Macdonald polynomials, Laumon spaces and perverse coherent sheaves

TL;DR

This paper proposes a geometric interpretation of Macdonald polynomials through perverse coherent sheaves on arc schemes, proves it for SL(N), and relates it to Laumon resolutions and K-theoretic results.

Contribution

It introduces a new geometric conjecture linking Macdonald polynomials to perverse coherent sheaves and proves it for SL(N) using Laumon resolutions.

Findings

Conjecture formulated for geometric interpretation of Macdonald polynomials.

Proved conjecture for G=SL(N) using Laumon resolution.

Provided a K-theoretic version of a previous main result.

Abstract

Let $G$ be an almost simple simply connected complex Lie group, and let $G/U_-$ be its base affine space. In this paper we formulate a conjecture, which provides a new geometric interpretation of the Macdonald polynomials associated to $G$ via perverse coherent sheaves on the scheme of formal arcs in the affinization of $G/U_-$. We prove our conjecture for $G=SL(N)$ using the so called Laumon resolution of the space of quasi-maps (using this resolution one can reformulate the statement so that only "usual" (not perverse) coherent sheaves are used). In the course of the proof we also give a $K$-theoretic version of the main result of arXiv/0811.4454.