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Strong Macdonald Theory and the Brylinski Filtration for Affine Lie Algebras


The strong Macdonald theorems state that, for L reductive and s an odd variable, the cohomology algebras H*(L[z]/zN) and H*(L[z, s]) are freely generated, and describe the co-homological, s-, and z-degrees of the generators. The resulting identity for the z-weighted Euler characteristic is equivalent to Macdonald's constant term identity for a finite root system. The proof of the strong Macdonald theorems, due to Fishel, Grojnowski, and Teleman, uses a Laplacian calculation for the (continuous) cohomology of L[[z]] with coefficients in the symmetric algebra of the (continuous) dual of L[[z]].

Our main result is a generalization of this Laplacian calculation to the setting of a general parahoric p of a (possibly twisted) loop algebra g. As part of this result, we give a detailed exposition of one of the key ingredients in Fishel, Grojnowski, and Teleman's proof, a version of Nakano's identity for infinite-dimensional Lie algebras.

We apply this Laplacian result to prove new strong Macdonald theorems for H*(p/zNp) and H*(p[s]), where p is a standard parahoric in a twisted loop algebra. We show that H(p/zNp) contains a parabolic subalgebra of the coinvariant algebra of the fixed-point subgroup of the Weyl group of L, and thus is no longer free. We also prove a strong Macdonald theorem for H*(b; S* n* ) and H(b /zN n) when b and n are Iwahori and nilpotent subalgebras respectively of a twisted loop algebra. For each strong Macdonald theorem proved, taking z-weighted Euler characteristics gives an identity equivalent to Macdonald's constant term identity for the corresponding affine root system. As part of the proof, we study the regular adjoint orbits for the adjoint action of the twisted arc group associated to L, proving an analogue of the Kostant slice theorem.

Our Laplacian calculation can also be adapted to the case when g is a symmetrizable Kac-Moody algebra. In this case, the Laplacian calculation leads to a generalization of the Brylinski identity or affine Kac-Moody algebras. In the semisimple case, the Brylinski identity states that, at dominant weights, the q-analog of weight multiplicity is equal to the Poincare series of the principal nilpotent filtration of the weight space. This filtration is known as the Brylinski filtration. We show that this identity holds in the affine case, as long as the principal nilpotent filtration is replaced by the principal Heisenberg. 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, and give some partial results for indefinite Kac-Moody algebras.

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