Special classes of homomorphisms between generalized Verma modules for ${{\mathscr{U}}}_{q}(su(n,n))$

We study homomorphisms between quantized generalized Verma modules M(Vλ) φλ,λ1→ M(Vλ1 ) for uq (su(n; n)). There is a natural notion of degree for such maps, and if the map is of degree k, we write φkλ,λ1. We examine when one can have a series of such homomorphisms φ1λn-1,λ n φ1λn-1,λ n ⋯ o φ1λ, λ1 = Detq, where Detq denotes the map M(Vλ)ϵ p → detq p ϵ 2 M(Vλn). If, classically, su(n; n)C = p ⊗(su(n) ⊗su(n) ⊗C) ⊗p+, then λ = (λL, λR,λ) and λn = (λL;λRλ+2). The answer is then that - must be one-sided in the sense that either λL = 0 or λR = 0 (non-exclusively). There are further demands on λ if we insist on Uq(gC) homomorphisms. However, it is also interesting to loosen this to considering only Uq (gC) homomorphisms, in which case the conditions on λ disappear. By duality, there result have implications on covariant quantized difierential operators. We finish by giving an explicit, though sketched, determination of the full set of Uq(gC) homomorphisms φ1λ, λ1. © 2019 Published under licence by IOP Publishing Ltd.