Infinitesimal character explained
of a semisimple Lie group
on a vector space
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
by two successive linearizations.
Formulation
The infinitesimal character is the linear form on the center
of the universal enveloping algebra of the Lie algebra of
that the representation induces. This construction relies on some extended version of Schur's lemma to show that any
in
acts on
as a scalar, which by abuse of notation could be written
.In more classical language,
is a differential operator, constructed from the infinitesimal transformations which are induced on
by the Lie algebra of
. The effect of Schur's lemma is to force all
in
to be simultaneous eigenvectors of
acting on
. 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
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
of the space
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
