Harish-Chandra isomorphism

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.

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.

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 V_λ, resp. V_μ be highest weight modules with highest weight λ, resp. μ.

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