I am currently aiming at estimating orbital integrals. Maybe surprizingly, I hope for some help in the compact case (ramified places), in proving the usual formula



> 
> $$O\_\gamma(f) = \int\_G f(x^{-1}\gamma x)dx = \Theta\_\pi(\gamma) dim(\pi)^{-1}$$
> 
> 
> 


where $f$ is a suitably normalized matrix coefficient for $\pi$. There are general proofs of this fact (for instance in Arthur or Kottwitz, using Peter-Weyl theorem or properties of supercuspidal representations), yet I would rather find an easiest way to do so in the compact case.


For instance, it is quite easy to prove 


$$O\_\gamma(f) = \Theta\_\pi (f) \Theta\_\pi (\gamma)$$


so is there any way to see that $\Theta\_\pi(f) = dim(\pi)^{-1}$ (if true) ?


Thank you in advance for any clue or reference.



Orbital integral for matrix coefficients
You said in comments that you are OK with using orthogonality of matrix coefficients, but let's suppose you changed your mind. Let $\langle\cdot, \cdot\rangle$ be a $G$-invariant pairing on the space $V$ of $\pi$, and let $\mathrm dg$ be the Haar measure on $G$ normalised to give it total mass $1$. What follows is all very standard, so I assume it's not what you want, but let's have something out here so that you can clarify where it falls short.


Note that, for each $b, w \in V$, we have that $v \otimes v' \mapsto {\displaystyle\int} \langle g b, v'\rangle\cdot\overline{\langle g w, v\rangle}\mathrm dg$ is an element of $\operatorname{Hom}\_G(V \otimes V, \mathbb C)$, hence, by Schur's lemma, a scalar multiple of $\langle\cdot, \cdot\rangle$; and that the map sending $b \otimes w$ to this scalar is itself is an element of $\operatorname{Hom}\_G(V \otimes V, \mathbb C)$, hence again a scalar multiple of $\langle\cdot, \cdot\rangle$. Call this scalar $c$. (Note the illustration of Yemon Choi's [claim](https://mathoverflow.net/questions/235809/orbital-integral-for-matrix-coefficients/235855#comment583565_235809) that Schur's lemma lies at the bottom of everything!)


Now
$$
\Theta\_\pi(g) = \sum\_{b \in B} \langle g b, b\rangle,
$$
where $B$ is an orthonormal basis of the space $V$ of $\pi$; so, if we write $f\_{v, w}$ for the matrix coefficient $g \mapsto \langle v, g w\rangle$, then
\begin{align\*}
\Theta\_\pi(f\_{v, w})
& {}= \sum\_{b \in B} \int \langle g b, b\rangle\cdot\langle v, g w\rangle\mathrm dg \\
& {}= \sum\_{b \in B} \int \langle g b, b\rangle\cdot\overline{\langle g w, v\rangle}\mathrm dg \\
& {}= \sum\_{b \in B} c\langle b, w\rangle\cdot\overline{\langle b, v\rangle} \\
& {}= c\langle v, w\rangle = c f\_{v, w}(1).
\end{align\*}
On the other hand,
\begin{align\*}
f\_{v, w}(1)
& {}= \int \langle g v, g w\rangle\mathrm dg \\
& {}= \int \sum\_{b \in B} \langle g v, b\rangle\cdot\overline{\langle g w, b\rangle}\mathrm dg \\
& {}= \sum\_{b \in B} c\langle v, w\rangle\cdot\langle b, b\rangle \\
& {}= c\cdot\lvert B\rvert\cdot f\_{v, w}(1) = c\cdot\dim(\pi)f\_{v, w}(1).
\end{align\*}
Since this is true for all $v, w \in V$, we have that $c = \dim(\pi)^{-1}$.


