Distinction of depth-zero representations

Let π be a depth-zero irreducible admissible representation of a connected reductive p-adic group G. Let H be the group of fixed points of an involution θ of G. We relate H-distinction of π to existence of minimal K-types of π that exhibit particular symmetry properties relative to θ. In addition, we show that when π is H-distinguished, then (up to conjugacy) the support of π   is of the form (M,τ)(M,τ) where M is a θ-stable Levi subgroup of G and τ is a depth-zero irreducible supercuspidal representation of M. Moreover, τ contains a minimal K  -type (Mx,ρ)(Mx,ρ) such that MxMx is a θ-stable maximal parahoric subgroup of M and ρ   is the inflation of a distinguished cuspidal representation of the quotient of MxMx by its pro-unipotent radical.