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


Koppensteiner, C

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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.

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Journal Article