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

Consider a crystallographic root system together with its Weyl group W acting on the weight lattice �. Let Z(�) W and S(�) W be the W-invariant subrings of the integral group ring Z(�) and the symmetric algebra S(�) respectively. A celebrated result by Chevalley says that Z(�) W is a polynomial ring in classes of funda- mental representations �1;㨺㨻�n and S(�) W Q is a polynomial ring in basic polynomial invariants q1;㨺㨻qn. In the present paper we es- tablish and investigate the relationship betweeni's and qi's over the integers. As an application we provide estimates for the torsion of the Grothendieck -filtration and the Chow groups of some twisted flag varieties up to codimension 4.

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Basic Polynomial Invariants, Fundamental Representations and the Chern Class Map

In the present notes we provide a new uniform way to compute a canonical p-dimension of a split algebraic group G for a torsion prime p using degrees of basic polynomial invariants described by V. Kac. As an application, we compute the canonical p-dimensions for all split simple algebraic groups. The notion of a canonical dimension of an algebraic structure was introduced by Berhuy and Reichstein [1]. For a split algebraic group G and its torsion prime p the canonical p-dimension of G was studied by Karpenko and Merkurjev in [3] and [4]. In particular, this invariant was shown to be related with the size of the image of the characteristic map φG : S ∗(T ) → CH∗(X), (1) where T is the character group of a maximal split torus T , X = G/B is the variety of complete flags and S stands for the symmetric algebra. Namely, one has the following formula for the canonical p-dimension of a group G cdp(G) = min{i | Chi(X)}, (2) where Chi(X) stands for the image of φG in the Chow ring Chi(X) = CHi(X;Z/pZ) with Z/pZ-coefficients. Using this remarkable fact together with the explicit description of the image Rp = Ch(X) Karpenko and Merkurjev computed the canonical p-dimensions for all classical algebraic groups G. The goal of the present notes is to relate the work by V. Kac [2] devoted to the study of the p-torsion of the Chow ring of an algebraic group G with the canonical p-dimensions of G. In particular we provide a different and uniform approach of computing cdp(G).

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Canonical p‐dimensions of algebraic groups and degrees of basic polynomial invariants

We prove a version of Knebusch's Norm Principle for finite \'etale extensions of semi-local Noetherian domains with infinite residue fields of characteristic different from 2. As an application we prove Grothendieck's conjecture on principal homogeneous spaces for the spinor group of a quadratic space.

On the norm principle for quadratic forms

In the present notes we provide a new uniform way to compute a canonical p-dimension of a split algebraic group G for a torsion prime p using degrees of basic polynomial invariants described by V.Kac. As an application, we compute the canonical p-dimensions for all split exceptional algebraic groups.

Kirill Zainoulline

Papers

Basic Polynomial Invariants, Fundamental Representations and the Chern Class Map

Canonical p‐dimensions of algebraic groups and degrees of basic polynomial invariants

On the norm principle for quadratic forms

Canonical p-dimensions of algebraic groups and degrees of basic polynomial invariants

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