= Solution
The usual <Weyl character formula> requires \b[a <compact> <connected> <Lie group>]. Choose a <maximal torus> $T$, a positive <root system> $\Phi^+$, <Weyl group> $W$, and $\rho=\frac12\sum_{\alpha\in\Phi^+}\alpha$. Irreducible representations are indexed by dominant integral highest weights $\lambda$ belonging to the <character> <lattice> of the actual <group> $T$. For $X\in L(T)$ such that $\exp X$ is regular,
$$
\boxed{\chi_\lambda(\exp X)=
\frac{\sum_{w\in W}\det(w)e^{w(\lambda+\rho)(X)}}
{\sum_{w\in W}\det(w)e^{w\rho(X)}}}.
$$
Weights are written as linear functionals on the complexified <torus> algebra; their values on the <compact> real <torus> algebra are imaginary. The denominator is $e^{\rho(X)}\prod_{\alpha>0}(1-e^{-\alpha(X)})$. If $\rho$ is not itself a <character> of $T$, the individual expressions are interpreted on its covering algebra; their quotient is single-valued and descends to $T$. The apparent singularities at nonregular elements are removable, and conjugacy into a <maximal torus> determines the <character> on the whole <connected> <group>.
The formula explicitly determines all irreducible characters from their highest weights, and its limit at the identity gives the dimension formula
$$
\dim V_\lambda=\prod_{\alpha>0}\frac{\langle\lambda+\rho,\alpha^\vee\rangle}{\langle\rho,\alpha^\vee\rangle}.
$$
<Character> multiplication and <orthogonality> then compute tensor-product multiplicities. For $SU_2$, highest weights are integers $m\ge0$, and at $\operatorname{diag}(z,z^{-1})$,
$$
\chi_m=\frac{z^{m+1}-z^{-m-1}}{z-z^{-1}}=z^m+z^{m-2}+\cdots+z^{-m},\qquad\dim V_m=m+1.
$$
<Connectedness> cannot simply be omitted from this version. For a nontrivial finite <group> the <maximal torus> is trivial, yet characters can differ away from the identity. The root data on that <torus> alone cannot supply the characters of all disconnected <compact> groups.
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