We compute the Hochschild–Kostant–Rosenberg decomposition of the Hochschild cohomology of generalised Grassmannians, i.e., partial flag varieties associated to maximal parabolic subgroups in a simple algebraic group, in terms of representation-theoretic data.We explain how the decomposition is concentrated in global sections for the (co)minuscule and (co)adjoint generalised Grassmannians, and conjecture that for (almost) all other cases the same vanishing of the higher cohomology does not hold. Our methods give an explicit partial description of the Gerstenhaber algebra structure for the Hochschild cohomology of cominuscule and adjoint generalised Grassmannians. We also consider the case of adjoint partial flag varieties in type A, which are associated to certain submaximal parabolic subgroups.
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Pieter Belmans, Maxim Smirnov, Hochschild cohomology of generalised Grassmannians. Doc. Math. 28 (2023), no. 1, pp. 11–53DOI 10.4171/DM/912