Parabolically induced representations of p-adic $G_2$ distinguished by $SO_4$, I

We consider the parabolically induced representations of the symmetric space $SO_4\backslash G_2$ over a p-adic field using the geometric lemma when the inducing parabolic is $P_β$. Using an explicit description of the embedding of $G_2$ in $GL_8$, we characterize precisely the induced representations which are $(SO_4, χ)$-distinguished, given a certain type of involutions is chosen.

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