# Automorphic forms annihilated by $I_1$ but not $I_2$ for finite codim ideals $I_1 \subsetneq I_2$

Suppose $G$ is a connected real reductive Lie group, $\mathfrak{g} = \text{Lie}(G)$, and $\mathcal{Z} = \mathcal{Z}[U(\mathfrak{g})]$ the center of the universal enveloping algebra of $\mathfrak{g}$.

Let $A$ denote the space of automorphic forms on $G$, which, for the sake of argument, I'll assume are functions $f:G_\mathbb{Q}\backslash G \to \mathbb{C}$ that are $K$-finite for some maximal compact $K$, $\mathcal{Z}$-finite, and of uniform moderate growth, for each $f \in A$. For each $d \in \mathbb{N}_0$, let $A_d$ denote the subspace of $A$ consisting of those automorphic forms $f$ for which there exists $C > 0$ such that $|Xf(g)| \leq C\|g\|^d$ for all $X \in U(\mathfrak{g}_\mathbb{C})$.

Every $f \in A$ is annihilated by a finite codimensional ideal $I$ in $\mathcal{Z}$. My question is about how strong a converse exists for this statement. Specifically, given finite codimensional ideals $I_1, I_2 \subset \mathcal{Z}$, if $I_1 \subsetneq I_2$, can one always find an automorphic form $f \in A$ that is annihilated by $I_1$ but not by $I_2$? What about finding $f \in A_d$? If not, are there stronger conditions one can impose on the ideals to ensure that this holds?

• If a given automorphic form has a continuous parameter, e.g., is an Eisenstein series (possibly with cuspidal data, etc.), its derivative with repect to that continuous parameter is only annihilated by a strictly higher power of the ideal annihilating the original. This is very easy, and/but probably exactly what you don't want to consider... ? – paul garrett Jul 22 '16 at 14:58
• @paulgarrett: Thanks for your response! Is there a way to begin with a finite codim ideal $I$ in $\mathcal{Z}$ and construct an Eisenstein series annihilated by $I$? I'm mostly interested in whether or not a characterization of the annihilators of automorphic forms exists that distinguishes them in the finite codimensional ideals of $\mathcal{Z}$, and (better yet!) characterizations that allow for distinguishing annihilators of automorphic forms of a fixed growth rate. – Dan Jul 22 '16 at 15:50
• First, if you allow Eisenstein series of minimal parabolics, taking derivatives gives a lot of different ideals. E.g., for $GL_n$ the Harish-Chandra isom is easy to understand. No global constraints, just local. If you constrain growth, this tends to discretize things, but apart from Arthur/Selberg type conjectures, I don't think we know what can appear as cuspidal spectrum, for example. – paul garrett Jul 22 '16 at 16:27