Representation with a highest weight vector

A linear representation (cf. Representation of a Lie algebra) of a finite-dimensional semi-simple split Lie algebra over a field of characteristic zero with a split Cartan subalgebra , having the following properties.

1) In the space of there is a cyclic vector (i.e. is the smallest -invariant subspace containing ).

2) for all , where is some fixed linear form on with values in .

3) If is a system of simple roots, defined by a lexicographical order on the set of all roots of relative to (cf. Root system), and if are the vectors from the Chevalley basis of corresponding to , , then for all . Thus, is a weight relative to the restriction of to (cf. Weight of a representation of a Lie algebra); it is called a highest weight. The space is called a cyclic -module with highest weight and generator , and is called a highest weight vector.

There exists for every linear form on a unique, up to equivalence, irreducible representation of with highest weight . The -module determined by is a direct sum of weight subspaces relative to the restriction of to . Their weights have the form

where the are non-negative integers. The weight subspace of weight is finite-dimensional, spanned over by vectors of the form

and for any the restriction of to is the operator of scalar multiplication by . The space is one-dimensional; the weight is the only highest weight of and can be characterized as the unique weight of the -module such that any other weight has the form

where the are non-negative integers.

A representation is finite-dimensional if and only if is a dominant linear form on , i.e. is a non-negative integer for . Every irreducible finite-dimensional linear representation of has the form for some dominant linear form on (hence all such representations are classified, up to equivalence, by the dominant linear forms on ). The set of all weights of a finite-dimensional representation relative to is invariant relative to the Weyl group of (regarded as a group of linear transformations of ), and if weights and belong to one orbit of the Weyl group, then the dimensions of the spaces and are equal. For every weight and every root the number is an integer; if, moreover, is also a weight, then

(here is the element in corresponding to and is the root vector of ).

References

[1]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[2]	, Theórie des algèbres de Lie. Topologie des groupes de Lie , Sem. S. Lie , Ie année 1954–1955 , Secr. Math. Univ. Paris (1955)
[3]	D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)
[4]	E. Cartan, "Les tenseurs irréductibles et les groupes linéaires simples et semi-simples" Bull. Sci. Math. , 49 (1925) pp. 130–152
[5]	Harish-Chandra, "On some applications of the universal enveloping algebra of a semisimple Lie algebra" Trans. Amer. Math. Soc. , 70 (1951) pp. 28–96