Generalized $F$-signatures of the rings of invariants of finite group schemes

Let $k$ be a perfect field of prime characteristic $p$, $G$ a finite group scheme over $k$, and $V$ a finite-dimensional $G$-module. Let $S=V$ be the symmetric algebra with the standard grading. Let $M$ be a $\mathbb{Q}$-graded $S$-finite $S$-free $(G,S)$-module, and $L$ be its $S$-reflexive graded $(G,S)$-submodule. Assume that the action of $G$ on $V$ is small in the sense that there exists some $G$-stable Zariski closed subset $F$ of $V$ of codimension two or more such that the action of $G$ on $V\setminus F$ is free. Generalizing the result of P. Symonds and the first author, we describe the Frobenius limit $(L^G)$ of the $S^G$-module $L^G$. In particular, we determine the generalized $F$-signature $s(M,S^G)$ for each indecomposable gradable reflexive $S^G$-module $M$. In particular, we prove the fact that the $F$-signature $s(S^G)=s(S^G,S^G)$ equals $1\mathrm{/}\dim k[G]$ if $G$ is linearly reductive (already proved by Watanabe–Yoshida, Carvajal-Rojas–Schwede–Tucker, and Carvajal-Rojas) and $0$ otherwise (some important cases has already been proved by Broer, Yasuda, Liedtke–Martin–Matsumoto).