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?

In On construction of holomorphic cusp forms of half integral weight 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} \lvert\det Q\rvert^{-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?