The theta series for a lattice $\Lambda$ is defined by $$\displaystyle \Theta_\Lambda(q) = \sum_{x \in \Lambda} q^{x \cdot x}.$$ Setting $q=e^{-\pi\tau}$ yields the (maybe more usual) related theta series $$\displaystyle \theta_\Lambda(\tau) = \sum_{x \in \Lambda} e^{-\tau\pi\|x\|^2}.$$ By Poisson formula one gets the functional equation fulfilled by $\theta$: $\theta_\Lambda(\tau) = \frac{1}{\tau^{n/2}\textrm{det}(\Lambda)}\theta_{\Lambda^\lor}\left(\frac{1}{\tau}\right)$. It allows to derive some decay information on $\theta_\Lambda$ such as: $\theta_\Lambda(\tau)\leq \tau^{-n/2}\theta_\Lambda(1)$.
If we want to generalise the definition of the theta series to sublattices (instead of only vectors) we could think of something like: $$\displaystyle \theta^{(r)}_\Lambda(\tau) = \sum_{\begin{aligned}\ell~ \textrm{su}&\textrm{blattice of}~\Lambda\\ &\textrm{rk}(\ell) = r\end{aligned}} e^{-\tau\pi\cdot\textrm{det}(\ell)^2}, $$ for a fixed integer $1\leq r\leq \textrm{rk}(\Lambda)$. One could also restrict the latter sum to only primitive sublattices.
However since the set of sub-lattices of fixed rank is not a lattice itself (it is only embedded in the r-exterior power lattice constructed from $\Lambda$), harmonic analysis such as Poisson formula seems compromised.
Is it possible to derive a functional equation similar to the classical one for regular $\theta$? Or maybe in a direct manner get some decay bounds ?
Note: in the case of primitive sublattice one can relate the $\theta^{(r)}_\Lambda$ series with $\theta^{\textrm{rk}(\Lambda)-r}_{\Lambda^\lor}$ by relating a rank $r$ primitive sub-lattice $\ell\subset\Lambda$ with a sub-lattice of corank $r$ in $\Lambda^\lor$ of determinant $\det(\Lambda)\cdot\det(\ell)$
