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](https://mathoverflow.net/questions/238100/is-it-true-that-irreducible-generic-representations-of-g-2f-are-self-dual#comment589145_238100) and [Jim Humphreys](https://mathoverflow.net/questions/238100/is-it-true-that-irreducible-generic-representations-of-g-2f-are-self-dual#comment589936_238100) for their comments and [Jeffrey Adams](https://mathoverflow.net/a/238376) 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? 



Is it true that irreducible generic representations of $G_2(F)$ are self-dual?
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](https://arxiv.org/abs/1512.04347) (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](https://arxiv.org/abs/1203.1901) (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.


