Groupes et algébres de lie

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/63B506EB85D6875966CE63BA0336BC2F/S0025557200190515a.pdf/div-class-title-span-class-italic-lie-groups-and-lie-algebras-chapters-1-3-span-by-n-bourbaki-pp-450-dm-98-1989-isbn-3-540-50218-1-springer-div.pdf

Lie groups and Lie algebras (chapters 1-3 ), by N. Bourbaki. Pp 450. DM 98. 1989. ISBN 3-540-50218-1 (Springer)

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Reflection Groups And Coxeter Groups

Quadratic and hermitian forms over rings

Soit G un groupe algebrique lineaire semi-simple de type interieur sur un corps F et soit X un G-espace homogene projectif tel que le groupe G soit deploye sur le point generique de X. Nous introduisons le J-invariant de G qui caracterise le comportement motivique de X et generalise le J-invariant defini par A. Vishik dans le cadre des formes quadratiques. Nous utilisons cet invariant pour obtenir les decompositions motiviques de tous les G-espaces homogenes projectifs qui sont generiquement deployes, par exemple les varietes de Severi-Brauer, les quadriques de Pfister, la grassmannienne des sous-espaces totalement isotropes maximaux d'une forme quadratique, la variete des sous-groupes de Borel de G. Nous discutons egalement les relations avec les indices de torsion, la dimension canonique et les invariants cohomologiques du groupe G.

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J-invariant of linear algebraic groups

In the present notes we generalize the classical work of Demazure [Invariants symetriques entiers des groupes de Weyl et torsion] to arbitrary oriented cohomology theories and formal group laws. Let G be a split semisemiple linear algebraic group over a field and let T be its split maximal torus. We construct a generalized characteristic map relating the so called formal group ring of the character group of T with the cohomology of the variety of Borel subgroups of G. The main result of the paper says that the kernel of this map is generated by W-invariant elements, where W is the Weyl group of G. As one of the applications we provide an algorithm (realized as a Macaulau2 package) which can be used to compute the ring structure of an oriented cohomology (algebraic cobordism, Morava $K$-theories, connective K-theory, Chow groups, K_0, etc.) of a complete flag variety.

Invariants, torsion indices and oriented cohomology of complete flags

http://www.crm.umontreal.ca/K/pdf/zainoulline.pdf

Oriented cohomology and motivic decompositions of relative cellular spaces

/pdf/invariants-torsion-indices-and-oriented-cohomology-of-3d0lgl6x2p.pdf

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.

Kirill Zainoulline

Papers

J-invariant of linear algebraic groups

Invariants, torsion indices and oriented cohomology of complete flags

Oriented cohomology and motivic decompositions of relative cellular spaces

Invariants, torsion indices and oriented cohomology of complete flags

Formal Hecke algebras and algebraic oriented cohomology theories