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.

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.

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

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.