Symmetric coinvariant algebras and local Weyl modules at a double point

Let Sn be the symmetric group acting on C[x1, . . . , xn]. The classical symmetric coinvariant algebra C[x1, . . . , xn]Sn is the quotient of C[x1, . . . , xn] by the ideal generated by symmetric polynomials vanishing at (0, . . . , 0). According to a classical result, it is isomorphic to C[Sn] as Sn-module. In [Comm. Math. Phys. 251 (2004), no. 3, 427–445; MR2102326 (2005m:17005)], B. L. Fe˘ıgin and S. A. Loktev defined the symmetric coinvariant algebra A n Sn , where A is the coordinate ring of an affine variety M over C. In the paper under review the author deals with A = C[x, y]/(xy), the coordinate ring ofM = {(x, y) 2 C2: xy = 0}. In this case the symmetric coinvariant algebra is Rn = A n/Jn, where Jn is the ideal of A n generated by the elementary symmetric polynomials ei = ei(x1, . . . , xn), fi = fi(x1, . . . , xn), 1 i n. He introduces a generalization of Rn, Rn i,j = A n/In i,j , for 1 i, j n, and gives its Snmodule structure when i+j n+1. This description is then used to show that, as Sn-module, Rn = C[Sn] (n−1)IndSn S2L1,1, where IndSn S2L(1,1) = C[Sn] C[S2] L(1,1) and L(1,1) is the sign representation of S2. This result, combined with a theorem contained in the paper by Fe˘ıgin and Loktev quoted above, gives another description of the slr+1-module structure of the local Weyl module at the double point 0 ofM for slr+1 A.