Cristian Lenart (Albany, NY)
Abstract
The first connection between (specialized) Macdonald polynomials and the quantum K-theory of flag manifolds was found by Braverman-Finkelberg, via their q-Whittaker functions. Kato-Naito-Sagaki and their collaborators discovered another connection, via the level-zero extremal weight modules over quantum affine algebras. In both cases, the semi-infinite flag manifolds were also involved. I survey these connections, and present joint work with S. Naito and D. Sagaki on the quantum K-theory side. Among them is a combinatorial multiplication formula (of Chevalley-type) in the equivariant quantum K-theory of flag manifolds. This is based on the so-called quantum alcove model, which is also relevant to the representation theory mentioned above. The talk will be largely self-contained.