Topological realization of rings of quasi-invariants of finite reflection groups

Yuri Berest (Cornell)

27-Oct-2021, 20:00-21:00 (2 years ago)

Abstract: Quasi-invariants are natural geometric generalizations of classical invariant polynomials of finite reflection groups. They first appeared in mathematical physics in the early 1990s, and since then have found applications in a number of other areas (most notably, representation theory, algebraic geometry and combinatorics).

In this talk, I will explain how the algebras of quasi-invariants can be realized topologically: as (equivariant) cohomology rings of certain spaces naturally attached to compact connected Lie groups. Our main result can be viewed as a generalization of a well-known theorem of A. Borel that realizes the algebra of invariant polynomials of a Weyl group W as the cohomology ring of the classifying space BG of the corresponding Lie group G. Replacing equivariant cohomology with equivariant K-theory gives a multiplicative (exponential) analogues of quasi-invariants of Weyl groups. But perhaps more interesting is the fact that one can also realize topologically the quasi-invariants of some non-Coxeter groups: our `spaces of quasi-invariants' can be constructed in a purely homotopy-theoretic way, and this construction extends naturally to (p-adic) pseudoreflection groups. In this last case, the compact Lie groups are replaced by p-compact groups (a.k.a. homotopy Lie groups). The talk is based on joint work with A. C. Ramadoss.

representation theory

Audience: researchers in the topic


MIT Lie groups seminar

Series comments: Description: Research seminar on Lie groups

This seminar will take place entirely online: Zoom Meeting Link. You should be able to watch live video at this link. Your microphone will be muted, but you are welcome to unmute it (microphone icon on the lower left of the Zoom window, perhaps visible only when you put your mouse near there) to ask a question.

Organizers: André Lee Dixon*, Ju-Lee Kim, Roman Bezrukavnikov*
*contact for this listing

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