2002/12/31 by B. Feigin, S. Loktev · 1 citation
Mathematics · #math.QA #math.RT
paper · pdf · doi:10.1007/s00220-004-1166-8
published as Comm. Math. Phys. 251 (2004), no. 3, 427-445 · LaTeX, 13 pages; more detail added. To appear at Comm. Math. Phys
arxiv created 2004/03/05 · arxiv updated 2015/06/26
The Weyl modules in the sense of V.Chari and A.Pressley [CP] over the current Lie algebra on an affine variety are studied. We show that local Weyl modules are finite-dimensional and generalize the tensor product decomposition theorem from [CP]. More explicit results are stated for currents on a non-singular affine variety of dimension d with coefficients in the Lie algebra slr. The Weyl modules with highest weights proportional to the vector representation one are related to the multi-dimensional analogs of harmonic functions. The dimensions of such local Weyl modules are calculated in the following cases. For d=1 we show that the dimensions are equal to powers of r. For d=2 we show that the dimensions are given by products of the higher Catalan numbers (the usual Catalan numbers for r=2). We finally formulate a conjecture for an arbitrary d and r=2.