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?
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?