The mapping class group of a Heegaard splitting is the group of
automorphisms of the ambient manifold that take the Heegaard surface
onto itself.  There is a canonical homomorphism from this group into
the mapping class group of the 3-manifold.  I will outline a proof
that for high distance Heegaard splittings this homomorphism is an
isomorphism, then describe examples of low distance, irreducible
Heegaard splittings for which the kernel is infinite.