This question is closely related to the following MO question http://mathoverflow.net/questions/185126/characterizing-the-real-analytic-eisenstein-series


Let $\mathfrak{h}=\{z=x+iy\in\mathbf{C}\}$ be the Poincare upper half plane endowed with its Poincare metric. Let $w\in\mathbf{Z}$ be a weight and define the "weight $w$ Laplacian" on $\mathfrak{h}$ by 
$$
\Delta[w]=-y^2(\partial_x^2+\partial_y^2)+i\cdot w\cdot y\cdot\partial_x.
$$
Here $i=\sqrt{-1}$. In general, $\Delta[w]$ will commute with the $|_{w}$ right action of $GL_2^+(\mathbf{R})$ on Maass forms as is explained on page 130 of Bump's book on automorphic forms and representations.

Let $g\in\mathbf{Z}_{\geq 1}$ be a fixed integer and consider the symmetric space $\mathfrak{h}^g$. 
Choose (arbitrarily) a totally real field $K$ of degree $g$. Choose also (arbitrarily) a discrete subgroup $\Gamma\leq GL_2^+(K)$ which acts discontinuously on $\mathfrak{h}^g$ (by Moebius transformation through the distinct embeddings of $K$) and which has finite covolume. The specific choices of $K$ and $\Gamma$ **are not important** here.

Let $\underline{z}=(z_j)_{j=1}^g\in\mathfrak{h}^g$ where $z_j=x_j+iy_j$. We let 
$D_j[w_j]$ be the weight $w_j$ Laplacian with respect to the variable $z_j$.
Let $\underline{w}=(w_1,w_2,\ldots,w_g)\in\mathbf{Z}^g$ be an integral weight vector. For
$\gamma=\left(\begin{matrix} a & b \\ c& d\end{matrix}\right)\in \Gamma$ and 
$\underline{z}\in\mathfrak{h}^g$ we let
$$
j(\gamma,\underline{z})=(c^{(j)}z_j+d^{(j)})_{j=1}^g
$$
be the usual $1$-cocycle of $\Gamma$ taking values in the ring of analytic functions going from $\mathfrak{h}^g$ to $\mathfrak{h}^g$. We also let 
$$
\omega_{\underline{w}}(j(\gamma,\underline{z})):=\prod_{j=1}^g |c^{(j)}z_j+d^{(j)}|^{2s-w_j}
$$
be a (convenient) automorphic factor of weight $\underline{w}$. Now consider a function
$$
(\star) \;\;\;\;\;\;\;\;\;\;\;\;\;\; F(Im(\underline{z}),s):=A(s)\prod_{j=1}^g y_j^{s-w_j/2}+B(s)\prod_{j=1}^g y_j^{1-s+w_j/2}
$$
where $A(s)$ and $B(s)$ are say (for the sake of being precise) holomorphic functions in $s$. Here $Im(\underline{z})=\underline{y}=(y_1,y_2,\ldots,y_g)$.

Then the function $F(Im(\underline{z}),s)$ satisfy the following two properties

**(1)** For $1\leq j\leq g$, the linear differental operator 
$$
D_j:=\Big(\Delta_j[w_j]-(s-w_j/2)(1-s+w_j/2)\Big)
$$
kill $F(Im(\underline{z}),s)$.

**(2)** For all $\gamma\in \Gamma$, $F(Im(\gamma \underline{z}),s)=\omega_{\underline{w}}(\gamma,\underline{z})F(Im(\underline{z}),s)$.

**Question 1:** Let $G(Im(\underline{z}),s)$ be a real analytic function in $\underline{y}$ and holomorphic function in $s$ which satisfy (1) and (2); Does it follow that $G(Im(\underline{z}),s)$ has the same form as the expression in $(\star)$?

**Question 2:** Assuming that the answer to question 1 is positive, what is the natural context to phrase this question? It seems to me that similar questions could be phrased for other symmetric spaces acted on by arithmetic subgroups of finite covolume.

**Remark:** 

(a) We note that condition (1) is independent of the specific choices of $K$
and $\Gamma$. 

(b) The answer to Question 1 is positive if $g=1$. In this case one has a homogenous linear ODE of order $2$ which can be easily solved. In fact, condition (1) alone is enough to guarantee that the solution is as in $(\star)$. 

(c) The fact that $F(Im(\underline{z}),s)$ does not depend on the variables 
$(x_1,x_2,\ldots,x_g)$ simplifies the way that $\Delta_j[w_j]$ acts on $F(Im(\underline{z}),s)$. But I thought it was more natural to keep $\Delta[w]$ as I originally defined it, i.e., as the weight $w$ Laplacian.  

(d) If one only looks at condition (1), then $\bigcap_{j=1}^g\ker(\Delta_j[w_j])$ is infinite dimensional if $g\geq 2$. This was my motivation for adding the "artificial" condition (2) in the hope of obtaining uniqueness. But there are probably better ways of doing this.