In mathematics, the infinitesimal character of an

irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...

ρ of a

semisimple Lie group In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals and homomorphisms, i ...

''G'' on a vector space ''V'' is, roughly speaking, a mapping to scalars that encodes the process of first differentiating and then

diagonalizing In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) ...

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 Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...

''Z'' of the

universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representati ...

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

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 of a group ' ...

to show that any ''z'' in ''Z'' acts on ''V'' as a scalar, which by

abuse of notation In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors an ...

could be written ρ(''z''). In more classical language, ''z'' is a

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 that accepts a function and return ...

, constructed from the

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 ske ...

s which are induced on ''V'' by the

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

of ''G''. The effect of Schur's lemma is to force all ''v'' in ''V'' to be simultaneous

eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

s 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 isomorphism In mathematics, the Harish-Chandra isomorphism, introduced by , is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps the center \mathcal(U(\mathfrak)) of the universal enveloping algebra U(\mathfr ...

, the center ''Z'' can be identified with the subalgebra of elements of the

symmetric algebra In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...

of the

Cartan subalgebra In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by � ...

''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 In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...

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

Formulation

See also