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Towards the Jantzen conjecture. II

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Towards the Jantzen conjecture. II COMPOSITIO MATHEMATICA A. JOSEPH Towards the Jantzen conjecture. II Compositio Mathematica, tome 40, no 1 (1980), p. 69-78. <http://www.numdam.org/item?id=CM_1980__40_1_69_0> © Foundation Compositio Mathematica, 1980, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 69 TOWARDS THE JANTZEN CONJECTURE II A. Joseph* COMPOSITIO MATHEMATICA, Vol. 40, Fasc. 1, 1980, pag. 69-78 (c) 1980 Sijthoff &#x26; Noordhoff International Publishers - Alphen aan den Rijn Printed in the Netherlands Abstract A recently developed additivity principle for Goldie rank is com- bined with the results of the first paper of this series to establish a very slightly weaker form of the Jantzen conjecture for the primitive spectrum of a complex simple Lie algebra of type An. Precise in- formation on the primitive spectra in other simple Lie algebras and on the Goldie ranks of the associated quotient algebras is also obtained. 1. Introduction Let q be a complex semisimple Lie algebra, U(g) its enveloping algebra and Prim U(g) the set of primitive ideals of U(g). The classification of Prim U(g) for A simple of type An-t (Cartan notation) for n _ 6 (and several other low rank cases) was given by Borho and Jantzen [2], [3], and from their results Jantzen [1], 5.9 guessed its solution for general n. In [8], 8.2, 11.8, we suggested what form this conjecture should take for an arbitrary semisimple Lie algebra and indicated how this should be related to the Goldie ranks of the quotient algebras U (

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