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

We study homomorphisms between quantized generalized Verma modules \(M({V}_{\Lambda })\mathop{\to }\limits^{\phi \Lambda,{\Lambda }_{1}}M({V}_{{\Lambda }_{1}})\) for \({{\mathscr{U}}}_{q}(su(n,n))\). There is a natural notion of degree for such maps, and if the map is of degree k, we write \({\phi }_{\Lambda,{\Lambda }_{1}}^{k}\). We examine when one can have a series of such homomorphisms \({\phi }_{{\Lambda }_{n-1},{\Lambda }_{n}}^{1}\circ {\phi }_{{\Lambda }_{n-2},{\Lambda }_{n-1}}^{1}\circ \cdots \circ {\phi }_{\Lambda,{\Lambda }_{1}}^{1}={{\rm{Det}}}_{q}\), where Det q denotes the map \(M({V}_{\Lambda})\ni p\to {\det }_{q}\cdot p\in M({V}_{{\Lambda }_{n}})\). If, classically, \(su{(n,n)}^{{\mathbb{C}}}={{\mathfrak{p}}}^{-}\oplus (su(n)\oplus su(n)\oplus {\mathbb{C}})\oplus {{\mathfrak{p}}}^{+}\), then Λ = (Λ L , Λ R , λ) and Λ n = (Λ L , Λ R , λ + 2). The answer is then that Λ must be one-sided in the sense that either Λ L = 0 or Λ R = 0 (non-exclusively). There are further demands on λ if we insist on \({{\mathscr{U}}}_{q}({{\mathfrak{g}}}^{{\mathbb{C}}})\) homomorphisms. However, it is also interesting to loosen this to considering only \({{\mathscr{U}}}_{q}^{-}({{\mathfrak{g}}}^{{\mathbb{C}}})\) homomorphisms, in which case the conditions on λ disappear. By duality, there result have implications on covariant quantized differential operators. We finish by giving an explicit, though sketched, determination of the full set of \({{\mathscr{U}}}_{q}({{\mathfrak{g}}}^{{\mathbb{C}}})\) homomorphisms \({\phi }_{\Lambda,{\Lambda }_{1}}^{1}\).