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

Universal enveloping algebra

In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...

 of the Lie algebra of G that the representation induces. This construction relies on some extended version of Schur's lemma

Schur's lemma

In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations...

 to show that any z acts on V as a scalar, which by abuse of notation

Abuse of notation

In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition . Abuse of notation should be contrasted with misuse of notation, which should be avoided...

 could be written ρ(z).

In more classical language, z is a differential operator

Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...

, constructed from the infinitesimal transformation

Infinitesimal transformation

In mathematics, an infinitesimal transformation is a limiting form of small transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 skew-symmetric matrix A...

s which are induced on V by the Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

 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

Harish-Chandra homomorphism

In mathematics, the Harish-Chandra isomorphism, introduced by ,is an isomorphism of commutative rings constructed in the theory of Lie algebras...

, the center Z can be identified with the subalgebra of elements of the symmetric algebra

Symmetric algebra

In mathematics, the symmetric algebra S on a vector space V over a field K is the free commutative unital associative algebra over K containing V....

 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

Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...

W of the space a^*⊗ C of complex linear functions on the Cartan subalgebra.