Cyclic homology and the Macdonald conjectures

Summary LetA+(k) denote the ring [t]/t^k+1 and letG be a reductive complex Lie algebra with exponentsm₁, ...,m n. This paper concerns the Lie algebra cohomology ofGA₊(k) considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we callweight, is inherited from the obvious grading ofGA₊(k)). We conjecture that this Lie algebra cohomology is an exterior algebra withk+1 generators of homological degree 2m_s+1 fors=1,2, ...,n. Of thesek+1 generators of degree 2m_s+1, one has weight 0 and the others have weights (k+1)m_s+t fort=1,2, ...,k.It is shown that this conjecture about the Lie algebra cohomology of A₊(k) implies the Macdonald root system conjectures. Next we consider the case thatG is a classical Lie algebra with root systemA_n,B_n,C_n, orD_n. It is shown that our conjecture holds in the limit onn asn approaches infinity which amounts to the computation of the cyclic and dihedral cohomologies ofA+(k). Lastly we discuss the relevance of this limiting case to the case of finiten in this situation.Partially supported by NSF grant number MCS-8401718 and a Bantrell Fellowship