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

Spectral equivalence of smooth group schemes over principal ideal local rings