The elliptic Weyl character formula

Abstract

We calculate equivariant elliptic cohomology of the partial flag variety G/H, where H \subseteq G are compact connected Lie groups of equal rank. We identify the RO(G)-graded coefficients Ell_G^* as powers of Looijenga's line bundle and prove that transfer along the map {\pi}: G/H -\rightarrow pt is calculated by the Weyl-Kac character formula. Treating ordinary cohomology, K-theory and elliptic cohomology in parallel, this paper organizes the theoretical framework for the elliptic Schubert calculus of [N.Ganter and A.Ram, Elliptic Schubert calculus. In preparation].Comment: 44 page

Elliptic and K-theoretic stable envelopes and Newton polytopes

“…where ∼ is induced by gluing ι I E A (pt) = E n and ι s a,b (I) E A (pt) = E n along the diagonal ∆ I,s a,b (I) = {x ∈ E n : x a = x b }, for any I, a, b, cf. [GKV], [GKM], [Ga,Example 4.4]. The isomorphism between the two descriptions of E A (F λ ) is induced by the map…”

Elliptic stable envelope for Hilbert scheme of points in the plane

“…. , r -the elliptic Chern roots of V. For the definition of c see Section 1.8 in [8] or Section 5 in [5]. For Nakajima varieties we have a map given by the Chern classes of the tautological bundles c :…”

“…Let λ = (λ 1 ≥ · · · ≥ λ k ) be a partition of n = λ 1 + · · · + λ k and l(λ) = k. It will be convenient to use the standard representations for a partition as a Young diagram in its French and Russian (a diagram rotated by 45 • ) realization as in the Fig.1. 5 If a box = (i, j) they are defined as…”

“…(1 − x −1 i ) = e K (V), see Sections 7.1-7.2 of [5] for discussion of relations between elliptic, K-theoretic and cohomological versions Euler class. In this section we use this limit to obtain the explicit formulas for stable envelops in the equivariant K-theory of H. Recall that in the K-theory the stable envelope is a locally constant functions of a slope parameter [26,28,25]:…”

“…We refer to Section 7 of[5] where definitions of the elliptic Thom sheaf and the elliptic Euler class are discussed.…”