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.

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.

Lie algebra Coxeter number h Dual Coxeter number Degrees of fundamental invariants
R 0 0 1
An n + 1 n + 1 2, 3, 4, ..., n + 1
Bn 2n 2n − 1 2, 4, 6, ..., 2n
Cn 2n n + 1 2, 4, 6, ..., 2n
Dn 2n − 2 2n − 2 n; 2, 4, 6, ..., 2n − 2
E6 12 12 2, 5, 6, 8, 9, 12
E7 18 18 2, 6, 8, 10, 12, 14, 18
E8 30 30 2, 8, 12, 14, 18, 20, 24, 30
F4 12 9 2, 6, 8, 12
G2 6 4 2, 6

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 \mathfrak{g} be a semisimple Lie algebra, \mathfrak{h} its Cartan subalgebra and \lambda, \mu\in\mathfrak{h}^* be two elements of the weight space and assume that a set of positive roots \Phi^%2B have been fixed. Let V_\lambda, resp. V_\mu be highest weight modules with highest weight \lambda, resp. \mu.

Central characters

The \mathfrak{g}-modules V_\lambda and V_\mu are representations of the universal enveloping algebra \mathfrak{U}(\mathfrak{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 V_\lambda and x\in Z(\mathfrak{U}(\mathfrak{g})),

x\cdot v:=\chi_\lambda(x)v

and similarly for V_\mu.

The functions \chi_\lambda, \,\chi_\mu are homomorphims to scalars called central characters.

Statement of Harish-Chandra theorem

For any \lambda,\mu\in\mathfrak{h}^*, the characters \chi_\lambda=\chi_\mu if and only if \lambda and \mu are on the same orbit of the Weyl group of \mathfrak{g} under the affine action (corresponding to the choice of the positive roots \Phi^%2B).

Another closely related formulation is that the Harish-Chandra homomorphism from the centrum of the universal enveloping algebra Z(\mathfrak{U}(\mathfrak{g})) to S(\mathfrak{h})^W (invariant polynomials over the Cartan subalgebra fixed by the affine action of 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 moules (a homomorphism of such modules preserves central character). A simple consequence is that for Verma modules or generalized Verma modules V_\lambda with highest weight \lambda, there exist only finitely many weights \mu such that a nonzero homomorphism V_\lambda\to V_\mu exists.

See also

References