Distinguished positive regular representations

Let $G$ be a tamely ramified reductive $p$-adic‎ ‎group‎. ‎We study distinction of a class of irreducible admissible representations‎ ‎of $G$ by the group of fixed points $H$ of an involution‎ ‎of $G$‎. ‎The representations correspond to $G$-conjugacy classes of‎ ‎pairs $(T,phi)$‎, ‎where $T$ is a‎ ‎tamely ramified maximal torus of $G$ and $phi$ is a quasicharacter‎ ‎of $T$ whose restriction to the maximal pro-$p$-subgroup‎ ‎satisfies a regularity condition‎. ‎Under mild restrictions on the residual characteristic of‎ ‎$F$‎, ‎we derive necessary conditions for $H$-distinction of‎ ‎a representation corresponding to $(T,phi)$‎, ‎expressed in terms of properties of $T$ and $phi$‎ ‎relative to the involution‎. ‎We prove that if an $H$-distinguished representation arises from‎ ‎a pair $(T,phi)$ such that $T$ is stable under the involution and‎ ‎compact modulo $(Tcap H)Z$ (here‎, ‎$Z$ is the centre of‎ ‎$G$)‎, ‎then the representation is $H$-relatively supercuspidal‎.