Let $L$ be an even lattice and $L'$ its dual lattice . Consider the sum $$\sum_{\delta\in L'/L}\sum_{\mu\in L'/L}e^{-2\pi i (\mu,\delta)},$$ where $(\mu,\delta)=Tr(\overline \mu \delta)=\overline \mu \delta+\mu \overline \delta .$ If $\delta=0$, then the sum is equal to $|L'/L|$. Why is the sum equal to $0$ if $\delta \ne 0$ ? My idea is to consider representatives $r_1,...r_n$ of $L'/L $ and to use the geometric series , but somehow it does not work .
Thanks for the help .
For $A\in GL_n(\Bbb{R})$, $L = A \Bbb{Z}^n=\{ Av,v\in \Bbb{Z}^n\}$ is a lattice in $\Bbb{R}^n$, the norm is $ \|Av\|^2= vA^\top Av$,
We assume that $M=A^\top A\in \frac12 M_n(\Bbb{Z})$ and the diagonal coefficients are integers so that $\forall Av\in L,\|Av\|^2\in \Bbb{Z}$. Even lattice means that $M \in M_n(\Bbb{Z})$.
let $$L' = \{ x\in \Bbb{R}^n, \forall Av\in L, x^\top Av\in \Bbb{Z}\} = A^{-\top} \Bbb{Z}^n= A M^{-1} \Bbb{Z}^n$$ We find $L'' =(A^{-\top})^{-\top}\Bbb{Z}^n= L$ so that $$\forall A^{-\top} v\in L', (A^{-\top} w)^\top A^{-\top} v \in \Bbb{Z} \iff A^{-\top} w\in L'' \iff A^{-\top} w \in L$$ For $A^{-\top}w\in L', \not \in L$ then $$\exists A^{-\top} u\in L',\lambda=\exp(2i\pi (A^{-\top} w)^\top (A^{-\top} u))\ne 1$$ And since $M\in M_n(\Bbb{Z}) \implies L\subset L'$ $$S=\sum_{A^{-\top} v\in L'/L} \exp(2i\pi (A^{-\top} w)^\top (A^{-\top} v))=S\lambda= 0$$
$$\sum_{A^{-\top} w\in L'/L,} \sum_{A^{-\top} v\in L'/L} \exp(2i\pi (A^{-\top} w)^\top (A^{-\top} v))= |L'/L|= |\det(M)|$$