论文标题
平滑组方案的光谱等效性超过主要理想本地环
Spectral equivalence of smooth group schemes over principal ideal local rings
论文作者
论文摘要
令$ \ Mathcal {G} $为有限类型的平滑线性组方案。对于任何积极的整数$ k $和有限字段$ \ mathbb {f} $，让$ w_k（\ mathbb {f}）$是长度$ k $ over $ \ mathbb {f} $的witt vectors的环。 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} $具有足够大的特征。我们还证明，如果$ \ mathrm {char} \ mathbb {f} $足够大，则设置$ \ {\ dimCt \ big |ρ\ in \ mathrm {inrrm {ilrm {inrcal {hratcal {g}（g}（g}）（g}（g}））
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 $\mathbb{F}$.
