2 1 Fe b 20 06 On the hermiticity of q - differential operators and forms on the quantum Euclidean spaces R Nq

We show that the complicated ⋆-structure characterizing for positive q the Uqso(N)-covariant differential calculus on the non-commutative manifold Rq boils down to similarity transformations involving the ribbon element of a central extension of Uqso(N) and its formal square root ṽ. Subspaces of the spaces of functions and of p-forms on Rq are made into Hilbert spaces by introducing non-conventional “weights” in the integrals defining the corresponding scalar products, namely suitable positive-definite qpseudodifferential operators ṽ′±1 realizing the action of ṽ±1; this serves to make the partial q-derivatives antihermitean and the exterior coderivative equal to the hermitean conjugate of the exterior derivative, as usual. There is a residual freedom in the choice of the weight m(r) along the ‘radial coordinate’ r. Unless we choose a constant m, then the square-integrables functions/forms mustfulfill an additional condition, namely their analytic continuations to thecomplex r plane can have poles only on the sites of some special lattice.Among the functions naturally selected by this condition there are q-special functions with ‘quantized’ free parameters. MSC-class: 81R50; 81R60; 16W10; 16W30; 20G42.