Infinitesimal character

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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 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:

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, the center Z 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:

a*⊗C/W

the orbits under the Weyl group W of the space a*⊗C of complex linear functions on the Cartan subalgebra.

## References

* Knapp, Anthony W. Lie Groups Beyond an Introduction. 1st ed., Vol. 140. Boston: Birkhäuser, 1996.

## See also

* Harish-Chandra isomorphism

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