A degenerate version of Brion's formula

Let $\mathfrak{p} \subset V$ be a polytope and $\xi \in V_{\mathbb{C}}^*$. We obtain an expression for $I(\mathfrak{p}; \alpha) := \int_{\mathfrak{p}} e^{\langle \alpha, x \rangle} dx$ as a sum of meromorphic functions in $\alpha \in V^*_{\mathbb{C}}$ parametrized by the faces $\mathfrak{f}$ of $\mathfrak{p}$ on which $\langle \xi, x \rangle$ is constant. Each term only depends on the local geometry of $\mathfrak{p}$ near $\mathfrak{f}$ (and on $\xi$) and is holomorphic at $\alpha = \xi$. When $\langle \xi, \cdot \rangle$ is only constant on the vertices of $\mathfrak{p}$ our formula reduces to Brion's formula. Suppose $\mathfrak{p}$ is a rational polytope with respect to a lattice $\Lambda$. We obtain an expression for $S(\mathfrak{p}; \alpha) := \sum_{\lambda \in \mathfrak{p} \cap \Lambda} e^{\langle \alpha, \lambda \rangle}$ as a sum of meromorphic functions parametrized by the faces $\mathfrak{f}$ on which $e^{\langle \xi, x \rangle} = 1$ on a finite index sublattice of $\text{lin}(\mathfrak{f}) \cap \Lambda$. Each term only depends on the local geometry of $\mathfrak{p}$ near $\mathfrak{f}$ (and on $\xi$ and $\Lambda$) and is holomorphic at $\alpha = \xi$. When $e^{\langle \xi, \cdot \rangle} \neq 1$ at any non-zero lattice point on a line through the origin parallel to an edge of $\mathfrak{p}$, our formula reduces to Brion's formula, and when $\xi = 0$, it reduces to the Ehrhart quasi-polynomial. Our formulas are particularly useful for understanding how $I(\mathfrak{p}(h); \xi)$ and $S(\mathfrak{p}(h); \xi)$ vary in a family of polytopes $\mathfrak{p}(h)$ with the same normal fan. When considering dilates of a fixed polytope, our formulas may be viewed as polytopal analogues of Laplace's method and the method of stationary phase. Such expressions naturally show up in analysis on symmetric spaces and affine buildings.