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.



Integrals of representations over geodesics
I had made a silly error in the first version of this answer.


To try to answer your first question, notice that the image of $\gamma$ is a circle subgroup $T$, say, of $G$ and that the integral $I(\rho,\gamma)$ is (with your normalisation of the measure on the circle) the projector onto the subspace $V^T$ of $T$-invariants in the restriction to $T$ of the representation $\rho: G \to GL(V)$. Hence $I(\rho,\gamma)$ is invertible if and only if $V^T=V$. (Previously I had written $V^T=0$ and hence reached the wrong conclusion.) As Venkataramana points out in the comment, this cannot happen if $\rho$ is a faithful representation, hence for simple $G$.


You could take $G = G\_1 \times G\_2$ and consider $\rho$ such that the $G\_2$ subgroup acts trivially and have $\gamma(S^1)$ be a circle subgroup of the form $(e,\gamma\_2(t)) \in G\_1 \times G\_2$, with $e \in G\_1$ the identity element.


