Commutative avatars of representations of semisimple Lie groups

Abstract

Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra, called big algebra, attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are commutative finite flat algebras over the cohomology of the classifying space of the group. They are isomorphic with the equivariant intersection cohomology of affine Schubert varieties, endowing them with a new ring structure. Study of the finer aspects of the structure of the big algebras will also furnish the stalks of the intersection cohomology with ring structure, thus ringifying Lusztig's q-weight multiplicity polynomials i.e. affine Kazhdan-Lusztig polynomials.

arXiv commentoriginal title "Big algebras" changed to match published version; 4.2 is replaced for visualisations and clarifications; 15 pages, 9 figures

References in corpus (6)

• Gaudin Model, Bethe Ansatz and Critical Level
• Gaudin models with irregular singularities
• Opers with irregular singularity and spectra of the shift of argument subalgebra
• Very stable Higgs bundles, equivariant multiplicity and mirror symmetry
• Fomenko-Mischenko Theory, Hessenberg Varieties, and Polarizations
• The noncommutative Poisson bracket and the deformation of the family algebras

Cited by in corpus (2)

• Dynkin automorphism actions on Gaudin algebras
• On involutions of minuscule Kirillov algebras induced by real structures