  The determination of the density functions for products of random elements
from specified classes of matrices is a basic problem in random matrix theory
and is also of interest in theoretical physics. For connected simple Lie groups
of $2\times 2$ matrices and conjugacy and spherical classes a complete solution
is given here. The problem/solution can be re-stated in terms of the structure
of certain Hecke algebras attached to groups of $2\times 2$ matrices.
