Traces, Schubert calculus, and Hochschild cohomology of category $\mathcal{O}$

TL;DR

This paper computes the Hochschild cohomology of the category al by relating it to the trace of its Serre functor and applying geometric methods, providing explicit results including the Euler characteristic in type A.

Contribution

It introduces a novel approach to compute Hochschild cohomology of category al using traces of Serre functors and geometric techniques, linking algebraic and topological invariants.

Findings

Hochschild cohomology of al is expressed as the trace of its Serre functor.

The Hochschild cohomology of al is computed as the compactly supported cohomology of an associated space.

The Euler characteristic of Hochschild cohomology in type A is explicitly obtained.

Abstract

We discuss how the Hochschild cohomology of a dg category can be computed as the trace of its Serre functor. Applying this approach to the principal block of the Bernstein--Gelfand--Gelfand category $\mathcal{O}$, we obtain its Hochschild cohomology as the compactly supported cohomology of an associated space. Equivalently, writing $\mathcal{O}$ as modules over the endomorphism algebra $A$ of a minimal projective generator, this is the Hochschild cohomology of $A$. In particular our computation gives the Euler characteristic of the Hochschild cohomology of $\mathcal{O}$ in type A.