Lie algebra case

Let ${\displaystyle {\mathfrak {g}}}$ be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra ${\displaystyle {\mathfrak {h}}}$. Let ${\displaystyle R}$ be the associated root system. We then say that an element ${\displaystyle \lambda \in {\mathfrak {h}}^{*}}$ is integral^[5] if

${\displaystyle 2{\frac {\langle \lambda ,\alpha \rangle }{\langle \alpha ,\alpha \rangle }}}$

is an integer for each root ${\displaystyle \alpha }$. Next, we choose a set ${\displaystyle R^{+}}$ of positive roots and we say that an element ${\displaystyle \lambda \in {\mathfrak {h}}^{*}}$ is dominant if ${\displaystyle \langle \lambda ,\alpha \rangle \geq 0}$ for all ${\displaystyle \alpha \in R^{+}}$. An element ${\displaystyle \lambda \in {\mathfrak {h}}^{*}}$ dominant integral if it is both dominant and integral. Finally, if ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ are in ${\displaystyle {\mathfrak {h}}^{*}}$, we say that ${\displaystyle \lambda }$ is higher^[6] than ${\displaystyle \mu }$ if ${\displaystyle \lambda -\mu }$ is expressible as a linear combination of positive roots with non-negative real coefficients.

A weight ${\displaystyle \lambda }$ of a representation ${\displaystyle V}$ of ${\displaystyle {\mathfrak {g}}}$ is then called a highest weight if ${\displaystyle \lambda }$ is higher than every other weight ${\displaystyle \mu }$ of ${\displaystyle V}$.

The theorem of the highest weight then states:^[2]

• If ${\displaystyle V}$ is a finite-dimensional irreducible representation of ${\displaystyle {\mathfrak {g}}}$, then ${\displaystyle V}$ 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 ${\displaystyle \lambda }$, there exists a finite-dimensional irreducible representation with highest weight ${\displaystyle \lambda }$.

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 ${\displaystyle K}$ be a connected compact Lie group with Lie algebra ${\displaystyle {\mathfrak {k}}}$ and let ${\displaystyle {\mathfrak {g}}:={\mathfrak {k}}+i{\mathfrak {k}}}$ be the complexification of ${\displaystyle {\mathfrak {g}}}$. Let ${\displaystyle T}$ be a maximal torus in ${\displaystyle K}$ with Lie algebra ${\displaystyle {\mathfrak {t}}}$. Then ${\displaystyle {\mathfrak {h}}:={\mathfrak {t}}+i{\mathfrak {t}}}$ is a Cartan subalgebra of ${\displaystyle {\mathfrak {g}}}$, and we may form the associated root system ${\displaystyle R}$. 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 ${\displaystyle \lambda \in {\mathfrak {h}}}$ is analytically integral^[7] if

${\displaystyle \langle \lambda ,H\rangle }$

is an integer whenever

${\displaystyle e^{2\pi H}=I}$

where ${\displaystyle I}$ is the identity element of ${\displaystyle K}$. Every analytically integral element is integral in the Lie algebra sense,^[8] but there may be integral elements in the Lie algebra sense that are not analytically integral. This distinction reflects the fact that if ${\displaystyle K}$ is not simply connected, there may be representations of ${\displaystyle {\mathfrak {g}}}$ that do not come from representations of ${\displaystyle K}$. On the other hand, if ${\displaystyle K}$ is simply connected, the notions of "integral" and "analytically integral" coincide.^[3]

The theorem of the highest weight for representations of ${\displaystyle K}$^[9] is then the same as in the Lie algebra case, except that "integral" is replaced by "analytically integral."