Eisenstein series over a definite division algebra

Let $D$ be the definite quaternion division algebra over $\mathbb{Q}$. $\mathcal{O}$ is a maximal order inside $D$, let's fix $\mathcal{O}$ to be the Hurwitz quaternion. Let $\Gamma=PGL_2(\mathcal{O})$ and $\Gamma_\infty$ is the stabilizer of the unique cusp of $\Gamma\backslash PGL_2(\mathbb{H})/K$, where $\mathbb{H}=D\otimes_\mathbb{Q}\mathbb{R}$ the Hamilton's quaternions.

I want to find a literature regarding analytic continuation and functional equation for the Eisenstein series related to $\Gamma_\infty\backslash\Gamma$. Intuitively it may look like a mimic proof of the cases $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})$ or $SL_2(\mathbb{Z[i]})\backslash SL_2(\mathbb{C})$, but a significant problem is arising regarding non-commutativity of $\mathcal{O}$.

One could possibly get away using Langlands's work, but we are still in "rank $1$ and unique cusp" case I am expecting an easier way than Langlands, something like Colin de Verdière's work for $SL_2(\mathbb{R})$.

Thanks for any help and reference.

• Please help me to understand what the cusp is. I recall that quaternion division algebras give rise to compact quotients, hence no cusps and Eisenstein series. The trace formula becomes simple and one gets the Jacquet-Langlands correspondence with forms with at least two square-integrable components. – Marc Palm Jun 27 '14 at 13:45
• Suppose $G$ is the algebraic group defined by its rational points $G(\mathbb{Q})=GL_2(D)$ and $P$ is the standard $\mathbb{Q}$-parabolic subgroup of $G$. For an arithmetic subgroup $\Gamma\subset G(\mathbb{Q})$ the cosets $\Gamma\backslash G(\mathbb{Q})/P(\mathbb{Q})$ are called set of $\Gamma$-cusps. – Subhajit Jana Jun 27 '14 at 19:04
• Using strong approximation (i.e. class number of $G$ respect to $\prod GL_2(\mathcal{O}_p)$ is one) one can prove that there is a unique cusp. – Subhajit Jana Jun 27 '14 at 20:57
• Ah okay, I see my mistake. I was thinking $GL(1)$ over a division algebra, you are considering $GL(2)$. Moeglin-Waldspurger do not consider this setting in their book? – Marc Palm Jun 30 '14 at 12:15

If you are asking about right $K$-invariant Eisenstein series, then Y. Colin de Verdiere's approach is easily applied, using the functional $\eta_a$ (the analogue called $T_a$ in Colin de Verdiere) that evaluates constant term, and evaluates at some fixed height $a$. Thus, this functional is an integration over a codimension-one sub-object of $\Gamma\backslash SL_2(\mathbb H)/K$, so is in $H^{-1/2-\epsilon}$ for every $\epsilon>0$, and the set-up for the Friedrichs extension (as in Lax-Phillips and in CdV) will work. The same argument for the compactness of the inclusion $H^1\to L^2$ (all cut off at height $a$) succeeds as in Lax-Phillips, using Fourier expansions along the unipotent radical.