  Let $G$ be a compact Lie group, $N\geq 1$ and $L>0$. The random geometric
graph on $G$ is the random graph $\Gamma(N,L)$ whose vertices are $N$ random
points $g_1,\ldots,g_N$ chosen under the Haar measure of $G$, and whose edges
are the pairs $\{g_i,g_j\}$ with $d(g_i,g_j)\leq L$, $d$ being the distance
associated to the standard Riemannian structure on $G$. In this paper, we
describe the asymptotic behavior of the spectrum of the adjacency matrix of
$\Gamma(N,L)$, when $N$ goes to infinity. If $L$ is fixed and $N \to + \infty$
(Gaussian regime), then the largest eigenvalues of $\Gamma(N,L)$ converge after
an appropriate renormalisation towards certain explicit linear combinations of
values of Bessel functions. If $L = O(N^{-\frac{1}{\dim G}})$ and $N \to
+\infty$ (Poissonian regime), then the random geometric graph $\Gamma(N,L)$
converges in the local Benjamini-Schramm sense, which implies the weak
convergence in probability of the spectral measure of $\Gamma(N,L)$. In both
situations, the representation theory of the group $G$ provides us with
informations on the limit of the spectrum, and conversely, the computation of
this limiting spectrum involves many classical tools from representation
theory: Weyl's character formula and the weight lattice in the Gaussian regime,
and a degeneration of these objects in the Poissonian regime. The
representation theoretic approach allows one to understand precisely how the
degeneration from the Gaussian to the Poissonian regime occurs, and the article
is written so as to highlight this degeneration phenomenon. In the Poissonian
regime, this approach leads us to an algebraic conjecture on certain
functionals of the irreducible representations of $G$.
