A Brylinski filtration for affine Kac-Moody algebras

TL;DR

This paper proves a conjecture relating Lusztig's q-analog of weight multiplicity to a principal Heisenberg filtration in affine Kac-Moody algebras, extending the geometric Satake isomorphism framework.

Contribution

It demonstrates that the conjectured equality holds with the principal Heisenberg filtration, providing a key Lie algebra cohomology vanishing result and exploring cases beyond affine types.

Findings

Equality holds for all affine Kac-Moody algebras with Heisenberg filtration.

Counterexample shows the nilpotent filtration does not always match the q-analog.

Partial results are provided for indefinite Kac-Moody algebras.

Abstract

Braverman and Finkelberg have recently proposed a conjectural analogue of the geometric Satake isomorphism for untwisted affine Kac-Moody groups. As part of their model, they conjecture that (at dominant weights) Lusztig's q-analog of weight multiplicity is equal to the Poincare series of the principal nilpotent filtration of the weight space, as occurs in the finite-dimensional case. We show that the conjectured equality holds for all affine Kac-Moody algebras if the principal nilpotent filtration is replaced by the principal Heisenberg filtration. The main body of the proof is a Lie algebra cohomology vanishing result. We also give an example to show that the Poincare series of the principal nilpotent filtration is not always equal to the q-analog of weight multiplicity. Finally, we give some partial results for indefinite Kac-Moody algebras.