Theorem of the highest weight explained

ak g .^[1] There is a closely related theorem classifying the irreducible representations of a connected compact Lie group K .^[2] The theorem states that there is a bijectionλ \mapsto [V ^{λ]} from the set of "dominant integral elements" to the set of equivalence classes of irreducible representations of ak g or K . The difference between the two results is in the precise notion of "integral" in the definition of a dominant integral element. If K is simply connected, this distinction disappears.

The theorem was originally proved by Élie Cartan in his 1913 paper.^[3] The version of the theorem for a compact Lie group is due to Hermann Weyl. The theorem is one of the key pieces of representation theory of semisimple Lie algebras.

Statement

Lie algebra case

Let

ak{g} be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra ak{h} . Let R be the associated root system. We then say that an element λ \in ak h ^{*} is integral^[4] if 2 (\langle λ ,\alpha\rangle)/(\langle\alpha,\alpha\rangle) is an integer for each root \alpha . Next, we choose a set R ^{+} of positive roots and we say that an element λ \in ak h ^{*} is dominant if \langle λ ,\alpha\rangle\geq 0 for all \alpha\in R ^{+} . An element λ \in ak h ^{*} dominant integral if it is both dominant and integral. Finally, if λ and \mu are in ak h ^{*} , we say that λ is higher^[5] than \mu if λ- \mu is expressible as a linear combination of positive roots with non-negative real coefficients.λ of a representation V of ak g is then called a highest weight if λ is higher than every other weight \mu of V .

The theorem of the highest weight then states:

• If

V is a finite-dimensional irreducible representation of ak{g} , then V has a unique highest weight, and this highest weight is dominant integral.

• If two finite-dimensional irreducible representations have the same highest weight, they are isomorphic.
• For each dominant integral element

λ , there exists a finite-dimensional irreducible representation with highest weight λ .

The most difficult part is the last one; the construction of a finite-dimensional irreducible representation with a prescribed highest weight.

The compact group case

Let

K be a connected compact Lie group with Lie algebra ak k and let ak g: = ak k + i ak k be the complexification of ak g . Let T be a maximal torus in K with Lie algebra ak t . Then ak h: = ak t + i ak t is a Cartan subalgebra of ak g , and we may form the associated root system R . The theory then proceeds in much the same way as in the Lie algebra case, with one crucial difference: the notion of integrality is different. Specifically, we say that an element λ \in ak h is analytically integral^[6] if \langle λ ,H\rangle is an integer whenevere ^{2\pi} = I where I is the identity element of K . Every analytically integral element is integral in the Lie algebra sense,^[7] but there may be integral elements in the Lie algebra sense that are not analytically integral. This distinction reflects the fact that if K is not simply connected, there may be representations of ak g that do not come from representations of K . On the other hand, if K is simply connected, the notions of "integral" and "analytically integral" coincide.

The theorem of the highest weight for representations of

K ^[8] is then the same as in the Lie algebra case, except that "integral" is replaced by "analytically integral."

Proofs

There are at least four proofs:

• Hermann Weyl's original proof from the compact group point of view,^[9] based on the Weyl character formula and the Peter–Weyl theorem.
• The theory of Verma modules contains the highest weight theorem. This is the approach taken in many standard textbooks (e.g., Humphreys and Part II of Hall).
• The Borel–Weil–Bott theorem constructs an irreducible representation as the space of global sections of an ample line bundle; the highest weight theorem results as a consequence. (The approach uses a fair bit of algebraic geometry but yields a very quick proof.)
• The invariant theoretic approach: one constructs irreducible representations as subrepresentations of a tensor power of the standard representations. This approach is essentially due to H. Weyl and works quite well for classical groups.

See also

• Classifying finite-dimensional representations of Lie algebras
• Representation theory of a connected compact Lie group
• Weights in the representation theory of semisimple Lie algebras

References

• • • • .

Notes and References

1. Theorems 9.4 and 9.5
2. Theorem 12.6
3. Reviewed work: Matrix Groups: An Introduction to Lie Group Theory, Andrew Baker; Lie Groups: An Introduction through Linear Groups, Wulf Rossmann. 3647845. Knapp. A. W.. The American Mathematical Monthly. 2003. 110. 5. 446–455. 10.2307/3647845.
4. Section 8.7
5. Section 8.8
6. Definition 12.4
7. Proposition 12.7
8. Corollary 13.20
9. Chapter 12