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 $D^\star\_A$.


Questions:


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?


Thanks.


[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]



Finite dimensional automorphic representations of a definite quaternion with prime discriminant and Hecke action
1) Yes, I think that's true. I guess it follows relatively easy from the statement that an automorphic representation of the algebraic group $D^\times$ is finite-dimensional iff it's 1-dimensional and factoring through the norm. This latter statement probably follows from strong approximation applied to (the adelic points of the algebraic group associated to) the norm 1 elements of $D^\times$, or you can apply Jacquet-Langlands and move to $GL(2)$ and basically use the standard classification of automorphic representations there.


2) Yes, I think it does. There's some analogue of the Petersson inner product here, right? It's just a combinatorial thing and you can use this. The space $S(1,n)$ is not a mysterious thing: multiplicity 1 etc is all true in this setting and the space is a sum of generalised eigenspaces just like classical spaces of cusp forms.


3) Yes. Use either Jacquet-Langlands and classical results, or mimic the standard proof for $GL(2)$ using the pairing on $S(1,N)$.


4) This question has a bit more meat to it. You seem to be asking what the Jacquet-Langlands correspondence is explicitly. I know something this but it's a bit messy. Let me write down some stuff; the answer isn't quite as "clean" as you'd like it to be though, perhaps.


In $S\_0(N)$ you're seeing automorphic representations which have a fixed vector under $R\_p^\*(1)$. These come in two flavours. The first are those which actually have a fixed vector under all of $R\_p^\*$. These contribute one dimension, locally, to $S\_1(N)$ and under J-L correspond to forms of level $Np$ which are new at $p$ with trivial character. The second sort are those without an $R\_p^\*$-fixed vector. These are 2-dimensional (I mean their $\pi\_p$ is 2-dimensional) [Edit: as Tomasso pointed out, this isn't true. There are some more 1-dimensional ones factoring through the local norm map and coming from tamely ramified but not unramified quasicharacters; let's ignore them in what follows.] and all of $\pi\_p$ is fixed by $R\_p^\*(1)$, so here you're getting 2-dimensional spaces in $S\_0(N)$ on which Hecke is acting as scalars (so a super-naive version of multiplicity 1 is failing; what's really going on is that the classical theory of oldforms and newforms works in a completely different way to the behaviour of $S\_0(N)$ at $p$). Applying JL, if memory serves, you see forms which are supercuspidal at $p$; their level will be something like $Np^2$, their character will be something like the norm of the character of $\mathbf{F}\_{p^2}$ associated to the form on $S\_0(N)$. So the answer to your question is "yes, you can write down a classical space of forms that corresponds to $S\_0(N)$" but it's not pleasant to write it down---you have to understand the possibilities of what forms of level $Np^2$ look like, and only choose certain of them. It's much easier to understand from the point of view of representation theory, either of $GL(2,\mathbf{Q}\_p)$ or the local Weil group. 


All this is very well-understood by lots of people here, so feel free to correct me or ask more.


