Global Weyl Modules for Equivariant Map Algebras

Equivariant map algebras are Lie algebras of algebraic maps from a scheme (or algebraic variety) to a target finite-dimensional Lie algebra (in the case of the current paper, we assume the latter is a simple Lie algebra) that are equivariant with respect to the action of a finite group. In the first part of this paper, we define global Weyl modules for equivariant map algebras satisfying a mild assumption. We then identify a commutative algebra |$\textbf {A}^{\lambda} _\Gamma $| that acts naturally on the global Weyl modules, which leads to a Weyl functor from the category of |$\textbf {A}^{\lambda} _{\Gamma} $|-modules to the category of modules for the equivariant map algebra in question. These definitions extend the ones previously given for generalized current algebras (i.e., untwisted map algebras) and twisted loop algebras. In the second part of the paper, we restrict our attention to equivariant map algebras where the group involved is abelian, acts on the target Lie algebra by diagram automorphisms, and freely on (the set of rational points of) the scheme. Under these additional assumptions, we prove that |$\textbf {A}^{\lambda}_{\Gamma} $| is finitely generated and the global Weyl module is a finitely generated |$\textbf {A}^{\lambda}_{\Gamma} $|-module. We also define local Weyl modules via the Weyl functor and prove that these coincide with the local Weyl modules defined directly in [18]. Finally, we show that |$\textbf {A}^{\lambda}_{\Gamma} $| is the algebra of coinvariants of the analogous algebra in the untwisted case.