Irreducible components in Hochschild cohomology of flag varieties

Let G be a simple, simply-connected complex algebraic group with Lie algebra g, and $G/B$ the associated complete flag variety. The Hochschild cohomology HH^•(G/B) is a geometric invariant of the flag variety related to its generalized deformation theory and has the structure of a g-module. We study this invariant via representation-theoretic methods; in particular, we give a complete list of irreducible subrepresentations in HH^•(G/B) when G=SLₙ(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(ρ) ⊗ V(ρ).