Non-archimedean Schur representations of $\mathrm{GL}(n,R)$ and their invariant lattices

Yassine EL Maazouz, Antonio Lerario

Published 2022-09-27Version 1

Let a $K$ be a non-archimedean discretely valued field and $R$ its valuation ring. Given a vector space $V$ of dimension $n$ over $K$ and a partition $\lambda$ of an integer $d$, we study the problem of determining the invariant lattices in the Schur module $S_\lambda(V)$ under the action of the group $\mathrm{GL}(n,R)$. When $K$ is a non-archimedean local field, our results determine the $\mathrm{GL}(n,R)$-invariant Gaussian distributions on $S_\lambda(V)$.