Alistair Savage

Irreducible finite-dimensional representations of equivariant map algebras

Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The corresponding equivariant map algebra is the Lie algebra M of equivariant regular maps from X to g. We classify the irreducible finite-dimensional representations of these algebras. In particu- lar, 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 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 classifica- tions of irreducible finite-dimensional representations of other types of equivariant map algebras, such as the generalized Onsager algebra.

Local Weyl modules for equivariant map algebras with free abelian group actions

Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. Examples include generalized current algebras and (twisted) multiloop algebras. Local Weyl modules play an important role in the theory of finite-dimensional represen- tations of loop algebras and quantum affine algebras. In the current paper, we extend the definition of local Weyl modules (previously defined only for generalized current algebras and twisted loop algebras) to the setting of equivariant map algebras where g is semisimple, X is affine of finite type, and the group is abelian and acts freely on X. We do so by defin- ing twisting and untwisting functors, which are isomorphisms between certain categories of representations of equivariant map algebras and their untwisted analogues. We also show that other properties of local Weyl modules (e.g. their characterization by homological prop- erties and a tensor product property) extend to the more general setting considered in the current paper.

Formal Hecke algebras and algebraic oriented cohomology theories

In the present paper, we generalize the construction of the nil Hecke ring of Kostant–Kumar to the context of an arbitrary formal group law, in particular, to an arbitrary algebraic oriented cohomology theory of Levine–Morel and Panin–Smirnov (e.g., to Chow groups, Grothendieck’s

Hecke algebras, finite general linear groups, and Heisenberg categorification

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Global Weyl Modules for Equivariant Map Algebras

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EXTENSIONS AND BLOCK DECOMPOSITIONS FOR FINITE-DIMENSIONAL REPRESENTATIONS OF EQUIVARIANT MAP ALGEBRAS

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