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A.l. General facts Let G be a simple linear algebraic group, fix a maximal torus and Borel subgroup T C B C G, and let W = N(T)/T be the Weyl group. Let R, R+, R~ , and A be the corresponding roots, positive roots, negative roots, and simple Assume that G is a connected reductive group, T is a fixed maximal torus, B is a fixed Borel subgroup, T C B C G, B is the opposite Borel subgroup, P is a parabolic subgroup containing B_, X(T) is the lattice of characters of T, and A G X(T) is Suppose that T C B C G and T C B C G are maximal tori, embedded in Borel subgroups of G and G. The Lgroup is then equipped with an isomorphism from T onto the complex dual torus X*(T) ® C* of T, which maps the 2 General automorphic Fixing a

pinning.T C B C G of G. there are 16 standard parabolic subgroups of G, corresponding to subsets of the set of simple roots. The geometric interpretation of these parabolic subgroups, as stabilizers of certain flags, can be expressed in  PLEASURE BIRD HATCHINC hILDLIFE AND PIRE PFCTC CRIVTNG FCR FLEASLRE SIGHTSEEING ATTENCINC SPCRTS EVENTS ATTENDINC CCNCERTS ICE SKATING SNOW SKIING SLEDDING ANC TCBCG. CTHEP CCLNTY L19 Propositions 1 and 2 give T C B C G, and the result follows by transitivity. □ Remarks. 1) A particular instance of Corollary 3 was established in 1977 by Dudley and Gutmann [2] : given Q as in Corollary 3 and a probability law /i on ]0, oo], there We fix a Borel subgroup B in G, and a maximal torus T in B.

We.denote by t C b C 0 the Lie algebras of T C B C G,by W = NG{T)/T the Weyl group, and by X = G/B the flag variety. 2.2. Standard example. To simplify the present review, I shall We restrict ourselves to the situation from the previous section, where T C B C G is the maximal torus of a complex, semisimple Lie group G, with a Borel subgroup B. Assume, for simplicity, that the group G is of 17 Equivariant cohomology Let T C B C G be a fixed maximal torus and Borel subgroup of G; put N = [B, B], and let W be the Weyl group of G. All of them are viewed as group schemes. Let L = Hom(Gm , T) be the coweight lattice of G; the Weyl group acts on L. Recall that Soient T C G un tore maximal de G et T C B C G un sousgroupe de Borel de

G.admettant T comme facteur de Levi. On note W le groupe de Weyl de (G , T ). On a alors un diagramme d'induction parabolique G (11.1) i \P G T ou (on a noté G