Equivariant Cohomology of Rationally Smooth Group Embeddings

Abstract

We describe the equivariant cohomology ring of rationally smooth projective embeddings of reductive groups. These embeddings are the projectivizations of reductive monoids. Our main result describes their equivariant cohomology in terms of roots, idempotents, and underlying monoid data. Also, we characterize those embeddings whose equivariant cohomology ring is obtained via restriction to the associated toric variety. Such characterization is given in terms of the closed orbits.

Localization in equivariant operational K-theory and the Chang–Skjelbred property

“…But then a simple counting argument would yield |R 1 | < |E 1 ||W |. This is impossible, by our assumptions and [G2,Lemma 4.14]. Hence, N = w∈W wY.…”

“…In [G2, Section 4], we identify explicitly the T × T -characters associated to these curves. With such data at our disposal, Theorem 5.4 yields an immediate translation of [G2,Theorem 4.10] into the language of equivariant operational K-theory. Furthermore, as Theorem 5.4 does not require any conditions on the singular locus, the result (Theorem 6.2) applies to all projective group embeddings.…”

“…. Finally, we apply Theorem 5.4, taking into account that: (i) the curves of type 1 and 2 in Theorem 6.1 are contained in e∈Λ 1 G[e]G and these curves describe K T ×T (G[e]G) via Example 5.5, (ii) the characters associated to the curves of type (3) give assertions (a) and (b), as in [G2,Theorem 4.10].…”

“…Proof. The argument here is an adaptation of [G2,Proof of Theorem 4.12]. First, consider the commutative diagram…”

Algebraic rational cells and equivariant intersection theory

“…The number of closed irreducible T × T -invariant curves in M is finite (all of them passing through 0), and it equals |R 1 |. Indeed, each closed T × T -curve of M can be written as T xT , where x ∈ R 1 , for they correspond to the T ×T -fixed points of P ǫ (M ), see [G2,Theorem 3.1]. It follows that dim M ≤ |R 1 |.…”