Push-pull operators on the formal affine Demazure algebra and its dual

Abstract

Contents 1. Introduction 1 2. Formal Demazure and push-pull operators 4 3. Two bases of the formal twisted group algebra 7 4. The Weyl and the Hecke actions 9 5. Push-pull operators and elements 11 6. The push-pull operators on the dual 13 7. Relations between bases coefficients 15 8. Another basis of the W Ξ -invariant subring 17 9. The formal Demazure algebra and the Hecke algebra 18 10. The algebraic restriction to the fixed locus on G/B 20 11. The algebraic restriction to the fixed locus on G/P 24 12. The push-pull operators on D ⋆ F 27 13. An involution 28 14. The non-degenerate pairing on the W Ξ -invariant subring 29 15. Push-forwards and pairings on D ⋆ F,Ξ 32 References 34

Motivic decompositions of twisted flag varieties and representations of Hecke-type algebras

“…In the present section we prove that if H is the Borel subgroup of a split semisimple linear algebraic group, then the convolution ring h H 2 (G) of Definition 4.3 can be identified with the subring of push-pull operators (Corollary 5.3). Our arguments are essentially based on the Bruhat decomposition of G stated using the G-orbits on the product G/H × k G/H and the resolution of singularities (8).…”

“…In the present section we identify the convolution ring h B 2 (G) with the formal affine Demazure algebra D F of [19] and show that it is self-dual with respect to the convolution product (Theorem 6.2). Our arguments are based on the results of [19], [7], [8] and, especially, [9]. We use the notation of [9].…”

“…If E is a generic G-torsor, then D F can be replaced by the formal affine Demazure algebra D F . The theory of such algebras and formal push-pull operators has been recently developed in [6], [19], [7], [8], [9] motivated by Bernstein-Gelfand-Gelfand [1], Demazure [11], [12], Bressler-Evens [2], [3], Kostant-Kumar [22], [21], Brion [4], Totaro [29] and Edidin-Graham [13]. The key properties of D F are -It is a free module over the T -equivariant oriented cohomology ring S = h T (k) of a point, where T is a split maximal torus in G [7].…”

Restriction formula for stable basis of the Springer resolution

“…In this section we recall the definition of the affine Hecke algebra in terms of the twisted group algebra of Kostant and Kumar [KK90], while following notions from [CZZ12,CZZ13]. The root system we consider will be the one associated to the group G.…”

“…We will frequently use the following identities, whose proof can be checked easily, or found in [CZZ13,§6].…”