On Harmonic Elements for Semi-simple Lie Algebras

AbstractLet g be a semi-simple complex Lie algebra and g=n−⊕h⊕n its triangular decomposition. Let U(g), resp. Uq(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 Uq(g), and the decomposition of Joseph and Letzter (cited above). For g=sln+1, we describe the specialization of quantum harmonic space in the N-filtered algebra U(sln+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(sln+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 C0 of integral points of a convex cone