Modularity of certain theta series associated to hyperbolic lattice

Question

Let $L$ be an even hyperbolic lattice, i.e. a free $\mathbb{Z}$-module with a non-degenerate inner product $\cdot$ valued in $\mathbb{Z}$ of signature $(1,n)$ such that the norm of every vector is even. Let $O(L)$ denote the isometries of $L$ and let $\mathcal{C}^+$ denote the set of vectors in $L$ of positive norm. Is it known if there are modular properties of the following series:

$$\Theta_L(q):=\sum_{v\in O(L)\backslash L\cap\mathcal{C}^+} \frac{1}{|Stab(v)|}q^{(v\cdot v)/2}$$

where the sum is over a set of orbit representatives. Note that generally hyperbolic lattices will have infinitely many vectors of a given positive norm, but there should be finitely many orbits of vectors of positive norm. So the above series "renormalizes" the usual sum taken for a positive definite lattice. For example, when $L=H$ is the lattice with quadratic form $2xy$, we get

$$\Theta_H(q)=\frac{1}{2}\sum_{a,b>0}q^{ab}= \frac{1}{2}\sum_{n>0} \sigma_0(n)q^n$$

where $\sigma_0$ is the number of divisors function. This seems similar to the Fourier expansion of Eisenstein series, though I have only ever seen $\sigma_i$ for odd $i$ appear. Since general lattices are hard to understand, I would also be curious about results with more restrictive hypotheses, such as $L$ being unimodular i.e. $L=II_{1,k}$.

Any references would be welcome, I googled a bit for series of the above form without success.

Answer 1

For the purpose of someone seeing this question, I'll describe what I've figured out so far, which I think elaborates on Paul's comment. Though a warning: This isn't my forte so there may be errors. An element $p\in \mathbb{H}^n$ determines a positive-definite line $V^+\subset L\otimes \mathbb{R}=\mathbb{R}^{1,n}$. Denote the orthogonal complement by $V^-$. One can form the following function on $\mathbb{H}^n\times \mathbb{H}$: $$\Theta(p,\tau):=\sum_{v\in L}q^{(v\cdot v)/2}|q|^{-v^-\cdot v^-}$$ where $v^-$ is the projection of $v$ onto $V^-$. Then $\Theta$ satisfies the nice transformation rule: $$\Theta(p,\gamma\cdot \tau)=\pm (c\tau +d)^{1/2}(c\overline{\tau}+d)^{n/2}\Theta(p,\tau).$$ Furthermore, $\Theta$ is absolutely invariant with respect to the action of $\Gamma:=O^+(L)$ on the $p$ variable. As far as I can tell, this is what is meant by a "theta correspondence"--one can either integrate $\Theta(p,\tau)$ against a modfular form $F(\tau)$ to produce a function of $p$, or one can integrate $\Theta(p,\tau)$ against a function of $p$ in a fundamental domain of $\Gamma\backslash \mathbb{H}^n$ to produce a function of $\tau$. To produce a series similar to the one above, one should integrate out the $p$ variable. Setting $$G(\tau):=\int_{\Gamma\backslash \mathbb{H}^n} \Theta(p,\tau)\,dp$$ where $dp$ is the hyperbolic volume form produces a relatively nice series $$G(\tau)=\sum_{v\in O^+(L)\backslash L}\frac{1}{|Stab(v)|} F(v\cdot v\,\textrm{Im}\,\tau)q^{(v\cdot v)/2}$$ where $F$ is a function of one variable which appears to undergo a rather drastic change of behavior on positive vs. negative values (details are left out, and in particular, the obits of isotropic vectors seem to be tricky, possibly requiring one to regularize the integral). In any case, the function $h(\tau):=\sqrt{\textrm{Im}\,\tau} \cdot \overline{G}(\tau)$ then satisfies the usual transformation rule for a weight $(n-1)/2$ modular form, but of course won't be holomorphic. Instead, I suspect that $h(\tau)$ may be a Maass form. Since the coefficients of $h$ of interest are those where $q$ has a negative power, it seems that the anti-holomorphic part of the Maass form is the relevant power series... Any thoughts or further elaboration on this speculation is welcome.