Abstract :
Biparabolic subalgebras of semisimple Lie algebras were introduced by V. Dergachev and A. Kirillov [V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2000) 331–343] under the name of Lie algebras of seaweed type. Let be such an algebra, its derived algebra, its nilradical and the symmetric algebra over . Now is algebraic, so by a result of Chevalley–Dixmier [J. Dixmier, Sur les représentations unitaires des groupes de Lie nilpotents. II, Bull. Soc. Math. France 85 (1957) 325–388], index . Here we give a combinatorial formula for index and use it to prove a conjecture of P. Tauvel and R.W.T. Yu [P. Tauvel R.W.T. Yu, Sur lʹindice de certaines algèbres de Lie, Ann. Inst. Fourier (Grenoble) 54 (2004) 1793–1810]. We also compute the Gelfand–Kirillov dimension of , an algebra we conjecture to be polynomial. This number is combinatorially more subtle than index . As a by-product we show that is always polynomial. The present methods are an adaption of those used in the study of similar questions for parabolic subalgebras in [F. Fauquant-Millet, A. Joseph, Sur les semi-invariants dʹune sous-algèbre parabolique dʹune algèbre enveloppante quantifiée, Transform. Groups 6 (2001) 125–142; F. Fauquant-Millet, A. Joseph, Semi-centre de lʹalgèbre enveloppante dʹune sous-algèbre parabolique dʹune algèbre de Lie semi-simple, Ann. Sci. École Norm. Sup. 38 (2005) 155–191; F. Fauquant-Millet, A. Joseph, La somme des faux degrés—un mystère en théorie des invariants, preprint, 2005]. Many problems remain open.
