Why is probability on Lie groups nice? The question may sound weird. I am a probabilist who has heard of works connecting probability with (compact) Lie groups. What is the motivation? Is it generalisation just for the sake of generalisation? 
Applebaum’s book is the classic but I am firstly interested to know why doing probability on Lie groups is a worthwhile effort. I know little lie theory from a differential geometric point of view. Is it enough for reading about connections between Lie groups and probability?
 A: Any compact (Hausdorff) topological group $G$ admits a left-invariant measure called Haar measure which is unique up to scale, and is unique if we require that it's a probability measure. This measure allows us to make sense of what it means to pick a random element of $G$, which is useful and interesting for various purposes. 
In representation theory, integrating with respect to Haar measure is used to generalize various results from the representation theory of finite groups to compact groups, and is crucial in proving results like the Peter-Weyl theorem. For example, if $V, W$ are two finite-dimensional representations with characters $\chi_V, \chi_W : G \to \mathbb{C}$, then we can show that
$$\dim \text{Hom}_G(V, W) = \int_G \overline{\chi_V}(g) \chi_W(g) \, d \mu$$
which exactly generalizes a finite group formula, where $d \mu$ is the Haar probability measure. 
If $\chi_V$ is real-valued, this formula implies that the moments of the random variable $\chi_V : G \to \mathbb{R}$ are given by the dimensions of the $G$-invariant subspaces of the tensor powers $V^{\otimes n}$. This implies, for example, that the trace of a random element of the Lie group $SU(2)$ has a Wigner semicircular distribution, and this is the simplest example I know of a "natural" random variable with such a distribution. This is related to various topics in random matrix theory (although the distributions on matrices studied in random matrix theory are usually not the Haar distributions) as well as to the Sato-Tate conjecture in number theory. 
