FOURIER-DELIGNE TRANSFORM AND REPRESENTATIONS OF THE SYMMETRIC GROUP

We calculate the Fourier-Deligne transform of the IC extension to Cn+1 of the local system L-Lambda on the cone over Conf(n)(P-1) associated with a representation Lambda of the symmetric group S-n, where the length n-k of the first row of the Young diagram of Lambda is at least vertical bar Lambda vertical bar-1/2. The answer is the IC extension to the dual vector space Cn+1 of the local system R-lambda on the cone over the kth secant variety of the rational normal curve in P-n, where R-lambda corresponds to the representation lambda of S-k, the Young diagram of which is obtained from the Young diagram of Lambda by deleting its first row. We also prove an analogous statement for S-n-local systems on fibers of the Abel-Jacobi map. We use our result on the Fourier-Deligne transform to rederive a part of a result of Michel Brion on Kronecker coefficients.