Irreducible components in Hochschild cohomology of flag varieties

TL;DR

We address the problem of classifying irreducible $\mathfrak{g}$-subrepresentations inside the Hochschild cohomology $HH^\bullet(G/B)$ of the complete flag variety. The approach builds a bridge to Kostant's conjecture on the tensor square $V(\rho)\otimes V(\rho)$ via an embedding into $HH^\bullet(G/B)$ and an HKR reinterpretation of Hochschild cohomology as the cohomology of polyvector fields, with Demazure operator tools governing Euler--Poincaré characteristics. A compact character formula for $V(\rho)\otimes V(\rho)$ is derived, and, under Kostant's conjecture, irreducible components of $HH^\bullet(G/B)$ are predicted to correspond exactly to weights $\lambda\le 2\rho$, each with multiplicity at least $m_\lambda$ (the tensor-square multiplicity). These results are established for $G=SL_n$ and exceptional types and provide a concrete conditional framework for all types, linking geometric invariants of $G/B$ to deep representation-theoretic structures and guiding computations of Hochschild cohomology in broader settings.

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 $V(ρ) \otimes V(ρ)$.

Theorems & Definitions (32)

• Conjecture 1.1
• Proposition 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Theorem 2.4
• Corollary 2.5
• Lemma 2.6
• proof