  We construct a family of Pfaffian point processes relevant for the harmonic
analysis on the infinite symmetric group. The correlation functions of these
processes are representable as Pfaffians with matrix valued kernels. We give
explicit formulae for the matrix valued kernels in terms of the classical
Whittaker functions. The obtained formulae have the same structure as that
arising in the study of symplectic ensembles of Random Matrix Theory.
  The paper is an extended version of the author's talk at Fall 2010 MSRI
Random Matrix Theory program.
