Let $D$ be a totally definite quaternion algebra over a totally real number field $F$. Let $U$ be an open compact subgroup of $D(\mathbb{A}_F)^\times$, maximal compact almost everywhere. Consider automorphic forms on $D$ as functions on the usual adelic double quotient (which here is a finite set). There are Hecke operators etc. as in the $GL_2$ setting.

Suppose $f$ is a Hecke eigenform, giving rise to an automorphic representation $\pi_f$ with local components $\pi_{f,\nu}$.

What is knows about these $\pi_{f,\nu}$?

I've read claims that they are always Steinberg (or unramified?). What would be a good reference for this?

More specific (but kind of sloppy): Let $p$ be a fixed rational prime, $\rho_f$ be the mod $p$ Galois representation associated to $f$ and call $\nu$ a Taylor-Wiles prime of $F$, if

- $\rho_f$ is unramified at $\nu$ with eigenvalues $\alpha\neq\beta$ of $\rho_f(Frob_\nu)$
- $D$ splits at $\nu$
- $U_\nu$ is contained in the Iwahori subgroup and $det(U_\nu)\equiv 1 $ in the $p$-power quotient of $k_\nu^\times$
- the Hecke operator at $\nu$ has eigenvalue $A_\nu$ on $f$; where $A_\nu$ is a lift of one of $\alpha$.

Then, supposedly, at Talor-Wiles primes $\nu$ it holds that $\pi_{f,\nu}$ is not Steinberg but tamely ramified principal series.

How is this? And how does this fit together with the unramifiedness asserted above?

notsatisfied. In this way, the possibility of Steinberg at the TW primes is ruled out. (Look in any discussion of the TW method, e.g. the one in theFermat's Last Theorem book, and you will see this explained.) Regards, $\endgroup$1more comment