Fourier-Deligne transform and representations of the symmetric group

Abstract

We calculate the Fourier-Deligne transform of the IC extension to ${\C}^{n+1}$ of the local system ${\mathcal{L}}_{\Lambda }$ on the cone over $\Conf_n(¶^1)$ associated to a representation $\Lambda$ of $S_n$, where the length $n-k$ of the first row of the Young diagram of $\Lambda$ is at least $\frac{|\Lambda |-1}{2}$. The answer is the IC extension to the dual vector space ${\C}^{n+1}$ of the local system ${\mathcal{R}}_{\lambda }$ on the cone over the $k$-th secant variety of the rational normal curve in ${\P }^n$, where ${\mathcal{R}}_{\lambda }$ 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.