In mathematics, the Harish-Chandra isomorphism, introduced by Harish-Chandra (1951), is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps the center Z(U(g)) of the universal enveloping algebra U(g) of a reductive Lie algebra g to the elements S(h)W of the symmetric algebra S(h) of a Cartan subalgebra h that are invariant under the Weyl group W.
Contents
Fundamental invariants
Let n be the rank of g, which is the dimension of the Cartan subalgebra h. H. S. M. Coxeter observed that S(h)W is a polynomial algebra in n variables (see Chevalley–Shephard–Todd theorem for a more general statement). Therefore, the center of the universal enveloping algebra of a reductive Lie algebra is a polynomial algebra. The degrees of the generators are the degrees of the fundamental invariants given in the following table.
For example, the center of the universal enveloping algebra of G2 is a polynomial algebra on generators of degrees 2 and 6.
Examples
Introduction and setting
Let g be a semisimple Lie algebra, h its Cartan subalgebra and λ, μ ∈ h* be two elements of the weight space and assume that a set of positive roots Φ+ have been fixed. Let Vλ, resp. Vμ be highest weight modules with highest weight λ, resp. μ.
Central characters
The g-modules Vλ and Vμ are representations of the universal enveloping algebra U(g) and its center acts on the modules by scalar multiplication (this follows from the fact that the modules are generated by a highest weight vector). So, for v in Vλ and x in Z(U(g)),
and similarly for Vμ.
The functions
Statement of Harish-Chandra theorem
For any λ, μ ∈ h*, the characters
Another closely related formulation is that the Harish-Chandra homomorphism from the center of the universal enveloping algebra Z(U(g)) to S(h)W (the elements of the symmetric algebra of the Cartan subalgebra fixed by the Weyl group) is an isomorphism.
Applications
The theorem may be used to obtain a simple algebraic proof of Weyl's character formula for finite-dimensional representations.
Further, it is a necessary condition for the existence of a nonzero homomorphism of some highest weight modules (a homomorphism of such modules preserves central character). A simple consequence is that for Verma modules or generalized Verma modules Vλ with highest weight λ, there exist only finitely many weights μ such that a nonzero homomorphism Vλ → Vμ exists.