In mathematics, the infinitesimal character of an irreducible representation
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
The infinitesimal character is the linear form on the center
This construction relies on some extended version of Schur's lemma to show that any
as a scalar, which by abuse of notation could be written
is a differential operator, constructed from the infinitesimal transformations which are induced on
The effect of Schur's lemma is to force all
Calling the corresponding eigenvalue: the infinitesimal character is by definition the mapping: There is scope for further formulation.
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 complex linear functions on the Cartan subalgebra.