Fourier transform, parabolic induction, and nilpotent orbits
The author proves that in the symmetric space setting the functors of Fourier transforms in the sense of Deligne and parabolic induction in the sense of Lusztig commute. Two consequences are deduced. The first is a new proof of Lusztig's description of the intersection cohomology of nilpotent orbit closures for \(\text{GL}_n({\mathbb C})\), see [\textit{G. Lusztig}, Adv. Math. 42, 169--178 (1981; Zbl 0473.20029)]. The second is an analogous description for the symmetric space \(\text{GL}_{2n}({\mathbb C})/ \text{Sp}_{2n}({\mathbb C})\). Namely, the nilpotent \(\text{Sp}_{2n}({\mathbb C})\)-orbits in the invariant complement of \(\mathfrak{sp}_{2n}({\mathbb C})\) in \(\mathfrak{gl}_{2n}({\mathbb C})\) are considered. Like the nilpotent \(\text{GL}_n({\mathbb C})\)-orbits in \(\mathfrak{gl}_{n}({\mathbb C})\), these are parametrized by partitions of \(n\). It is shown that the closures of these orbits have the same local intersection cohomology as the ones for \(\text{GL}_n({\mathbb C})\) but with all degrees doubled.
scientific article; zbMATH DE number 1760661