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To fix notation, if $G$ is a compact Lie group, $Rep(G)$ denotes the set of continuous irreducible unitary representations of G, and $\widehat{G}$ denotes the quotient $Rep(G)/\sim$, which identifies isomorphic representations. It is known that every element in $Rep(G)$ has a finite dimension. It is also known that for abelian compact Lie groups, every element of $Rep(G)$ must have dimension 1, which is the case for example for the torus $G=\mathbb{T}^n$. In particular, there is an upper bound for the dimension of elements of $Rep(G)$ in this case. On the other hand, in the case $G=SU(2)$, for each $\ell \in \frac{1}{2}\mathbb{N}_0$ there is exactly one element $t^\ell \in \widehat{SU(2)}$ which have dimension $2\ell+1$, so there is no upper bound for the dimensions of elements in $Rep(SU(2))$.

First question: if $G$ is a compact Lie group and $n \in \mathbb{N}$, when is possible to guarantee that there is $\phi \in Rep(G)$ with $\dim \phi=n$?

Second question: which kinds of compact Lie groups $G$ (besides abelian ones) have this kind of upper bound: there exists $N \in \mathbb{N}$ such that $\dim \phi \leq N$ for all $\phi \in Rep(G)$?

I know that for the second question there is a positive answer in the context of finite groups, and since compact Lie groups have some analogy with finite groups, I wonder if there is at least a partial answer for the second question.

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    $\begingroup$ The first question seems delicate but see the Weyl dimension formula. The answer to the second question should be that $G$ is virtually abelian, e.g. it could be a semidirect product $T^n \rtimes H$ of a torus by a finite group, but if the Lie algebra $\mathfrak{g}$ has a nontrivial simple factor there should be irreducibles of arbitrarily large dimension. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 6 at 0:32
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    $\begingroup$ It was shown in C.C. Moore: Groups with finite dimensional irreducible representations. Trans. AMS 166, 401–410 (1972) that a locally compact group has unitary representations of bounded degree iff it has a closed abelian subgroup of finite index. For compact groups this goes back to Kaplansky. $\endgroup$
    – Benjamin Steinberg
    Commented Dec 6 at 2:46

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Compact Lie groups have nice classification, see e.g. https://en.wikipedia.org/wiki/Compact_group#Compact_Lie_groups and references therein.

Theorem: Every connected compact Lie group is the quotient by a finite central subgroup of a product of a simply connected compact Lie group and a torus.

The simply connected compact Lie groups are finite products of simple factors which are isomorphic to either one of the five exceptional cases or to $Sp(n)$, $Spin(n)$ or $SU(n)$.

I think the moral is: "Once your group has a $SU(2)$ subgroup, the set of possible dimensions is unbounded."

For the simple factors, the set of possible dimensions is the image of the Weyl dimension formula which in each instance is a polynomial in nonnegative integer variables. E.g. for $SU(3)$ the representations $V_{m_1,m_2}$ are determined by two natural numbers $m_1, m_2\in\mathbb{N}$ and their dimensions are $$ \dim(V_{m_1,m_2})=\frac{1}{2}(m_1+1)(m_2+1)(m_1+m_2+2). $$

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