# Moscow Mathematical Journal

Volume 15, Issue 3, July–September 2015 pp. 407–423.

Sheaves on Nilpotent Cones, Fourier Transform, and a Geometric Ringel Duality

**Authors**:
Pramod N. Achar (1) and Carl Mautner (2)

**Author institution:**(1) Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A.

(2) Department of Mathematics, University of California, Riverside, 900 University Ave., Riverside, CA 92521, U.S.A.

**Summary: **

Given the nilpotent cone of a complex reductive Lie algebra, we consider its equivariant constructible derived category of sheaves
with coefficients in an arbitrary field. This category and its subcategory 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 GL_{n}, we show that this autoequivalence
can be regarded as a geometric version of Ringel duality for the Schur
algebra.

2010 Math. Subj. Class. Primary 17B08, 14F05; Secondary 20G43.

**Keywords:**Nilpotent cone, Fourier transform, Ringel duality, Schur algebra, Springer theory.

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