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Let $G$ be the group $GL(n,F)$, where $F$ is a p-adic field, and $I$ its standard Iwahori subgroup. Let $\pi$ be an irreducible smooth representation of $G$ with nonzero $I$-fixed vectors. It is well known from the Borel-Matsumoto theorem that such a representation embeds into an unramified principal series representation.

My question concerns the following unicity problem: Is it true that there is a unique, up to isomorphisms, irreducible smooth representation of $G$ (with trivial central character) with nonzero $I$-fixed vectors and with no nonzero $P$-fixed vectors, where $P$ is a fixed fixator of a panel of the affine building $X$ of $G$ (or any fixator because the fixators of the panels of $X$ are all conjugate).

The Steinberg representation of $G$ gives the existence of such a representation.

Is there any reference about this question ?

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    $\begingroup$ The fixators of panels are not conjugate in general! $\endgroup$
    – Paul Broussous
    Commented Nov 14 at 9:05
  • $\begingroup$ Yes, as example the group $SL(n,F)$ $\endgroup$
    – Jacky 1962
    Commented Nov 14 at 10:39
  • $\begingroup$ But $GL(n,F)$ acts transitively on the panels of its affine building so that the fixators of panels are $GL(n,F)$-conjugate $\endgroup$
    – Jacky 1962
    Commented Nov 14 at 10:44
  • $\begingroup$ Indeed, you're right, sorry! $\endgroup$
    – Paul Broussous
    Commented Nov 14 at 12:35

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