Discreteness in deSitter Space and Quantization of Kähler Manifolds

Recently, it has been proposed that the dimension of the Hilbert space of quantum gravity in deSitter space is finite and moreover it is expressed in terms of the coupling constants by using the entropy formula. A weaker conjecture would be that the coupling constant in deSitter space should take only discrete values not necessarily given by the entropy formula. We discuss quantization of the horizon in deSitter space by using Berezin’s functorial quantization of Kähler manifolds and argue that the weak conjecture is valid for Euclidean deSitter space. Moreover it can be valid for a class of bounded complex symmetric spaces.