Cyclic homology and the Macdonald conjectures
Let \(\mathfrak g\) be a complex semisimple Lie algebra of rank \(n\) with exponents \(m_ 1,\ldots,m_ n\) and for a positive integer \(k\) denote by \(A_+(k)\) the truncated polynomial ring \(\mathbb C[t]/(t^{k+1})\). In Section 1 the following Conjecture is formulated. The Lie algebra cohomology of \(\mathfrak g\otimes A_+(k)\) is an exterior algebra with \(n(k+1)\) generators. For each \(s\), there are \(k+1\) generators of degree \(2m_ s+1\). The weight of these generators -- what is inherited from the grading of \(\mathfrak g\otimes A_+(k)\) -- is 0, \((k+1)m_ s+t\) with \(t=1,2,\ldots,k\). It is shown that this conjecture implies the Macdonald root system conjectures. Section 2 is a summary on cyclic and dihedral homology of a graded \(\mathbb C\)-algebra. In Section 3 the cyclic and dihedral homologies of \(A_+(k)\) are computed. In Section 4 the case of classical Lie algebras is studied with \(\mathfrak g_ n=\mathfrak{sl}_ n(\mathbb C)\), \(\mathfrak{o}_ n(\mathbb C)\) and \(\mathfrak{sp}_{2n}(\mathbb C)\). With the help of computations from Section 3 it is shown that the stated Conjecture holds in the limit on \(n\) as \(n\) approaches infinity. Finally, the relevance of this limiting case to the case of finite \(n\) is discussed.
scientific article; zbMATH DE number 3975166
scientific article