Received XXX

In the articles [3], [1], [2], [4], [5], and [6] it was shown how intertwining differential operators on spaces of vector valued holomorphic functions on hermitian symmetric spaces of the non-compact type are intimately connected both with homomorphisms between generalized Verma modules and with singular unitary representations through what was called the “missing k types”. It was clear from the outset that the construction was applicable in a much broader context for classical (semi-simple) Lie algebras. Later, it was natural to extend it further to quantum groups, in particular after the classification of unitarizable highest weight modules was obtained [7]. In connection with the generalization to quantum groups of the above, one needs of course to find good replacements of hermitian symmetric spaces, holomorphic functions, and, finally, differential operators. We believe that we through the discovery of quadratic algebras associated to hermitian symmetric spaces, see e.g. [8], have found the right setting. In a series of papers, c.f. [10], [11], and references cited therein, Dobrev has studied a quantization of the d’Alembert operator and its invariance properties and has also addressed the issue of relevant spaces. The quantized wave operator is also discussed in [9] which has some overlap with Dobrev’s work, both mathematical and notational, but in fact, Dobrev’s methods have much more in common with the earlier work cited above. Even more so, the methods in those references reach much further as we will illustrate below by giving “the quantization of the Dirac operator” in the setting of quantum groups. Our approach shows that the quantized wave operator is a major part of the quantized Dirac operator; in some sense the term that represents the most “non-commutative” part.