Before stating the questions that I have, which are very specific and probably not so interesting to someone who has never thought about these things, I need to introduce some notation.
Let $p$ be any prime, and let $D$ be the quaternion algebra over $Q$ ramified precisely at $p$ and infinity. Choose a maximal order $R$ inside $D$. For any prime $\ell\neq p$, fix an isomorphism of $D_\ell:=D\otimes Q_\ell$ with the algebra $M_2(Q_\ell)$ in such a way that the maximal compact subring $R_\ell:=R\otimes Z_\ell$ corresponds to $M_2(Z_\ell)$.
If $N$ is an integer $>0$ not divisible by $p$, let $U_\ell(N)$ be the subgroup of $R_\ell^\star\simeq GL_2(Z_\ell)$ given by those matrices whose bottom row is congruent to $(0$ $1)$ modulo $N$ (equivalently, modulo the highest power of $\ell$ dividing $N$).
The ring $R_p:=R\otimes Z_p$ has a unique maximal, two-sided principal ideal $(\pi)$ generated by any uniformizer $\pi$, the residue field $R/(\pi)$ is a finite field with $p^2$ elements. We let $R_p^\star(1)$ denote the subgroup of $R_p^\star$ given by the units that are congruent to $1$ modulo $(\pi)$.
Let now $D^\star$ be the multiplicative group of $D$, viewed as an algebraic group over $Q$. For any integer $N>0$ not divisible by $p$, we are going to define an open subgroup $U(1,N)$ of the group $D^\star_A$ of points of $D^\star$ valued in $A$, the adele ring of $Q$. Namely $U(1,N)$ is the product of all the $U_\ell(N)$, for $\ell\neq p$; of $R_p^\star(1)$; and of the full (connected) component at infinity $D^\star_\infty$.
Consider the space $S(1,N)$ of complex valued functions on the double coset $D^\star\backslash D^\star_A/U(1,N)$, which is known to be finite. For any prime $\ell\neq p$ there is an Hecke operator $T_\ell$ acting on $S(1,N)$ that can be defined in terms of double cosets in the usual way.
(Let $\alpha_\ell$ be the matrix whose top row is $(\ell$ $0)$ and whose bottom is $(0$ $1)$; decompose $U_\ell(N)\alpha_\ell U_\ell(N)$ as a finite union of left cosets $\gamma_i U_\ell(N)$; for $f\in S(1,N)$ define $T_\ell(f)(x)=\sum f(x\gamma_i)$).
Let $V$ be the vector space of locally constant, complex valued functions on $D^\star_A$ that are left invariant by $D^\star$. Observe that $S(1,N)$ can be viewed as a finite dimensional subspace of $V$. Right translation defines an admissible representation of $D^\star_A$ on $V$ which is known to be completely decomposable into a discrete direct sum of irreducible admissible representations of $D^*_A$. If $f\in S(1,N)$, then denote by $V_f$ the smallest subspace of $V$ that is stable by $D^\star_A$.
1) Let $f\in S(1,N)$, for some $N$. Is it true that the space $V_f$ is finite dimensional if and only if $f:D^\star_A\rightarrow C$ factors through the reduced norm map $Nr:D^\star_A\rightarrow A^\star$?
2) Does the subspace of $S(1,N)$ given by those functions that factor through the reduced Norm admit an Hecke stable complement?
3) Is the action of $T_\ell$ on $S(1,N)$, for $\ell\nmid pN$, semisimple?
4) Assuming that 2) holds, and letting $S_0(1,N)$ be such complement, how do we relate the C-subalgebra $T_0(N)$ of End($S_0(1,N)$) generated by the Hecke operators $T_\ell$, with $\ell\nmid pN$, to a C-algebra of Hecke operators acting on weight 2 cusps forms of a certain level? What I mean is: out of the J-L correspondence, can we read off an isomorphism between $T_0(N)$ and some Hecke algebra coming from classical modular forms?
[EDIT: In the 2nd and 3rd lines above "Questions:" I should have probably have said "discrete Hilbert direct sum", the "direct sum" being only dense in V]