Digne–Michel conjecture on Shintani descents and Fourier matrices

Let $G$ be a finite reductive group with generalized Frobenius map $F$, let $\boldsymbol{G}$ be the corresponding algebraic group, and let $\boldsymbol{G}^{F^{\delta}}$ and $\boldsymbol{G}^F$ denote the associated finite groups. Let $\mathcal{U}(\boldsymbol{G}^{F^{\delta}})$ be the set of unipotent characters, and suppose that $\chi\in\mathcal{U}(\boldsymbol{G}^{F^{\delta}})$ satisfies $\chi^F=\chi$. Choose an extension $\widetilde{\chi}$ of $\chi$ to $\boldsymbol{G}^{F^{\delta}}\rtimes\langle F\rangle$. For a family $\mathcal{F}$ of unipotent characters of $\boldsymbol{G}^F$, write $\omega_V$ for the associated roots of unity. Digne–Michel conjecture. The irreducible constituents of

$$\operatorname{Sh}_{F^{\delta}/F}\widetilde{\chi}$$

are unipotent characters of $\boldsymbol{G}^F$ and lie in the same family $\mathcal{F}$. Moreover, there exists a root of unity $u$ such that

$$\pm u\operatorname{Sh}_{F^{\delta}/F}\widetilde{\chi}=\sum_{V\in\mathcal{F}}a_V\omega_VV.$$

In this case, the coefficients $a_V$ give, up to a sign, a row of the Fourier matrix associated with $\mathcal{F}$. The conjecture connects Shintani descent with the family structure and Fourier matrices of unipotent characters; the supplied text gives no resolution status, so it is recorded as open.