Integrating adjoint representation over compact Lie group

Question

Let $ G $ be a compact Lie group. Let $ M \in Lie(G) $. Is there a well known formula for $$ \int_G Ad_g(M) $$ where $ \int_G $ is an integral with respect to the Haar measure? For example can this integral be expressed in terms of $ M $ in some way? For example a formula like $$ \int_G Tr(kg) Ad_g(M)=[k,M] $$

Answer 1

This may not be a "formula" per se, but it may be useful to understand the expression.

Let $G$ be a compact Lie group acting on a vector space $V$, and let $V_0\subseteq V$ be the subspace of elements fixed by this $G$-action. Define a function $\pi_0:V\to V$ by $$ \pi_0(v)=\int_Gg\cdot v\ dg $$ (where $dg$ is the normalized Haar measure) and note that $\pi_0(v)$ is always $G$-invariant. Using this, you can show that $V$ splits into a direct sum of subrepresentations $V=V_0+V_1$, and that $\pi_0$ is exactly the projection onto the first factor.

Using this description to comupute $\pi_0$ in concrete cases is somewhat circular, but it gives some useful information when applied to the adjoint representation. For instance, we know that $\pi_0$ is a projection onto $\operatorname{Lie}(Z(G))$, which uniquely specifies it for some simple cases like $T^n$ and $SO(n)$.