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 acts on V as a scalar, which by abuse of notation could be written ρ(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
- λ = λ(z),
the infinitesimal character is by definition the mapping
- z → λ(z).
There is scope for further formulation. By the Harish-Chandra homomorphism, 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*⊗ C/W,
the orbits under the Weyl group W of the space a*⊗ C of complex linear functions on the Cartan subalgebra.
Read more about this topic: Infinitesimal Character
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