# MIT Lie Groups Seminar

Wednesday, October 27, 2021 at 4:00pm to 5:00pm

Virtual Event**Speaker: Yuri Berest - Cornell University**

**Title: **Topological realization of rings of quasi-invariants of finite reflection groups

**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.

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