Year: 2025
Type: preprint
Abstract: Let $X$ and $\mathfrak{a}$ be an affine scheme and (respectively) a finite-dimensional associative algebra over an algebraically-closed field $\Bbbk$, both equipped with actions by a linearly-reductive linear algebraic group $G$. We describe the simple finite-dimensional modules over the algebra of $G$-equivariant maps $X\to \mathfrak{a}$ in terms of the representation theory of the fixed-point subalgebras $\mathfrak{a}^x:=\mathfrak{a}^{G_x}\le \mathfrak{a}$, $G_x$ being the respective isotropy groups of closed-orbit $k$-points $x\in X$. This answers a question of E. Neher and A. Savage, extending an analogous result for (also linearly-reductive) finite-group actions. Moreover, the full category of finite-dimensional modules admits a direct-sum decomposition indexed by closed orbits.
Source: arXiv (Cornell University)
Author: Chirvasitu, Alexandru
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416692446
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