Sheaves on nilpotent cones, fourier transform, and a geometric ringel duality
Given the nilpotent cone of a complex reductive Lie alge- bra, we consider its equivariant constructible derived category of sheaves with coefficients in an arbitrary field. This category and its subcate- gory of perverse sheaves play an important role in Springer theory and the theory of character sheaves. We show that the composition of the Fourier–Sato transform on the Lie algebra followed by restriction to the nilpotent cone gives an autoequivalence of the derived category of the nilpotent cone. In the case of GLn, we show that this autoequivalence can be regarded as a geometric version of Ringel duality for the Schur algebra.
Publication Source (Journal or Book title)
Moscow Mathematical Journal
Achar, P., & Mautner, C. (2015). Sheaves on nilpotent cones, fourier transform, and a geometric ringel duality. Moscow Mathematical Journal, 15 (3), 407-423. https://doi.org/10.17323/1609-4514-2015-15-3-407-423