The weighted composition operators as intertwining operators for holomorphic Lie group representations

Let D1D1 and D2D2 be domains in CnCn and let ζ, η   be holomorphic functions on D1D1 such that η(D1)⊂D2η(D1)⊂D2 and ζ:D1→Cζ:D1→C. In this paper, we determine necessary and sufficient conditions on ζ, η   in order that the weighted composition operator Wζ,ηWζ,η induced by ζ and η   be an intertwining operator of holomorphic Lie group representations having the form (Tg(j)F)(z)=hg(j)(z)F(kg(j)(z)), j=1,2j=1,2, where hg(j):Dj→C and kg(j):Dj→Dj are holomorphic on DjDj and g is an element of the Lie group G. Furthermore, we examine conditions on ζ, η   to ensure that Wζ,ηWζ,η is also an intertwining operator for the infinitesimal representation of TgTg given by(ρ(j)(v)F)(z)=ddϵ|ϵ=0Tgϵ(j)F(z), where (gϵ)ϵ∈R(gϵ)ϵ∈R is a smooth one-parameter subgroup of G   such that v=ddϵ|ϵ=0gϵ belongs to the Lie algebra of G.