Irreducible modules for equivariant map superalgebras and their extensions

Let $\mathrm{\Gamma }$ be a group acting on a scheme $X$ and on a Lie superalgebra $\mathfrak{g}$, both defined over an algebraically closed field of characteristic zero $\mathbb{k}$. The corresponding equivariant map superalgebra $M(\mathfrak{g},X{)}^{\mathrm{\Gamma }}$ is the Lie superalgebra of equivariant regular maps from $X$ to $\mathfrak{g}$. In this paper we complete the classification of finite-dimensional irreducible $M(\mathfrak{g},X{)}^{\mathrm{\Gamma }}$-modules when $\mathfrak{g}$ is a finite-dimensional simple Lie superalgebra, $X$ is of finite type and $\mathrm{\Gamma }$ is a finite abelian group acting freely on the rational points of $X$, by classifying these $M(\mathfrak{g},X{)}^{\mathrm{\Gamma }}$-modules in the case where $\mathfrak{g}$ is a periplectic Lie superalgebra. We also describe extensions between irreducible modules in terms of homomorphisms and extensions between modules for certain finite-dimensional Lie superalgebras.