On basic Cycles of A_n, B_n, C_n and D_n

Abstract

In this paper, we investigate a conjecture of Dixmier [ 2 ] on the structure of basic cycles. Our interest in basic cycles arises primarily from the fact that the irreducible modules of a simple Lie algebra L having a weight space decomposition are completely determined by the irreducible modules of the cycle subalgebra of L . The basic cycles form a generating set for the cycle subalgebra. First some notation: F denotes an algebraically closed field of characteristic 0, L a finite dimensional simple Lie algebra of rank n over F , H a fixed Cartan subalgebra, U(L) the universal enveloping algebra of L , C(L) the centralizer of H in U(L) , Φ the set of nonzero roots in H *, the dual space of H , Δ = { α 1 , …, α n } a base of Φ, and Φ + = { β 1 , …, β m } the positive roots corresponding to Δ.