Irreducible components in Hochschild cohomology of flag varieties

Sam Jeralds

Abstract

Let \(G\) be a simple, simply-connected complex algebraic group with Lie algebra \(\mathfrak{g}\), and \(G/B\) the associated complete flag variety. The Hochschild cohomology \(HH^\bullet(G/B)\) is a geometric invariant of the flag variety related to its generalized deformation theory and has the structure of a \(\mathfrak{g}\)-module. We study this invariant via representation-theoretic methods; in particular, we give a complete list of irreducible subrepresentations in \(HH^\bullet(G/B)\) when \(G=SL_n(\mathbb{C})\) or is of exceptional type (and conjecturally for all types) along with nontrivial lower bounds on their multiplicities. These results follow from a conjecture due to Kostant on the structure of the tensor product representation \(V(\rho) \otimes V(\rho)\).

Keywords: Hochschild cohomology, flag varieties, polyvector fields, Kostant conjecture.

AMS Subject Classification: Primary 14M15; secondary 17B10, 14F43.