HarishChandra isomorphism
It has been suggested that this article be merged into HarishChandra homomorphism. (Discuss) Proposed since June 2017.

In mathematics, the HarishChandra isomorphism, introduced by HarishChandra (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 
A_{n}  n + 1  n + 1  2, 3, 4, ..., n + 1 
B_{n}  2n  2n − 1  2, 4, 6, ..., 2n 
C_{n}  2n  n + 1  2, 4, 6, ..., 2n 
D_{n}  2n − 2  2n − 2  n; 2, 4, 6, ..., 2n − 2 
E_{6}  12  12  2, 5, 6, 8, 9, 12 
E_{7}  18  18  2, 6, 8, 10, 12, 14, 18 
E_{8}  30  30  2, 8, 12, 14, 18, 20, 24, 30 
F_{4}  12  9  2, 6, 8, 12 
G_{2}  6  4  2, 6 
For example, the center of the universal enveloping algebra of G_{2} is a polynomial algebra on generators of degrees 2 and 6.
Examples
 If g is the Lie algebra sl(2, R), then the center of the universal enveloping algebra is generated by the Casimir invariant of degree 2, and the Weyl group acts on the Cartan subalgebra, which is isomorphic to R, by negation, so the invariant of the Weyl group is simply the square of the generator of the Cartan subalgebra, which is also of degree 2.
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 V_{λ}, resp. V_{μ} be highest weight modules with highest weight λ, resp. μ.
Central characters
The gmodules V_{λ} and V_{μ} 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 V_{λ} and x in Z(U(g)),
and similarly for V_{μ}.
The functions are homomorphims to scalars called central characters.
Statement of HarishChandra 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 halfsum of the positive roots.^{[1]}
Another closely related formulation is that the HarishChandra 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 finitedimensional 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 V_{λ} with highest weight λ, there exist only finitely many weights μ such that a nonzero homomorphism V_{λ} → V_{μ} exists.
See also
Notes
 ^ Humphreys (1972), p.130
References
 HarishChandra (1951), "On some applications of the universal enveloping algebra of a semisimple Lie algebra", Transactions of the American Mathematical Society, 70: 28–96, doi:10.2307/1990524, ISSN 00029947, JSTOR 1990524, MR 0044515
 Humphreys, James (1972). Introduction to Lie algebras and Representation Theory. Springer. ISBN 0387900535.
 Humphreys, James E. (2008), Representations of semisimple Lie algebras in the BGG category O, AMS, p. 26, ISBN 9780821846780
 Knapp, Anthony W.; Vogan, David A. (1995), Cohomological induction and unitary representations, Princeton Mathematical Series, 45, Princeton University Press, ISBN 9780691037561, MR 1330919
 Knapp, Anthony, Lie groups beyond an introduction, Second edition, pages 300–303.