Special classes of homomorphisms between generalized Verma modules for ${\mathcal U}_q(su(n,n))$

Abstract

We study homomorphisms between quantized generalized Verma modules $M(V_Λ)\stackrel{ϕ_{Λ,Λ_1}}{\rightarrow}M(V_{Λ_1})$ for ${\mathcal U}_q(su(n,n))$. There is a natural notion of degree for such maps, and if the map is of degree $k$, we write $ϕ^k_{Λ,Λ_1}$. We examine when one can have a series of such homomorphisms $ϕ^1_{Λ_{n-1},Λ_{n}} \circ ϕ^1_{Λ_{n-2}, Λ_{n-1}} \circ\cdots\circ ϕ^1_{Λ,Λ_1} = \textrm{Det}_q$, where $\textrm{Det}_q$ denotes the map $M(V_Λ)\ni p\rightarrow \textrm{Det}_q\cdot p\in M(V_{Λ_n})$. 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 ${\mathcal U}_q({\mathfrak g}^{\mathbb C})$ homomorphisms. However, it is also interesting to loosen this to considering only ${\mathcal U}^-_q({\mathfrak g}^{\mathbb C})$ 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 ${\mathcal U}_q({\mathfrak g}^{\mathbb C})$ homomorphisms $ϕ^1_{Λ,Λ_1}$.