Global Lie Symmetries of the Heat and Schrödinger Equation

We examine solutions to a family of differential equations, including the heat and Schrdinger equations, that are globally invariant under the action of the corresponding Lie symmetry group.The solution space is realized in a nonstandard parabolically induced representation space as the kernel of a linear combination of Casimir operators of certain distinguished subgroups.Composition series provide a complete description of this kernel and, for special inducing parameters, the oscillator representation is realized in a natural and explicit way as a subspace of solutions to the Schrdinger equation.