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?
