# 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*.

## 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 *g*-modules *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 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.[1]

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 *V*_{λ} with highest weight λ, there exist only finitely many weights μ such that a nonzero homomorphism *V*_{λ} → *V*_{μ} exists.

## Notes

- Humphreys (1972), p.130

## References

- Harish-Chandra (1951), "On some applications of the universal enveloping algebra of a semisimple Lie algebra",
*Transactions of the American Mathematical Society*,**70**(1): 28–96, doi:10.2307/1990524, ISSN 0002-9947, JSTOR 1990524, MR 0044515 - Humphreys, James (1972).
*Introduction to Lie algebras and Representation Theory*. Springer. ISBN 978-0387900537. - Humphreys, James E. (2008),
*Representations of semisimple Lie algebras in the BGG category O*, AMS, p. 26, ISBN 978-0-8218-4678-0 - Knapp, Anthony W.; Vogan, David A. (1995),
*Cohomological induction and unitary representations*, Princeton Mathematical Series,**45**, Princeton University Press, ISBN 978-0-691-03756-1, MR 1330919 - Knapp, Anthony,
*Lie groups beyond an introduction*, Second edition, pages 300–303.