Irreducible modules for equivariant map superalgebras and their extensions

Tiago Macedo, Lucas Calixto

Published 2016-04-06Version 1

Let $\Gamma$ be a group acting on a scheme $X$ and on a Lie superalgebra $\mathfrak{g}$. The corresponding equivariant map superalgebra $M(\mathfrak{g}, X)^\Gamma$ is the Lie superalgebra of equivariant regular maps from $X$ to $\mathfrak{g}$. In this paper we complete the classification of finite-dimensional irreducible $M(\mathfrak{g}, X)^\Gamma$-modules when $\mathfrak{g}$ is a finite-dimensional simple Lie superalgebra, $X$ is of finite type, and $\Gamma$ is a finite abelian group acting freely on the rational points of $X$. We also describe extensions between these irreducible modules in terms of extensions between modules for certain finite-dimensional Lie superalgebras. As an application, when $\Gamma$ is trivial and $\mathfrak{g}$ is of type $B(0,n)$, we describe the block decomposition of the category of finite-dimensional $M(\mathfrak{g}, X)^\Gamma$-modules in terms of spectral characters for $\mathfrak{g}$.