  This paper constructs a new interacting particle system on a two--dimensional
lattice with geometric jumps near a boundary which partially reflects the
particles. The projection to each horizontal level is Markov, and on every
level the dynamics match stochastic matrices constructed from pure alpha
characters of $Sp(\infty)$, while on every other level they match an
interacting particle system from Pieri formulas for $Sp(2r)$. Using a
previously discovered correlation kernel, asymptotics are shown to be the
Discrete Jacobi and Symmetric Pearcey processes.
