 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

the infinitesimal character is by definition the mapping

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

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