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Zhao and the second author (2013) constructed a functor from o(k)-Mod to o(k+ 2)-Mod. In this paper, we use the functor successively to obtain an universalfirst-order differential operator realization for any highest-weightrepresentation of o(2n + 3) in (n + 1)^2 variables and that of o(2n + 2) in n(n+ 1) variables. When the highest weight is dominant integral, we determine thecorresponding finite-dimensional irreducible module explicitly. One can use theresult to study tensor decompositions of finite-dimensional irreducible modulesby solving certain first-order linear partial differential equations, andthereby obtain the corresponding physically interested Clebsch-Gordancoefficients and exact solutions of Knizhnik-Zamolodchikov equation in WZWmodel of conformal field theory. We also find an equation of counting thedimension of an irreducible o(k + 2)-module in terms of certain alternating sumof the dimensions of irreducible o(k)-modules. In the case of the Steinbergmodules, we obtain new combinatorial identities of classical type.

Full Conformal Oscillator Representations of Orthogonal Lie Algebras and Combinatorial Identities