A Voronoĭ summation formula is an equality between a weighted sum of Fourier coefficients of an automorphic form twisted by an additive character and a dual weighted sum of Fourier coefficients of the dual form twisted by Kloosterman sums. It is an important tool in analytic number theory. This paper deals with a Voronoĭ summation formula in the automorphic forms setting. It gives the conceptual understanding of the formula, which turns out to be rather simple–it is just a Fourier inversion formula. It is easy to get examples of Voronoĭ summation formula in classical formulation from the adelic formulations given in this paper.
Let \(\varphi\) be a cusp form on \(\mathrm{GL}(n)\), \(N\subset \mathrm{GL}(n)\) the standard maximal unipotent subgroup of \(\mathrm{GL}(n)\) with \(\psi\) a non-degenerate character of \(N\). Take \(Y\subset N\) consisting of elements whose \((1,2)\) entry is \(0\). Let \(X=\left\{\left(\begin{smallmatrix}{1}&{0}\\{x}&{I_{n-1}}\end{smallmatrix}\right): x_{n-1}=0\right\}\) and \(w=\left(\begin{smallmatrix}{1}&{0}\\{0}&{w_{n-1}}\end{smallmatrix}\right)\) where \(w_{n-1}\) is the longest element in Weyl’s group in \(\mathrm{GL}(n-1)\). Then Fourier inversion formula shows \[ \int_{Y(F)\backslash Y(A)}\varphi(u)\psi(u)\;du=\int_{X(A)}\left(\int_{Y(F)\backslash Y(A)} \varphi^{\iota}(uxw)\psi^{-1}(u)\, du\right)\;dx. \] Here \(\varphi^{\iota}(g)=\varphi((g^t)^{-1})\).
Next one evaluates the formula for \(\varphi=\varphi'(\cdot g_0)\) where \(g_{0,v}=\left(\begin{smallmatrix}{n(\xi_v)}&{0}\\{0}&{I_{n-2}}\end{smallmatrix}\right)\) with \(n(\xi_v)=\left(\begin{smallmatrix}{1}&{\xi_v}\\{0}&{1}\end{smallmatrix}\right)\) if \(\varphi'_v\) is an unramified vector, and \(g_{0,v}=I_{n}\) otherwise. One further assumes that \(W_{\varphi,v}\left(\left(\begin{smallmatrix}{y}&{0}\\{0}&{I_{n-1}}\end{smallmatrix}\right)\right)\) is a smooth compactly supported function \(w_v\) on \(F^*_v\) if \(\varphi'_v\) is not unramified. Calculating both sides yield the Voronoĭ summation formula: Let \(S\) be the places where \(\varphi'_v\) is not unramified, \(R\) be the places where \(\varphi_v'\) is unramified and \(g_{0,v}\not=I_{n}\), \[ \begin{split} \sum_{\gamma\in F^*}\psi(\gamma\xi)\prod_{v\not\in S}W_{0,v}\left(\left(\begin{smallmatrix}{\gamma}&{0}\\{0}&{I_{n-1}}\end{smallmatrix}\right)\right)\prod_{v\in S}w_v(\gamma) \\ =\sum_{\gamma\in F^*}K_R(\gamma,\xi,W_{0,R})\prod_{v\not\in S\cup R}W_{0,v}\left(w_n\left(\begin{smallmatrix}{\gamma^{-1}}&{0}\\{0}&{I_{n-1}}\end{smallmatrix}\right)\right)\prod_{v\in S}\tilde w_v(\gamma). \end{split} \] Here \(W_{0,v}\) denote the unramified Whittaker function, and \(\tilde w_v\) is an integral transform of \(w_v\).
If furthermore \(\varphi\) is in the space of an irreducible cuspidal representation \(\pi\), then one can give a better description of \(\tilde w_v\) using local function equation. It satisfies the following identity: \[ \int_{F_v^*}\tilde w_v(y)\chi(y)^{-1}|y|_v^{s-\frac{n-1}{2}}\, dy =\chi(-1)^{n-1}\gamma(1-s,\pi_v\times\chi,\psi_v)\int_{F_v^*}w_v(y)\chi(y)|y|^{1-s-\frac{n-1}{2}}\, dy. \]