Fourier-Deligne transform and representations of the symmetric group

We calculate the Fourier-Deligne transform of the IC extension to $\mathbb{C}^{n+1}$ of the local system ${\mathcal L}_{\Lambda}$ on the cone over $\mathrm{Conf}_n(\mathbb{P}^1)$ 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 $\frac{|\Lambda|-1}{2}$. The answer is the IC extension to the dual vector space $\mathbb{C}^{n+1}$ of the local system ${\mathcal R}_{\lambda}$ on the cone over the $k$th secant variety of the rational normal curve in $\mathbb{P}^n$, 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.

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Published 13 June 2014