RTMP Seminar: Calculating Duistermaat-Heckman measures of a perfect basis

Seminar | February 4 | 4-5 p.m. | 891 Evans Hall

 Anne Dranowski, University of Toronto

 Department of Mathematics

By work of Lusztig, generic modules for the preprojective algebra give bases for representations of ADE groups. By work of Mirkovic and Vilonen, MV cycles give bases for these representations as well. While both bases were known to be crystal bases practically since their inception, only recently did we see polytope descriptions of their crystal structures. The crystal structure on MV cycles was modelled on their moment polytopes by Kamnitzer. The crystal structure on generic modules was modelled on their Harder-Narasimhan polytopes by Baumann, Kamnitzer and Tingley. Miraculously, these descriptions turned out to coincide (HN polytopes agreeing with MV polytopes). Polytopes being inherently geometric, the combinatorial coincidence called for an upgrade, and in recent work by Baumann, Kamnitzer and Knutson, the authors asked whether equal polytopes have equal equivariant volumes. By identifying open subsets in MV cycles with generalized orbital varieties, we will show that the answer to this question in type A is no.