Mathematical Research Letters

Volume 25 (2018)

Number 2

Deformed cohomology of flag varieties

Pages: 649 – 657



Oliver Pechenik (Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A.; and Department of Mathematics, University of Michigan, Ann Arbor, Mi., U.S.A.)

Dominic Searles (Department of Mathematics, University of Southern California, Los Angeles, Ca., U.S.A.; and Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand)


This paper introduces a two-parameter deformation of the cohomology of generalized flag varieties. One special case is the Belkale–Kumar deformation (used to study eigencones of Lie groups). Another picks out intersections of Schubert varieties that behave nicely under projections. Our construction yields a new proof that the Belkale–Kumar product is well-defined. This proof is shorter and more elementary than earlier proofs.

OP was supported by an Illinois Distinguished Fellowship, an NSF Graduate Research Fellowship, and NSF MCTP grant DMS 0838434. DS was supported by a Golub Research Assistantship and a University of Illinois Dissertation Completion Fellowship.

This project was inspired by a talk of Sam Evens at the University of Illinois at Urbana-Champaign in April 2014. We are grateful for very helpful conversations with Sam Evens, William Haboush, Shrawan Kumar, Edward Richmond and Alexander Yong. We would also like to thank an anonymous referee for very helpful suggestions.

Received 21 January 2015

Accepted 14 June 2017

Published 5 July 2018