The open-orbit vanishing conjecture for global integrals

Let $H$ be a reductive group with parabolic subgroups $P$ and $P_{\mathcal O}$, and let $w_0$ and $\nu_0$ be the unique elements specified by the open-orbit type condition for the global integral

. Let $\int_{[\cdots]}$ denote the integral

, with Weyl-element parameter $w$ and double-coset parameter $\nu$.

Open-orbit vanishing conjecture. Suppose that the integral

$$is an open-orbit type integral. Then the integral$$

is zero unless $w=w_0$ and $\nu=\nu_0$. Moreover, $w_0$ can be chosen to be a Weyl element corresponding to the longest Weyl element in the space $P(F)\backslash H(F)/P_{\mathcal O}(F)$.

This claim predicts that, for an open-orbit type global integral, only the distinguished open orbit contributes, and that its Weyl representative may be taken from the longest Weyl position. The source presents it as an indication from experience; no resolution is supplied here.