, Volume 251, Issue 3, pp 427-445
Date: 14 Oct 2004

Multi-Dimensional Weyl Modules and Symmetric Functions

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Abstract

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 sl r . 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).

Communicated by L. Takhtajan
Acknowledgements This consideration is inspired by discussions with Vyjayanthi Chari at UC Riverside and MSRI. SL thanks V. Chari for the hospitality at UC Riverside. We are also grateful to V. V. Dotsenko, A. N. Kirillov and I. N. Nikokoshev for useful and stimulating discussions.
BF is partially supported by the grants RFBR-02-01-01015, RFBR-01-01-00906 and INTAS-00-00055.
SL is partially supported by the grants RFBR-02-01-01015, RFBR-01-01-00546 and CRDF RM1-2545-MO-03.