Infinitesimal character
In mathematics, the infinitesimal character of an irreducible representation of a semisimple Lie group ' on a vector space ' 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 by two successive linearizations.
Formulation
The infinitesimal character is the linear form on the center ' of the universal enveloping algebra of the Lie algebra of ' that the representation induces. This construction relies on some extended version of Schur's lemma to show that any ' in ' acts on ' as a scalar, which by abuse of notation could be written.In more classical language, ' is a differential operator, constructed from the infinitesimal transformations which are induced on ' by the Lie algebra of '. The effect of Schur's lemma is to force all ' in ' to be simultaneous eigenvectors of ' acting on '. Calling the corresponding eigenvalue:
the infinitesimal character is by definition the mapping:
There is scope for further formulation. By the Harish-Chandra isomorphism, the center ' 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:
the orbits under the Weyl group ' of the space of complex linear functions on the Cartan subalgebra.