Classification of distinguished representations of GL(2)

Let $F$ be a non-archimedean local field with odd residual characteristic, and let $H$ be a maximal torus of ${\mathrm{G}\mathrm{L}}_2(F)$. In this paper, we will classify the irreducible $H$-distinguished representations of ${\mathrm{G}\mathrm{L}}_2(F)$ by using Casselman’s criteria for $p$-adic symmetric spaces developed by Kato-Takano.