Irreducible modules for equivariant map superalgebras and their extensions

Abstract

Let Γ be a group acting on a scheme X and on a Lie superalgebra g, both defined over an algebraically closed field of characteristic zero k. The corresponding equivariant map superalgebra M(g,X)Γ is the Lie superalgebra of Γ-equivariant regular maps from X to g. In this paper we complete the classification of finite-dimensional irreducible M(g,X)Γ-modules when g is a finite-dimensional simple Lie superalgebra, X is of finite type and Γ is a finite abelian group acting freely on the rational points of X, by classifying these M(g,X)Γ-modules in the case where 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. As an application, one obtains the block decomposition of the category of finite-dimensional M(g,X)Γ-modules in terms of blocks and spectral characters of finite-dimensional Lie superalgebras.