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I am looking for a reference/idea, how this passage from Labesse's Snowbird Lecture "Introduction to endoscopy" pg.5 can be explained:

"We shall denote by $f_\pi$ a pseudo-coefficient for $\pi$, although it is highly non unique. But as regards invariant harmonic analysis this plays no role. In particular the orbital integrals are independent of the choice of the pseudocoefficient; they are also independent of the choice of the Haar measure on $G(F)$ but one has to use the canonical measure on the compact torus $G (F)$. The orbital integrals of $f_\pi$ are easily computed for regular semisimple: $$O_\gamma(f_\pi) =\begin{cases} \Theta_\pi(\gamma), & \gamma \mathrm{ elliptic}, \newline 0, &\mathrm{else},\end{cases} $$ where $\Theta_\pi(\gamma)$ is the character of $\pi$."

Here $G$ is a reductive group over a local field $F$ and $\pi$ a squareintegrable representation.

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up vote 3 down vote accepted

Existence of pseudo-coefficients for square-integrable representations (and the link with character values of the representations) is stated and proved in

D. Kazhdan, Cuspidal geometry of $p$-adic groups. J. Analyse Math. 47 (1986), 1–36.

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I have seen this reference, in fact Labesse gives it himself. The existence is not the issue. What I want to understand/reference is the statement about the orbital integral. Thanks nevertheless. –  plusepsilon.de May 14 '12 at 10:56
You also have the exact statement on the link bewteen orbital integral of a pseudo-coefficient and the character of the representation in Kazhdan's paper. –  Paul Broussous May 14 '12 at 13:32
Okay, I was searching the document for keywords, probably not the correct ones;) I will have a more careful look. Thank you. –  plusepsilon.de May 14 '12 at 16:16
I find Kazhdan's paper difficult to read ! –  Paul Broussous May 14 '12 at 20:27
I went back again. It is theorem K on page 7. –  plusepsilon.de Jun 12 '12 at 15:48
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