The Plancherel formula of Fourier integral operators: The case of Sp(1,1)

Abstract

In this paper, we consider the Plancherel formula for Fourier integral operator on the space [Formula: see text], where [Formula: see text] is the quaternion skew field. By using the metaplectic representations of a noncompact classical dual pair [Formula: see text] and combining with the Plancherel formula of Poisson transform, we obtain the direct integral decomposition of all irreducible unitary representations of Fourier integral operator.