主理想局部环上光滑群格式的谱等价性
Spectral equivalence of smooth group schemes over principal ideal local rings
摘要：
设$\mathcal{G}$是有限型光滑线性群格式.对任意正整数$k$和有限域$\mathbb{F}$，设$W_k(\mathbb{F})$是$\mathbb{F}$上长度为$k$的Witt向量环.证明了对于任意正整数$k$和特征足够大的有限域$\mathbb{F}$，$\mathcal{G}(\mathbb{F}[t]/(t^k))$和$\mathcal{G}(W_k(\mathbb{F}))$的群代数是同构的（即不可约表示维数的多个集合相等）.证明了如果$\mathrm{char}\mathbb{F}$足够大，则集合$\{\dimρ\big|ρ\in \mathrm{irr}(\mathcal{G}(\mathbb{F}))\}$的基数在$\mathbb{F}$中一致有界.
英文摘要：
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 $\{\dimρ\big|ρ\in \mathrm{irr}(\mathcal{G}(\mathbb{F}))\}$ is bounded uniformly in $\mathbb{F}$.
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