On the weight spaces of Lie algebra modules and their Jordan kernel

Abstract

Let L be a finite-dimensional Lie algebra over a field k of characteristic zero and let V be an L module. If S is a subset of L and i a function from S to k, we define the eigenspace V,(S) and the weight space V'(S) of V with respect to ;1 (and S) by vj,(s)= {UE V~vxES,XU=l"(x)u}, v"(S)= {VE VIvxES,3nEN, [x-A(x)]"u=O). Particularly, if S has only one element, we use the notations V,+,(s) and V"'"'(s) instead of V,({s}) and vi-({s}) and, if S coincides with L, we write V1 and V" instead of V,(L) and V'(L). If V'(S) is nonzero, we call 1 a weight of S in V. Remark that, if V is finite-dimensional, V'(S) is the set of all u E V such that [x-n(x)]" v = 0 for all x E L, where n is the dimension of V. It is clear that V, is a submodule of V, contained in V". Moreover, if V, is nonzero, then I must be a character of L (i.e., A is linear and ;1([L, L]) = 0). In [8] Smith asks wether Vi must be nonzero, if V" is non-zero. In case V is finite-dimensional, she answers this question affirmatively in [9] and she also proves that, in that situation, V" is a submodule of I'. These results may be generalized to arbitrary L modules V [Theorem 31. We are able to give a new characterization of these weight spaces [Theorem 11, Proposition 131 which greatly simplifies their actual computation. We also take special interest in applying these results to the case where V is either the universal enveloping algebra U(L) of L or its division ring of quotients D(L). In the latter case, each weight vector of D(L) can be written as a quotient of a weight vector of U(L) by a nonzero eigenvec-tor (semi-invariant) of U(L) [Proposition 173. Furthermore, we define the 28