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. 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. 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 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