Quasi-invariants are natural geometric generalizations of 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 is 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 multiplicative (exponential) analogues of quasi-invariants (and, in fact, quasi-invariants can be defined for an arbitrary (complex oriented) cohomology theory). But perhaps most interesting is the fact that one can also realize topologically the quasi-invariants of some non-Coxeter (p-adic pseudo-reflection) groups, in which case the compact Lie groups are replaced by p-compact groups --- remarkable homotopy-theoretic objects a.k.a. homotopy Lie groups.
(The talk is based on joint work with A. C. Ramadoss.)