4 problems

Uniqueness conjecture. (i) The group $C_G(X^*_{\mathrm{ss}})^\circ$ contains a maximal torus in $C_G(X^*)^\circ$. (ii) Every semisimple part of $X^*$ belongs to the…

Main conjecture. (i) The map

Let $G$ be a reductive group, let $g\in G_{\geqslant 0}$, and write its Jordan decomposition as $g=g_sg_u=g_ug_s$. Set $H=Z_G(g_s)$, the centralizer of the semisimple part. A weak…

Let ${\mathbf G}$ be a connected reductive group with Frobenius root $F$, let $s\in{\mathbf G}^{*F^*}$ be semisimple, and let $C_{{\mathbf G}^*}(s)$ satisfy condition $(*)$. Let…