Let G be a semisimple linear algebraic group of inner type over a field F and X be a projective homogeneous G-variety such that G splits over the function field of X. In the present paper we introduce an invariant of G called J-invariant which characterizes the motivic behavior of X. This generalizes the respective notion invented by A. Vishik in the context of quadratic forms. As a main application we obtain a uniform proof of all known motivic decompositions of generically split projective homogeneous varieties (Severi-Brauer varieties, Pfister quadrics, maximal orthogonal Grassmannians, G 2 -and F 4 -varieties) as well as provide new examples (exceptional varieties of types E 6 , E 7 and E 8 ). We also discuss relations with torsion indices, canonical dimensions and cohomological invariants of the group G. * The paper is based on the PhD Thesis of the first author. Partially supported by CNRS, DAAD A/04/00348, INTAS, SFB701, DFG GI706/1-1.Observe that the motive R p (G) depends only on G and p but not on the type of a parabolic subgroup defining X. Moreover, considered with Q-coefficients it always splits as a direct sum of twisted Tate motives.Our proof is based on two different observations. The first is the Rost Nilpotence Theorem. It was originally proven for projective quadrics by M. Rost and then generalized to arbitrary projective homogeneous varieties by P. Brosnan [Br05], V. Chernousov, S. Gille and A. Merkurjev [CGM]. Roughly speaking, this result plays a role of the Galois descent for motivic decompositions over a separable closureF of F . Namely, it reduces the problem to the description of idempotent cycles in the endomorphism group End(M(XF ; Z/p)) which are defined over F .To provide such cycles we use the second observation which comes from the topology of compact Lie groups. In paper [Kc85] V. Kac invented the notion of p-exceptional degrees -the numbers which relate the degrees of mod p basic polynomial invariants and the p-torsion part of the Chow ring of a compact Lie group. These numbers have combinatorial nature. By the result of K. Zainoulline [Za06] there is a strong interrelation between p-exceptional degrees and the subgroup of cycles in End(M(XF ; Z/p)) defined over F . To describe this subgroup we introduce the notion of the J-invariant of a group G mod p denoted by J p (G) (see Definition 4.6). In the most cases the values of J p (G) were implicitly computed by V. Kac in [Kc85] and can easily be extracted from Table 4.13.It follows from the proof that the J-invariant measures the 'size' of the motive R p (G) and, hence, characterizes the motivic decomposition of X. Observe that if the J-invariant takes its minimal possible non-trivial value J p (G) = (1), then the motive R p (G) ⊗ Q has the following recognizable decomposition (cf. [Vo03, §5] and [Ro06, §5])

Contents Part 3. Algebraic and geometric comparison. Applications. 27 13. Comparison results 27 14. Formulas for push-forwards 30 15. Algorithm for multiplying in h * (G/B) 32 16. Landweber-Novikov operations 32 17. Examples of computations 33 References 35

In the present paper we generalize the construction of the nil Hecke ring of Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory of Levine-Morel and Panin-Smirnov, e.g. to Chow groups, Grothendieck's K_0, connective K-theory, elliptic cohomology, and algebraic cobordism. The resulting object, which we call a formal (affine) Demazure algebra, is parameterized by a one-dimensional commutative formal group law and has the following important property: specialization to the additive and multiplicative periodic formal group laws yields completions of the nil Hecke and the 0-Hecke rings respectively. We also introduce a deformed version of the formal (affine) Demazure algebra, which we call a formal (affine) Hecke algebra. We show that the specialization of the formal (affine) Hecke algebra to the additive and multiplicative periodic formal group laws gives completions of the degenerate (affine) Hecke algebra and the usual (affine) Hecke algebra respectively. We show that all formal affine Demazure algebras (and all formal affine Hecke algebras) become isomorphic over certain coefficient rings, proving an analogue of a result of Lusztig.Comment: 28 pages. v2: Some results strengthened and references added. v3: Minor corrections, section numbering changed to match published version. v4: Sign errors in Proposition 6.8(d) corrected. This version incorporates an erratum to the published versio

Abstract. In the present paper, we generalize the coproduct structure on nil Hecke rings introduced and studied by Kostant and Kumar to the context of an arbitrary algebraic oriented cohomology theory and its associated formal group law. We then construct an algebraic model of the T -equivariant oriented cohomology of the variety of complete flags.

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