Irreducible components in Hochschild cohomology of flag varieties

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 $H{H}^{\bullet }(G/B)$ 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 $H{H}^{\bullet }(G/B)$ when $G=S{L}_n(\mathbb{C})$ 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 $V(\rho )\otimes V(\rho )$.

12 pages; v3 minor expositional edits, final version; v2 minor corrections, exposition around Question 4.12 rewritten with negative examples to previous version