Algebras of variable coefficient quantized differential operators

Abstract

In the framework of (vector valued) quantized holomorphic functions defined on non-commutative spaces, “quantized Hermitian symmetric spaces,” an obvious problem is to describe (quantum) holomorphically induced representations in terms of some manageable structures. An intimately related problem is to decide what the algebras of quantized differential operators with variable coefficients should be. It is an immediate point that even zeroth order operators, given as multiplications by polynomials, have to be specified as, e.g., left or right multiplication operators since the polynomial algebras are replaced by quadratic, non-commutative algebras. In the settings we are interested in, there are bilinear pairings that allow us to define differential operators as duals of multiplication operators. Indeed, there are different choices of pairings which lead to quite different results. We consider three different pairings while specializing to Uq(su(n,n)). The pairings are between quantized generalized Verma modules and quantized holomorphically induced modules. It is a natural demand that the corresponding representations can be expressed by (matrix valued) differential operators. We show that a quantum Weyl algebra Weylq(n,n) introduced by Hayashi [Commun. Math. Phys. 127(1), 129–144 (1990)] plays a fundamental role. In fact, for one pairing, the algebra of differential operators, though inherently depending on a choice of basis, is precisely matrices over Weylq(n,n). We determine explicitly the form of the (quantum) holomorphically induced representations and determine, for the different pairings, if they can be expressed by differential operators.