l)$. https://arxiv.org/abs/2610.15762

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