Kirillov conjecture

Let $F$ be a local field and $\pi$ an irreducible unitary representation of $\mathrm{GL}_n(F)$. Let $$ P_n(F) = \{ s \in \mathrm{GL}_n(F) : \text{last row}\,(s) = (0,0,\ldots,1) \} \ . $$

Then $\pi(P_n(F))$ is irreducible (cf. also Irreducible representation).

A related conjecture is that for two irreducible representations $\pi_1$ and $\pi_2$ of, respectively, $\mathrm{GL}_{n_1}(F)$ and $\mathrm{GL}_{n_2}(F)$, the product $$ \pi_1 \pi_2 = \mathrm{Ind}_{\mathrm{GL}(n_1,F)\times\mathrm{GL}(n_2,F)}^{\mathrm{GL}(n_1+n_2,F)} $$ is irreducible.

For $F$ non-Archimedean (cf. also Archimedean axiom), both conjectures are true (Bernstein's theorems).

For $F = \mathbf{C}$, these conjectures have been proved by S. Sahi [a1].

References

[a1]	S. Sahi, "On Kirillov's conjecture for Archimedean fields" Compositio Math. , 72 : 1 (1989) pp. 67–86 Zbl 0693.22006