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
as is explained on page 130 of Bump's book on automorphic forms and representations.

Now let $K$ be a totally real field of degree $g\in\mathbf{Z}_{\geq 1}$ and consider the symmetric space $\mathfrak{h}^g$. Let $\Gamma\leq GL_2^+(K)$ be a discrete subgroup which acts discontinuously on $\mathfrak{h}^g$ (by Moebius transformation through the distinct embeddings of $K$) and which has finite covolume.
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.

**Remark:** 

(a) The answer 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) is enough since after inspection, one gets (2) for free. 

(b) 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.  

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