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

TL;DR

This paper characterizes which parabolically induced representations of the p-adic group G_2, distinguished by SO_4, are explicitly identified using geometric methods and an embedding into GL_8.

Contribution

It provides a precise description of SO_4-distinguished induced representations of G_2 over p-adic fields, expanding understanding of symmetric space representations.

Findings

Explicit characterization of distinguished representations

Use of geometric lemma in the analysis

Embedding of G_2 into GL_8 facilitates the study

Abstract

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 $P_{\beta}$. Using an explicit description of the embedding of $G_2$ in $GL_8$, we characterize precisely the induced representations which are $(SO_4, \chi)$-distinguished, given a certain type of involutions is chosen.