On the homology of sln(tC[t]) and a theorem of Stembridge

In this paper we apply the Garland–Lepowsky Theorem to compute the homology of the Lie algebra N=sln(tC[t]). By suitably taking Euler characteristics we derive a theorem of Stembridge which gives a combinatorial decomposition of a virtual character of SLn(C). In his theorem, Stembridge indexes the Schur functions that appear in his decomposition by an integer vector of length n and a permutation in Sn. An additional feature of our approach is that we interpret the integer vector and permutation in terms of the affine Weyl group of the Kac–Moody Lie algebra of type A. A second result, given in the Garland–Lepowsky paper (extending an earlier result of Kostant), is an explicit formula for the eigenvalues of the Laplacian in the Koszul complex for computing the homology of N. We give an alternative quite different in nature description of those eigenvalues. The Garland–Lepowsky formula is algebraic, stated in terms of differences of squared norms. Our description is combinatorial, stated in terms of structural features of certain graphs. In order to make this combinatorial interpretation seemingly more natural, we link it directly with the paper of Kostant. We work in this part with the Lie algebra L=T N where T is the span of x 1 with x taken from the strictly upper triangular matrices in sln(C).