Identify medium algebras with centres of Kirillov algebras

Determine whether every medium algebra is equal to the centre of its corresponding Kirillov algebra.

Background

Medium algebras are defined as commutative subalgebras of Kirillov algebras generated by the medium operators associated with invariant polynomials. The paper uses their commutativity and their geometric realization via equivariant cohomology, but does not establish the stronger assertion that they exhaust the centres of the Kirillov algebras. Establishing this would clarify the algebraic significance of medium algebras and their relation to representation-theoretic constructions.

References

Hausel encountered those algebras in a different setting -- the study of Higgs bundle moduli -- where the spectra of medium algebras are proposed to describe Lagrangian closures of certain upward flows. Finally, medium algebras can be identified with the equivariant cohomology algebras of affine Schubert varieties, connecting them to geometric representation theory.

— Automorphisms of medium algebras and quasi-anisotropic real affine Grassmannians  (2610.18170 - Elkner, 13 Oct 2026) in Section 1, Introduction (first paragraph)