Extensions and block decompositions for finite-dimensional representations of equivariant map algebras

TL;DR

This paper studies the structure of finite-dimensional representations of equivariant map algebras, describing extensions and blocks, and provides new proofs and formulas for special cases like loop and multiloop algebras.

Contribution

It extends the understanding of extensions and block decompositions for equivariant map algebras, including new explicit formulas and a general spectral character framework.

Findings

Explicit description of extensions between irreducible representations.

Characterization of blocks of the representation category via spectral characters.

New formulas for extensions in (twisted) loop and multiloop algebras.

Abstract

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.