Multi-Dimensional Weyl Modules and Symmetric Functions

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).