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

TL;DR

This paper investigates the symmetry properties of finite group schemes, proves the triviality of the Knop character in various cases, and explores the implications for the $a$-invariant and Gorenstein properties of invariant rings.

Contribution

It establishes conditions under which the Knop character is trivial for various classes of finite and reductive group schemes, and relates these to the $a$-invariant and Gorenstein properties of invariant rings.

Findings

Knop character is trivial for finite, étale, constant, and reductive group schemes.

The $a$-invariant of the invariant ring satisfies $a(A) \,\leq\; -n$.

Equivalence of conditions related to Gorenstein and quasi-Gorenstein properties of the invariant ring.

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 $\lambda_{G,G}=H^0(\mathop{\mathrm{Spec}} k,e^*(\omega_{G_{\mathrm{ad}}}))$ the Knop character of $G$, where $e:\mathop{\mathrm{Spec}} k\rightarrow G_{\mathrm{ad}}$ 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 $k[G]^*$ is a symmetric algebra; (2) $G$ is finite and \'etale; (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 $H=\Bbb G_m$ be the one-dimensional torus, and let $V$ be of degree one as an $H$-module so that $S=\mathop{\mathrm{Sym}} V^*$ 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) $a(A)\leq -n$ in general, where $a(A)$ denotes the $a$-invariant. Moreover, the following are equivalent: (1) The action $G\rightarrow\mathrm{GL}(V)$ 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.