  The aim of this paper is to study some continuous-time bivariate Markov
processes arising from group representation theory. The first component (level)
can be either discrete (quasi-birth-and-death processes) or continuous
(switching diffusion processes), while the second component (phase) will always
be discrete and finite. The infinitesimal operators of these processes will be
now matrix-valued (either a block tridiagonal matrix or a matrix-valued
second-order differential operator). The matrix-valued spherical functions
associated to the compact symmetric pair $(\mathrm{SU}(2)\times \mathrm{SU}(2),
\mathrm{diag} \, \mathrm{SU}(2))$ will be eigenfunctions of these infinitesimal
operators, so we can perform spectral analysis and study directly some
probabilistic aspects of these processes. Among the models we study there will
be rational extensions of the one-server queue and Wright-Fisher models
involving only mutation effects.
