Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $GL_2(o/p)$.

share|improve this question

1 Answer 1

up vote 2 down vote accepted

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.

share|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.