  The analytic proof of mode stability of the Kerr black hole was provided by
Whiting. In his proof, the construction of a conserved quantity for unstable
mode was crucial. We extend the method of the analysis for the Kerr-de Sitter
geometry. The perturbation equations of massless fields in the Kerr-de Sitter
geometry can be transformed into Heun's equations which have four regular
singularities. In this paper we investigate differential and integral
transformations of solutions of the equations. Using those we construct a
conserved quantity for unstable modes in the Kerr-de Sitter geometry, and
discuss its property.
