By M. Duflo, M. Vergne, N. V. Pedersen (auth.)

ISBN-10: 1461244862

ISBN-13: 9781461244868

ISBN-10: 1461288401

ISBN-13: 9781461288404

Ever on the grounds that its creation round 1960 through Kirillov, the orbit strategy has performed a massive position in illustration thought of Lie teams and Lie algebras. This publication comprises the court cases of a convention held from August 29 to September 2, 1988, on the collage of Copenhagen, approximately "the orbit strategy in illustration theory." It includes ten articles, so much of that are unique study papers, by means of famous mathematicians within the box, and it displays the truth that the orbit approach performs an immense function within the illustration idea of semisimple Lie teams, solvable Lie teams, or even extra normal Lie teams, and likewise within the conception of enveloping algebras.

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Extra resources for The Orbit Method in Representation Theory: Proceedings of a Conference Held in Copenhagen, August to September 1988

Example text

On a dv = dtv - LtV. L'element -LtV est Ie poly nome X 1-+ -IIX d1 2, et dtv est element de t* ® A2t* tel que dtv(X)(Y, Z) = ([Y, Xh, Z) - ([Z, Xh, Y) + (Xl, [Y, Z]) pour X E t, Y, Z E t. En particulier, on a dtv(X)(Y, Z) = -2(ad t X(Y), Z) pour X E ~, Y, Z E t. La forme etdv(X) E S(t*) ® A(t)* s'ecrit donc etdv(X) = e-tIlXdl~ LI PI(X)tIII/2 II, OU PI est un polyn6me homogene de degre III/2. En particulier, si I = {I, ... /2(adX) = (-2)dnt/~(X). La fonction generalisee 'Yn(t,X) est combinaison lineaire de termes tie-tllXlll2 Pr(Xhl' (X) ou I est un ensemble d'indices de cardinal2j et If l'ensemble complementaire.

L'action adjointe. Le complexe de cohomologie relative (S(t*)~At*)H est muni d'une differentielle d t qui preserve Ie degre polynomial et on voit facilement que l'application Ve induit un isomorphisme du complexe Ae(M) muni de la differentielle de sur l'espace vectoriel (S(t*) ~ At*)H ~ C muni de la differentielle d = d t - Lt. On filtre Ie complexe «S(t*) ~ At*)H, d) par les sous-complexes Fit formes des elements dont Ie degre total (defini ici sans doubler Ie degre polynomial) est ~ k. • On trouvera dans Arabia [1] une autre demonstration de l'isomorphisme de la proposition 26.

On a DEMONSTRATION: La proposition 17 entraine que dans l'algebre Hx(M) on a i:M(a) = (fE/M a) EuIE(X), et donc • r a = JrM (rJE / M a) = JrM iM(a) EuIE(X)-l. 4 Formule de localisation. Dans ce paragraphe, G un groupe de Lie compact agissant sur une variete differentiable orientee M. Nous supposons que G est connexe, et nous fixons un element S dans Ie centre de g. On note Mo(S) l'ensemble des zeros du champ de vecteurs SM. Chaque composante connexe de Mo(S) est une sous-variete fermee de M, invariante par G.

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The Orbit Method in Representation Theory: Proceedings of a Conference Held in Copenhagen, August to September 1988 by M. Duflo, M. Vergne, N. V. Pedersen (auth.)


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