Representations and Stochastic Processes on Groups of Type-H

We find realizations of Lie algebras of "type-H" as vectors fields. These are used in a novel approach to representing group elements as products of one-parameter subgroups (splitting formula) and for finding polynomial matrix elements of representations for the Lie group, in particular irreducible representations. Special classes of polynomials, Appell systems, and polynomial solutions to heat equations (via a Feynman-Kac type formula) are found. Additionally, we show that the Lie algebra generates Gaussian random variables. The general form of stochastic processes on the group is given, and Brownian motion is discussed in particular.