REPRESENTATIONS OF THE WEYL GROUP IN SPACES OF SQUARE INTEGRABLE FUNCTIONS WITH RESPECT TO P-ADIC VALUED GAUSSIAN DISTRIBUTIONS

We construct a representation of the Weyl group in the p-adic Hilbert space of functions which are square integrable with respect to a p-adi c valued Gaussian distribution. The operators corresponding to positio n and momentum are determined by groups of unitary operators with para meters restricted to some balls in the field Q(p) of p-adic numbers. A surprising fact is that the radii of these balls are connected by 'an uncertainty relation' which can be considered as a p-adic analogue of the Heisenberg uncertainty relations. The p-adic Hilbert space repres entation of the Weyl group is the basis for a calculus of pseudo-diffe rential operators and for an operator quantization over p-adic numbers .