The Whittaker normalization conjecture for the Fourier–Mukai transform

Let $G=\operatorname{GL}_r$. Let $\mathcal{B}\subset(\mathrm{B}_{\operatorname{GL}_r}^{\mathrm{red}})^{\mathrm{cl}}$ be an open subset, and set

$$\mathcal{H}(\chi):=\mathrm{Higgs}_{\operatorname{GL}_r}(\chi)\times_{\mathrm{B}_{\operatorname{GL}_r}}\mathcal{B}.$$

Let $\mathcal{P}\in\operatorname{Coh}(\mathcal{H}(w)\times_{\mathcal{B}}\mathcal{H}(\chi))$ be the Cohen–Macaulay extension of the Poincaré line bundle, and let $\Phi$ be the induced Fourier–Mukai functor. Let $s:\mathcal{B}\to\mathcal{H}(\chi_0)$ be the Hitchin section, with $\chi_0=(r-r^2)(g-1)$, and let $s_!$ be the left adjoint of $s^*$. Whittaker normalization conjecture. There is an isomorphism

$$\Phi(\mathcal{O}_{\mathcal{H}(w)^{\mathrm{ss}}})\cong s_!\mathcal{O}_{\mathcal{B}}.$$

The conjecture concerns the normalization of the Fourier–Mukai transform by the Hitchin section. The supplied text does not state that it has been resolved; the related result cited in the paper proves it only together with an additional generation statement.