The object of this thesis is to discuss the Howe duality correspondence in the oscillator-spin representation of (O( p, q), osp (2, 2)) ⊂ osp (2(p + q), 2(p + q)). In this representation, the Casimir operators of O( p, q) and osp (2, 2) are exactly the same. We show that the Howe correspondence exists for the representations with nonzero eigenvalues for the Casimir operators.;When q = 1, we can even give the Howe duality correspondence of the representations with zero eigenvalue for the Casimir operators. Thus the Howe correspondence is completely proved in this case.;For p = 3, q = 1, the pair ( O(3, 1), osp (2, 2)) is closely related to Maxwell's equations. We use the structure results to give a representation-theoretic interpretation for a method of solving Maxwell's equations.
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