Infinitesimal character

Formulation

The infinitesimal character is the linear form on the center Z of the universal enveloping algebra of the Lie algebra of G that the representation induces. This construction relies on some extended version of Schur's lemma to show that any z in Z acts on V as a scalar, which by abuse of notation could be written \rho (z).

In more classical language, z is a differential operator, constructed from the infinitesimal transformations which are induced on V by the Lie algebra of G. The effect of Schur's lemma is to force all v in V to be simultaneous eigenvectors of z acting on V. Calling the corresponding eigenvalue:

:\lambda = \lambda (z)

the infinitesimal character is by definition the mapping:

:z \rightarrow \lambda (z)

There is scope for further formulation. By the Harish-Chandra isomorphism, the center Z can be identified with the subalgebra of elements of the symmetric algebra of the Cartan subalgebra a that are invariant under the Weyl group, so an infinitesimal character can be identified with an element of:

:a^* \otimes C / W

the orbits under the Weyl group W of the space a^* \otimes C of complex linear functions on the Cartan subalgebra.

References

• Knapp, Anthony W. Lie Groups Beyond an Introduction. 1st ed., Vol. 140. Boston: Birkhäuser, 1996.