Spectral equivalence of smooth group schemes over principal ideal local rings

Abstract

Let $\mathcal{G}$ be a smooth linear group scheme of finite type. For any positive integer $k$ and a finite field $\mathbb{F}$, let $W_k(\mathbb{F})$ be the ring of Witt vectors of length $k$ over $\mathbb{F}$. We show that the group algebras of $\mathcal{G}(\mathbb{F}[t]/(t^k))$ and $\mathcal{G}(W_k(\mathbb{F}))$ are isomorphic (i.e. the multi-sets of the dimensions of the irreducible representations are equal) for any positive integer $k$ and finite field $\mathbb{F}$ with large enough characteristic. We also prove that if $\mathrm{char}\mathbb{F}$ is large enough, then the cardinality of the set $\{\dim\rho\big|\rho\in \mathrm{irr}(\mathcal{G}(\mathbb{F}))\}$ is bounded uniformly in $\mathbb{F}$.