Harish-Chandra isomorphism


In mathematics, the Harish-Chandra isomorphism, introduced by,
is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps the center Z of the universal enveloping algebra U of a reductive Lie algebra g to the elements SW of the symmetric algebra S of a Cartan subalgebra h that are invariant under the Weyl group W.

Fundamental invariants

Let n be the rank of g, which is the dimension of the Cartan subalgebra h. H. S. M. Coxeter observed that SW is a polynomial algebra in n variables. 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.
Lie algebraCoxeter number hDual Coxeter numberDegrees of fundamental invariants
R001
Ann + 1n + 12, 3, 4,..., n + 1
Bn2n2n − 12, 4, 6,..., 2n
Cn2nn + 12, 4, 6,..., 2n
Dn2n − 22n − 2n; 2, 4, 6,..., 2n − 2
E612122, 5, 6, 8, 9, 12
E718182, 6, 8, 10, 12, 14, 18
E830302, 8, 12, 14, 18, 20, 24, 30
F41292, 6, 8, 12
G2642, 6

For example, the center of the universal enveloping algebra of G2 is a polynomial algebra on generators of degrees 2 and 6.

Examples

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 and its center acts on the modules by scalar multiplication. So, for v in Vλ and x in Z,
and similarly for Vμ.
The functions are homomorphisms to scalars called central characters.

Statement of Harish-Chandra theorem

For any λ, μ ∈ h*, the characters if and only if λ+δ and μ+δ are on the same orbit of the Weyl group of h*, where δ is the half-sum of the positive roots.
Another closely related formulation is that the Harish-Chandra homomorphism from the center of the universal enveloping algebra Z to SW 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 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.