# Algebras of Variable Coefficient Quantized Differential Operators

## TL;DR

This paper investigates the structure of algebras of variable coefficient quantized differential operators on non-commutative spaces, revealing their dependence on pairings and connections to quantum Weyl algebras.

## Contribution

It introduces a framework for defining quantized differential operators with variable coefficients using different pairings, linking them to quantum Weyl algebras and explicit representations.

## Key findings

- Differential operators depend on pairing choices.
- The algebra of differential operators can be expressed as matrices over quantum Weyl algebra.
- Explicit forms of holomorphic representations are determined for different pairings.

## Abstract

In the framework of (vector valued) quantized holomorphic functions defined on non-commutative spaces, ``quantized hermitian symmetric spaces'', we analyze what the algebras of quantized differential operators with variable coefficients should be. It is an emediate point that even $0$th 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 which allows 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. 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 ${\mathcal W}eyl_q(n,n)$ introduced by T. Hyashi (Comm. Math. Phys. 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 ${\mathcal W}eyl_q(n,n)$. 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.