Irreducible finite-dimensional representations of equivariant map algebras

Suppose a finite group acts on a scheme $X$ and a finite-dimensional Lie algebra $\mathfrak{g}$. The corresponding equivariant map algebra is the Lie algebra $\mathfrak{M}$ of equivariant regular maps from $X$ to $\mathfrak{g}$. We classify the irreducible finite-dimensional representations of these algebras. In particular, we show that all such representations are tensor products of evaluation representations and one-dimensional representations, and we establish conditions ensuring that they are all evaluation representations. For example, this is always the case if $\mathfrak{M}$ is perfect.

Our results can be applied to multiloop algebras, current algebras, the Onsager algebra, and the tetrahedron algebra. Doing so, we easily recover the known classifications of irreducible finite-dimensional representations of these algebras. Moreover, we obtain previously unknown classifications of irreducible finite-dimensional representations of other types of equivariant map algebras, such as the generalized Onsager algebra.