Chapter V. Finite-Dimensional Representations

Abstract

In any finite-dimensional representation of a complex semisimple Lie algebra $\mathfrak g$, a Cartan subalgebra $\mathfrak h$ acts completely reducibly, the simultaneous eigenvalues being called “weights.” Once a positive system for the roots $\Delta^+(\mathfrak g,\mathfrak h)$ has been fixed, one can speak of highest weights. The Theorem of the Highest Weight says that irreducible finite-dimensional representations are characterized by their highest weights and that the highest weight can be any dominant algebraically integral linear functional on $\mathfrak h$. The hard step in the proof is the construction of an irreducible representation corresponding to a given dominant algebraically integral form. This step is carried out by using “Verma modules,” which are universal highest weight modules.

All finite-dimensional representations of $\mathfrak g$ are completely reducible. Consequently the nature of such a representation can be determined from the representation of $\mathfrak h$ in the space of “$\mathfrak n$ invariants.” The Harish-Chandra Isomorphism identifies the center of the universal enveloping algebra $U(\mathfrak g)$ with the Weyl-group invariant members of $U(\mathfrak h)$. The proof uses the complete reducibility of finite-dimensional representations of $\mathfrak g$.

The center of $U(\mathfrak g)$ acts by scalars in any irreducible representation of $\mathfrak g$, whether finite dimensional or infinite dimensional. The result is a homomorphism of the center into $\mathbb C$ and is known as the “infinitesimal character” of the representation. The Harish-Chandra Isomorphism makes it possible to parametrize all possible homomorphisms of the center into $\mathbb C$, thus to parametrize all possible infinitesimal characters. The parametrization is by the quotient of $\mathfrak h^*$ by the Weyl group.

The Weyl Character Formula attaches to each irreducible finite-dimensional representation a formal exponential sum corresponding to the character of the representation. The proof uses infinitesimal characters. The formula encodes the multiplicity of each weight, and this multiplicity is made explicit by the Kostant Multiplicity Formula. The formula encodes also the dimension of the representation, which is made explicit by the Weyl Dimension Formula.

Parabolic subalgebras provide a framework for generalizing the Theorem of the Highest Weight so that the Cartan subalgebra is replaced by a larger subalgebra called the “Levi factor” of the parabolic subalgebra.

The theory of finite-dimensional representations of complex semisimple Lie algebras has consequences for compact connected Lie groups. One of these is a formula for the order of the fundamental group. Another is a version of the Theorem of the Highest Weight that takes global properties of the group into account. The Weyl Character Formula becomes more explicit, giving an expression for the character of any irreducible representation when restricted to a maximal torus.