In [this][1] paper by Shintani, in the last equation of page 96, an identity for the Kronecker delta for elements of $L^*/L$ is defined. Here, $L$ is an integral lattice and $L^*$ is its dual.

The identity can be written as follows: 
\begin{equation} 
    |\det Q|^{-1}\sum_{h''\in L^*/L }e^{2\pi i Q(h-h',h'')}
    =\delta_{h,h'}\;,
\end{equation}
where $Q$ is a quadratic form defined on $L$. 

I understand that this is a generalization of the geometric series representation of the Kronecker delta. How does one prove this identity?



  [1]: https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-58/issue-none/On-construction-of-holomorphic-cusp-forms-of-half-integral-weight/nmj/1118795445.full