The symmetry of finite group schemes, Watanabe type theorem, and the $a$-invariant of the ring of invariants

Abstract

Let $k$ be a field, and $G$ be a $k$-group scheme of finite type. Let $G_{\mathrm{ad}}$ be the $k$-scheme $G$ with the adjoint action of $G$. We call ${\lambda }_{G,G}=H^0(\mathrm{Spec}k,e^{\ast }({\omega }_{G_{\mathrm{ad}}}))$ the Knop character of $G$, where $e:\mathrm{Spec}k\to G_{\mathrm{ad}}$ is the unit element, and ${\omega }_{G_{\mathrm{ad}}}$ is the $G$-canonical module. We prove that ${\lambda }_{G,G}$ is trivial in the following cases: (1) $G$ is finite, and $k[G{]}^{\ast }$ is a symmetric algebra; (2) $G$ is finite and étale; (3) $G$ is finite and constant; (4) $G$ is smooth and connected reductive; (5) $G$ is abelian; (6) $G$ is finite, and the identity component $G^{\circ }$ of $G$ is linearly reductive; (7) $G$ is finite and linearly reductive. Let $V$ be a small $G$-module of dimension $n<\infty$. We assume that ${\lambda }_{G,G}$ is trivial. Let $H={\mathbb{G}}_m$ be the one-dimensional torus, and let $V$ be of degree one as an $H$-module so that $S=\mathrm{Sym}{V}^{\ast }$ is a $\tilde{G}$-algebra generated by degree one elements, where $\tilde{G}=G\times H$. We set $A=S^G$. Then we have (i) ${\omega }_A\cong {\omega }_S^G$ as $(H,A)$-modules; (ii) $a(A)\le -n$ in general, where $a(A)$ denotes the $a$-invariant. Moreover, the following are equivalent: (1) The action $G\to \mathrm{GL}(V)$ factors through $\mathrm{SL}(V)$; (2) ${\omega }_S\cong S(-n)$ as $(\tilde{G},S)$-modules; (3) ${\omega }_S\cong S$ as $(G,S)$-modules; (4) ${\omega }_A\cong A(-n)$ as $(H,A)$-modules; (5) $A$ is quasi-Gorenstein; (6) $A$ is quasi-Gorenstein and $a(A)=-n$; (7) $a(A)=-n$. This partly generalizes recent results of Liedtke--Yasuda arXiv:2304.14711v2 and Goel--Jeffries--Singh arXiv:2306.14279v1.

13 pages. Added a reference, and corrected some minor errors