Affine Hecke algebras and the Schubert calculus

Using a combinatorial approach which avoids geometry, this paper studies the ring structure of K_T(G/B), the T-equivariant K-theory of the (generalized) flag variety G/B. Here the data is a complex reductive algebraic group (or symmetrizable Kac-Moody group) G, a Borel subgroup B, and a maximal torus T, and K_T(G/B) is the Grothendieck group of T-equivariant coherent sheaves on G/B. We prove "Pieri-Chevalley" formulas for the products of a Schubert class by a homogeneous line bundle (dominant or anti-dominant) and for products of a Schubert class by a codimension 1 Schubert class. All of these Pieri-Chevalley formulas are given in terms of the combinatorics of the Littelmann path model. We give explicit computations of products of Schubert classes for the rank two cases and this data allows us to make a "positivity conjecture" generalizing the theorems of Brion and Graham, which treat the cases K(G/B) and H_T^*(G/B), respectively

The hyperbolic formal affine Demazure algebra

In the present paper we extend the construction of the formal (affine) Demazure algebra due to Hoffnung, Malag\'on-L\'opez, Savage and Zainoulline in two directions. First, we introduce and study the notion of an extendable weight lattice in the Kac-Moody setting and show that all the definitions and properties of the formal (affine) Demazure operators and algebras hold for such lattices. Second, we show that for the hyperbolic formal group law the formal Demazure algebra is isomorphic (after extending the coefficients) to the Hecke algebra.Comment: Final version. Accepted for publication in Algebras and Representation Theory. ALGE-D-15-0016

Equivariant quantum cohomology and Yang-Baxter algebras

There are two intriguing statements regarding the quantum cohomology of partial flag varieties. The first one relates quantum cohomology to the affinisation of Lie algebras and the homology of the affine Grassmannian, the second one connects it with the geometry of quiver varieties. The connection with the affine Grassmannian was first discussed in unpublished work of Peterson and subsequently proved by Lam and Shimozono. The second development is based on recent works of Nekrasov, Shatashvili and of Maulik, Okounkov relating the quantum cohomology of Nakajima varieties with integrable systems and quantum groups. In this article we explore for the simplest case, the Grassmannian, the relation between the two approaches. We extend the definition of the integrable systems called vicious and osculating walkers to the equivariant setting and show that these models have simple expressions in a particular representation of the affine nil-Hecke ring. We compare this representation with the one introduced by Kostant and Kumar and later used by Peterson in his approach to Schubert calculus. We reveal an underlying quantum group structure in terms of Yang-Baxter algebras and relate them to Schur-Weyl duality. We also derive new combinatorial results for equivariant Gromov-Witten invariants such as an explicit determinant formula.Comment: 60 pages, 11 figures; v2: statement about Schur-Weyl duality added and introduction slightly rewritten, see last paragraph on page 2 and first paragraph on page 3 as well as Theorem 1.3 in the introductio