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On Harmonic Elements for Semi-simple Lie Algebras

โœ Scribed by Philippe Caldero

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
237 KB
Volume
166
Category
Article
ISSN
0001-8708

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โœฆ Synopsis

Let g be a semi-simple complex Lie algebra and g=n -ร€ h ร€ n its triangular decomposition. Let U(g), resp. U q (g), be its enveloping algebra, resp. its quantized enveloping algebra. This article gives a quantum approach to the combinatorics of (classical) harmonic elements and Kostant's generalized exponents for g. A quantum analogue of the space of harmonic elements has been given by A. Joseph and G. Letzter (1994, Amer. J. Math. 116, 127-177). On the one hand, we give specialization results concerning harmonic elements, central elements of U q (g), and the decomposition of Joseph and Letzter (cited above). For g=sl n+1 , we describe the specialization of quantum harmonic space in the N-filtered algebra U(sl n+1 ) as the materialization of a theorem of A. Lascoux et al. (1995, Lett. Math. Phys. 35, 359-374). This enables us to study a Joseph-Letzter decomposition in the algebra U(sl n+1 ). On the other hand, we prove that highest weight harmonic elements can be calculated in terms of the dual of Lusztig's canonical basis. In the simply laced case, we parametrize a basis of n-invariants of minimal primitive quotients by the set C 0 of integral points of a convex cone.

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