Infinitesimal character

In mathematics, the infinitesimal character of an irreducible representation $\rho$ 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 $\rho$ 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 $\rho (z)$.

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:

$\lambda =\lambda (z)$

the infinitesimal character is by definition the mapping:

$z\rightarrow \lambda (z)$

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:

$a^{*}\otimes C/W$

the orbits under the Weyl group $W$ of the space $a^{*}\otimes C$ of complex linear functions on the Cartan subalgebra.

References

• Knapp, Anthony W., and Anthony William Knapp. Lie groups beyond an introduction. Vol. 140. Boston: Birkhäuser, 1996.

See also

• Harish-Chandra isomorphism