Paper detail

Fourier-Deligne transform and representations of the symmetric group

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