  Suppose a finite group acts on a scheme $X$ and a finite-dimensional Lie
algebra $\mathfrak{g}$. The associated equivariant map algebra is the Lie
algebra of equivariant regular maps from $X$ to $\mathfrak{g}$. The irreducible
finite-dimensional representations of these algebras were classified in
previous work with P. Senesi, where it was shown that they are all tensor
products of evaluation representations and one-dimensional representations. In
the current paper, we describe the extensions between irreducible
finite-dimensional representations of an equivariant map algebra in the case
that $X$ is an affine scheme of finite type and $\mathfrak{g}$ is reductive.
This allows us to also describe explicitly the blocks of the category of
finite-dimensional representations in terms of spectral characters, whose
definition we extend to this general setting. Applying our results to the case
of generalized current algebras (the case where the group acting is trivial),
we recover known results but with very different proofs. For (twisted) loop
algebras, we recover known results on block decompositions (again with very
different proofs) and new explicit formulas for extensions. Finally,
specializing our results to the case of (twisted) multiloop algebras and
generalized Onsager algebras yields previously unknown results on both
extensions and block decompositions.

Representation Theory;Rings and Algebras