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