# Orbital integrals of pseudo coefficients of supercuspidal reps

Let $\pi$ be a supercuspidal representation of $G =GL_2(F)$ for a non-archimedean local field $F$, then there exists a maximal subgroup $K$ of $G$, which is compact modulo the center, and a representation $\rho$ of $K$ such that $\pi = Ind_K^G \rho$.

It is possibly to show that $tr\; \sigma( \phi) \neq 0$ iff $\sigma \cong \pi$ for $\phi$ being equal to $tr(\rho)$ on $K$ and zero off $K$. This means $\phi$ is a constant multiple of a pseudo-matrix coeffient of $\pi$.

Now, one can compute that given an elliptic element $\gamma \in GL_2(F)$, i.e., the characteristic polynomial is irreducible, the corresponding elliptic orbital integral vanishes iff the conjugacy class of $\gamma$ doesn't meet $K$ and equals a constant multiple of $tr \rho(\gamma)$ with $\gamma$ conjugated inside $K$ otherwise.

There exists a classification/construction of those $\rho$'s respective $\pi$'s, see eg. Bushnell-Henniart --- Local Langlands conjecture for GL(2).

Question: Does there exists a reference for the explicit value of $tr \rho(\gamma)$ depending on the strata of $\rho$ and the characteristic polynomial of $\gamma$?

Remark: The depth-zero case is well documented in the representation theory of $GL_2(o/p)$.

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In fact $\phi$ is not only a (multiple of a) pseudo coefficient, but is a (multiple of a) coefficient of $\pi$. See e.g. Carayol's article "Représentations cuspidales du groupe linéaire", Ann. ENS.

Now to answer your main question, there is indeed a lot of such computations in e.g. the series of papers written by Bushnell and Henniart on explicit Jacquet-Langlands correspondence. For other references you may read the survey :

Sally, Paul J., Jr.; Spice, Loren Character theory or reductive $p$-adic groups. Ottawa lectures on admissible representations of reductive $p$-adic groups, 103–111, Fields Inst. Monogr., 26, Amer. Math. Soc., Providence, RI, 2009.

If youo are especially interested in supercuspidal representation of ${\rm GL}(2)$, you may read :

Kutzko, Phil; Pantoja, José Character formulas for supercuspidal representations of the groups ${\rm GL}_2,\ {\rm SL}_2$. Comm. Algebra 26 (1998), no. 6, 1679–1697.

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Sweet, I will have look. Thanks also for the comment about matrix vs pseudo coefficients of $\pi$. –  Marc Palm Apr 30 '13 at 11:02
I know that the for pseudo coefficients (also because of one of your excellent answers) that the value of the trace (seen as locally integrable function $\theta_\pi$) at $\gamma$ equals the orbital integral of $\gamma$ and an arbitrary pseudo coefficient. So I guess you suggest to extract the computation and the value of $\theta_\pi$ at $\gamma$ from the articles, or not? Is this really so easy that I can just quote the result or does it require a further computation? –  Marc Palm Apr 30 '13 at 11:12
No the computations are done. –  Paul Broussous Apr 30 '13 at 11:24
Sadly, the last reference omits the most complicated part. –  Marc Palm Apr 30 '13 at 17:11