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Let $G_2$ be the split exceptional group of type $G_2$ and $F$ be a p-adic field. Is it true that every irreducible smooth representation of $G_2(F)$ is self-contragredient? If the answer is Yes, can anybody give me a reference? If not, is there a way to describe the contragredient, for example, is there a MVW involution like the classical group case?

Edit:

I thank Paul Garrett and Jim Humphreys for their comments and Jeffrey Adams for his nice answer. According to Jeffrey Adams's answer, one expects that each L-packet of $G_2(F)$ is self-dual. On the other hand, according to the general philosophy of Gan–Gross–Prasad, in each L-packet, there should be at most one generic representation (GGP conjectured that in each generic local L-packet, there is at most one representation which has the given Bessel model or Fourier–Jacobi model for classical groups. I do not know if anybody conjectured this for exceptional groups. But I just think that we can expect this once we can define the corresponding model. In particular, one would expect the uniqueness of generic member in each L-packet. For the conjecture of uniqueness of generic element in each L-packet, there is probably early reference, but I learned it from GGP). Thus one would expect that:

Each generic smooth irreducible representation of $G_2(F)$ is self-dual.

Do we expect this or is this also false? If we do expect this, do we know anything related to this?

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    $\begingroup$ Surely not: (typical) non-unitarizable unramified principal series? Casselman's 1980 Compositio discussion of this applies to split $G_2$, among other things. $\endgroup$ – paul garrett May 5 '16 at 19:35
  • $\begingroup$ "MVW involution"? I'm mildly surprised that I can't easily guess what this acronym is... For regular characters, the spherical Weyl group gives intertwinings among unramified principal series, in any case. $\endgroup$ – paul garrett May 5 '16 at 22:28
  • $\begingroup$ @paulgarrett Thanks for your answer. MVW stands for Moeglin-Vigneras-Waldspurger. In their book "Correspondances de Howe sure un corps p-adique", they give a description of contragradient representations for unitary groups (which include symplectic, orthogonal,unitary...). For example, for symplectic groups, if $\pi$ is an irreducible smooth representation of $Sp_{2n}$ and $\delta$ is an element in $GSp_{2n}$ with similitude -1, then it is shown that $\tilde \pi$ is isomorphic to $\pi^\delta$. $\endgroup$ – Qing Zhang May 6 '16 at 0:43
  • $\begingroup$ @paulgarrett MVW also showed that irreducible smooth representations of $SO_{odd}$ are self-dual. I do not know what Casselman's 1980 Compositio say about $SO_{odd}$. Since $G_2$ can be embedded into $SO_7$, I thought the same would be true for $G_2$. $\endgroup$ – Qing Zhang May 6 '16 at 1:27
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    $\begingroup$ @JimHumphreys It's interesting to notice that the English translation of Bernstein-Zelevinski's classical paper " representations of GL(n,F), where $F$ is a non-archimedean local field" had the same mistake. See page 19 of that paper math1.tau.ac.il/~bernstei/Publication_list/publication_texts/… $\endgroup$ – Qing Zhang Jul 2 '16 at 1:58
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Since $-1$ is in the Weyl group (over $F$, not just the algebraic closure) you might expect every irreducible representation to be self-dual. This is the case over $\mathbb R$. It is false over a $p$-adic field, but subtle, and it is not easy to construct an example.

There are non-self dual cuspidal unipotent represenations of $G_2(k)$ where $k$ is the (finite) residue field. By the standard pull back and induction procedure these give rise to non-self-dual supercuspidal representations of $G_2(F)$. The same thing works for all exceptional groups.

Dipendra Prasad has a discussion of some closely related matters in A 'relative' local Langlands Correspondence (arXiv:1512:04347).

In terms of involutions, $G_2(F)$ (for example) has an involution $\tau$ such that $\tau(g)$ is conjugate to $g^{-1}$ over the algebraic closure. This is the "Chevalley involution", and it is inner for $G_2(F)$. However $\tau(g)$ cannot always be $G_2(F)$-conjugate to $g^{-1}$ (exactly because then every irreducible representation would be self-dual, which is false.) See The Real Chevalley Involution (arXiv:1203:1901), page 4.

Finally, because of the involution just mentioned, one would expect that every L-packet for $G_2(F)$ is self-dual, but the duality operation could be nontrivial on the packet.

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$\DeclareMathOperator\PGSp{PGSp}\DeclareMathOperator\PGL{PGL}$Generic representations of a $p$-adic $G_2$ are indeed self dual. It suffices to prove this for super-cuspidal representations. Observe that, for unitary representations, taking dual is the same as taking complex conjugate. Now all generic super-cuspidal lift one-to-one to generic representations of $\PGSp_6$ by the exceptional theta correspondence, see Savin–Weissman - Dichotomy for generic supercuspidal representations of $G_2$, Compositio Math (2011) (MSN). Since the exceptional theta correspondence commutes with complex conjugation, the statement follows from self-duality on $\PGSp_6$ side. In fact, since any representation of $G_2$ lifts either to $\PGSp_6$ or to a compact form of $\PGL_3$, one can completely classify representations of $G_2$ that are not self dual: the super-cuspidal representations that correspond to non-trivial representations of the compact form of $\PGL_3$.

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  • $\begingroup$ Would it be possible briefly to compare your answer to @JeffreyAdams's? For example, how does the depth-$0$ representation of $\operatorname G_2(F)$ that he mentioned correspond to the compact form of $\operatorname{PGL}_3$? (Also, I took the liberty of correcting the spelling of Marty's name.) $\endgroup$ – LSpice Sep 9 '19 at 10:21
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    $\begingroup$ Those supircuspidal representations of $G_2$ correspond to unramified cubic characters of the compact form of $PGL_3$, see my paper "A class of supercuspidal representations of $G_2$" Canadian Math Bulletin (1999). $\endgroup$ – Gordan Savin Sep 9 '19 at 13:54

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