La somme des faux degrés—un mystère en théorie des invariants

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• 作者：Florence Fauquant-Millet ; Anthony Joseph
• 关键词：Lie ; Alg&#xe8 ; bre enveloppante ; Centre ; Semi-centre ; Parabolique
• 刊名：Advances in Mathematics
• 出版年：2008
• 出版时间：1 March 2008
• 年：2008
• 卷：217
• 期：4
• 页码：1476-1520
• 全文大小：455 K

文摘

We continue the study of the polynomiality of the semicentre of the enveloping algebra of a parabolic subalgebra of a semisimple Lie algebra , motivated by its truth when is of type A or C [F. Fauquant-Millet, A. Joseph, Semi-centre de l'algèbre enveloppante d'une sous-algèbre parabolique d'une algèbre de Lie semi-simple, Ann. Sci. École Norm. Sup. (4) 38 (2) (2005) 155–191] and when , a Borel subalgebra [A. Joseph, A preparation theorem for the prime spectrum of a semisimple Lie algebra, J. Algebra 48 (1977) 241–289] and (Chevalley).

We construct a linear map of into and show it to be an isomorphism just in types A and C. We link this to the difficulty of proving the polynomiality of outside types A and C. It leads to “false degrees” defined by underlying combinatorial structure. These are the true degrees when the bounds in [F. Fauquant-Millet, A. Joseph, Semi-centre de l'algèbre enveloppante d'une sous-algèbre parabolique d'une algèbre de Lie semi-simple, Ann. Sci. École Norm. Sup. (4) 38 (2) (2005) 155–191] coincide and polynomiality ensues. We show that these false degrees always sum to which can fail for the true degrees when they are defined. Finally we prove the Tauvel–Yu conjecture on the index of a parabolic.