Exercise 5, § 4

Let $G$ be a compact group and let $\varrho$ be a faithful unitary finite-dimensional representation of G. Let $\pi \in \widehat{G}$.

a) There exist integers $a$ and $b$ such that $\pi$ is isomorphic to a subrepresentation of $\varrho^{\otimes a}\otimes \overline{\varrho}^{\otimes b}$. (One may use for example the Weierstrass–Stone theorem.)

Denote by $w(\pi )$ the minimum of $a+b$ when $(a, b)$ runs through the set of integers satisfying this property.

b) Let Λ be the weight lattice of $G^0$ and let $q$ be a norm on $\Lambda \otimes \mathbf{R}$. There exist real numbers C and D such that for every representation $\pi \in \widehat{G}$ whose restriction to $G^0$ admits the highest weights $(\lambda_1, . . . , \lambda_k)$, one has

$w(\pi )\leqslant C$ sup($q(\lambda_1), . . . , q(\lambda_k)$) $+ D$.

(Consider first the case of a connected semisimple group and apply a) to a suitably chosen finite family of representations of G).