Infinitesimal character

In mathematics, the infinitesimal character of an irreducible representation ρ of a semisimple Lie group "G" on a vector space "V" is, roughly speaking, a mapping to scalars that encodes the process of first differentiating and then diagonalizing the representation. It therefore is a way of extracting something essential from the representation ρ by two successive linearizations.

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.