View source for 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 [[#References|[a1]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Sahi,   "On Kirillov's conjecture for Archimedean fields"  ''Compositio Math.'' , '''72''' :  1  (1989)  pp. 67–86  {{ZBL|0693.22006}}</TD></TR>
</table>

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