On non-generic unramified genuine representations

We discuss Ginzburg’s conjecture relating non-generic unramified genuine representations to non-generic theta representations of Levi subgroups. On one hand, we give concrete counterexamples to the conjecture when stated in its original generality. On the other hand, we show that the conjecture does hold for unramified genuine representations whose Kazhdan–Lusztig–Reeder parameters are dictated by the regular orbits of dual Levi subgroups.