Let $G$ be a compact, connected Lie group and $\rho$ any of its irreducible, unitary representations. If $\gamma:S^1\to G$ is an injective homomorphism (a periodic geodesic passing through the identity), what can be said about the matrix valued integral $$ I(\rho,\gamma)=\int_{S^1}\rho(\gamma(t))dt? $$ More explicitly, I am interested in the following questions:

- Given $\rho$, can we find a $\gamma$ so that $I(\rho,\gamma)$ is invertible?
- Given $\rho$, can we find geodesics $\gamma_i$ and complex numbers $a_i$ so that $\sum_ia_iI(\rho,\gamma_i)$ is invertible?
- If a square matrix $A$ (of same dimension as $\rho$) satisfies $AI(\rho,\gamma)=0$, what can be said about $A$? This condition holds if $A=B\left.\frac{d}{dt}\rho(\gamma(t))\right|_{t=0}$ for some constant matrix $B$. Are all matrices $A$ satisfying this condition of this form?

The case of the torus $(\mathbb R/\mathbb Z)^n$ is quite simple. Then $\rho$ is parametrized by $\mathbb Z^n$ and $\gamma$ by $\mathbb Z^n\setminus\{0\}$, and $I(\rho_m,\gamma_k)=1$ if $m\cdot k=0$ and $I(\rho_m,\gamma_k)=0$ otherwise. (I have normalized the measure of $S^1$ to $1$.) Also, the irreducibles have dimension one, so we do not have a true matrix problem. Is there something as neat for nonabelian groups?

Although I am looking for a general answer, any examples for any groups are welcome.