A multiplicity one theorem for $\mathrm{GL}_n$ and $\mathrm{SL}_n$ over complete discrete valuation rings

Shiv Prakash Patel, Pooja Singla

Published 2018-09-24Version 1

Let $\mathfrak{o}$ be the ring of integers of a non-archimedian local field with maximal ideal $\mathfrak{p}$ and finite residue field of characteristic $p.$ For $\ell \geq 1,$ let $\mathfrak{o}_\ell = \mathfrak{o}/\mathfrak{p}^\ell,$ $\mathbf{G}(\mathfrak{o}_\ell) = {\rm GL}_{n}(\mathfrak{o}_\ell)$ or ${\rm SL}_{n}(\mathfrak{o}_{\ell})$ and $U(\mathfrak{o}_{\ell}) \subset \mathbf{G}(\mathfrak{o}_{\ell})$ be the subgroup of upper triangular unipotent matrices. We prove that the induced representation $\mathrm{Ind}^{\mathbf{G}(\mathfrak{o}_\ell)}_{U(\mathfrak{o}_\ell)}(\theta)$ of $\mathbf{G}(\mathfrak{o}_{\ell})$ obtained from a non-degenerate character $\theta$ of $U(\mathfrak{o}_\ell)$ is multiplicity free. This is analogous to the multiplicity one theorem in Gelfand-Graev representation for Chevalley groups. We also prove that for all $\mathbf{G}(\mathfrak{o}_\ell),$ except for ${\rm SL}_{n}(\mathfrak{o}_{\ell})$ with even $p$ or $p \mid n,$ an irreducible representation of $\mathbf{G}(\mathfrak{o}_\ell)$ is a constituent of $\mathrm{Ind}^{\mathbf{G}(\mathfrak{o}_\ell)}_{U(\mathfrak{o}_\ell)}(\theta)$ if and only if it is regular.