In this 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?