Distinguished positive regular representations

Let G be a tamely ramified reductive p-adicý ýgroupý. ýWe study distinction of a class of irreducible admissible representationsý ýof G by the group of fixed points H of an involutioný ýof Gý. ýThe representations correspond to G-conjugacy classes ofý ýpairs (T,ϕ)ý, ýwhere T is aý ýtamely ramified maximal torus of G and ϕ is a quasicharacterý ýof T whose restriction to the maximal pro-p-subgroupý ýsatisfies a regularity conditioný.
ýUnder mild restrictions on the residual characteristic ofý ýFý, ýwe derive necessary conditions for H-distinction ofý ýa representation corresponding to (T,ϕ)ý, ýexpressed in terms of properties of T and ϕý ýrelative to the involutioný.
ýWe prove that if an H-distinguished representation arises fromý ýa pair (T,ϕ) such that T is stable under the involution andý ýcompact modulo (T∩H)Z (hereý, ýZ is the centre ofýýG)ý, ýthen the representation is H-relatively supercuspidalý.