# Integral of the reflection operator in arbitrary symmetric spaces.

Just as the title says, suppose $X$ is a symmetric manifold and $\hat{S}(x)$ is the linear operator associated to $\sigma_x\in G$ for some unitary irreducible representation,
where $\sigma_x$ is the group element that performs reflections around $x$ (remember $X=G/H$ for $H\subset G$).

Now take the integral

$\int_X d\mu(x) \hat{S}(x)$,

where $d\mu(x)$ is the (normalized) reimannian measure in $X$.

Then, by covariance and using Schur's Lemmas, one can show that

$\int_X d\mu(x) \hat{S}(x)=c\hat I$

for some real value $c$.

Using heuristic arguments one can infer that $c=1/2$. I have calculated this integral explicitly for different cases ($X=\mathbb{R}^{2n},S^2,\mathbb{H}^2$) and it's always the same.
Of course it doesn't say the integral will take that value in the general case.

This looks like a very simple problem and it probably is, sadly I haven't been able to come up with a rigorous proof. Might as well be a well known result from group
theory, but a quick research in the literature gave no results. Group theory is not my expertise field, so any suggestions on this matter are most welcome.

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Actually, after repeating the evaluation of the integral on those manifolds more cautiously I found that for $S^2$ the integral is exactly the identity operator,
while for $\mathbb{H}^2$ the factor $1/2$ remains. For $\mathbb{R}^{2n}$ the situation changes too as $c=2^{-n}$.