发表机构
Louisiana State University; UCLA Mathematics Department; Université Clermont Auvergne, CNRS, LMBP（路易斯安那州立大学; 加州大学洛杉矶分校数学系; 克莱蒙奥弗涅大学，法国国家科学研究中心，LMBP）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为有限型仿射群概形建立范畴表示论形式体系，适用于任意基域，并证明正特征约化群表示范畴上恒等与Frobenius态射的范畴迹等同于伴随商及不动点上的Ind-凝聚层∞-范畴。
AI 中文摘要
本文提出了一种适用于有限型仿射群概形（在弱和强两种设定下）的范畴表示论形式体系，该体系适用于任意基域。这一形式体系基于为研究此类群概形的表示及Harish-Chandra（双）模而发展的适当的∞-范畴设定，这些表示与（双）模构成了此类结构的基本构件和基本例子。我们还在这一背景下研究了范畴迹，并特别证明了：在正特征代数闭域k上，连通约化代数群G的适当表示∞-范畴上，恒等态射（分别地，Frobenius态射）的范畴迹，在适当假设下，等同于伴随商G/G（分别地，Frobenius不动点G^F）上的Ind-凝聚层∞-范畴。
英文摘要
In this paper we present a formalism of categorical representation theory for affine group schemes of finite type over fields (both in the weak and strong settings) which applies to arbitrary base fields. This is based on the development of an appropriate $\infty$-categorical setting for the study of representations and Harish-Chandra (bi)modules of such group schemes, which constitute the building blocks and basic examples of such structures. We also study categorical traces in this context, and show in particular that the categorical trace of the identity morphism, resp.~Frobenius morphism, on the appropriate $\infty$-category of representations of a connected reductive algebraic group $G$ over an algebraically closed field $k$ of positive characteristic identifies (under suitable assumptions) with the $\infty$-category of Ind-coherent sheaves on the adjoint quotient $G/G$, resp.~of the fixed points $G^{\mathrm{F}}$ of the Frobenius.
Comments53 pages
