Cyclic homology and the Macdonald conjectures

Let A+(k) denote the ring l~[ t] / t k+t and let fr be a reductive complex Lie algebra with exponents m 1 . . . . . m,. This paper concerns the Lie algebra cohomology of ~| considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we call weight, is inherited from the obvious grading of f f | § (k)). We conjecture that this Lie algebra cohomology is an exterior algebra with k + 1 generators of homotogical degree 2ms + 1 for s = 1,2 . . . . . n. Of these k + 1 generators of degree 2ms + 1, one has weight 0 and the others have weights (k + 1)m~+ t for t = 1,2 . . . . . k. It is shown that this conjecture about the Lie algebra cohomology of f#| implies the Macdonald root system conjectures. Next we consider the case that f# is a classical Lie algebra with root system A,, Bn, C,, or Dn. It is shown that our conjecture holds in the limit on n as n approaches infinity which amounts to the computation of the cyclic and dihedral cohomologies of A § (k). Lastly we discuss the relevance of this limiting case to the case of finite n in this situation.