  The signatures of the inner product matrices on a Lie algebra's highest
weight representation are encoded in the representation's signature character.
We show that the signature characters of a finite-dimensional Lie algebra's
highest weight representations obey simple difference equations that have a
unique solution once appropriate boundary conditions are imposed. We use these
results to derive the signature characters of all $A_2$ and $B_2$ highest
weight representations. Our results extend, and explain, signature patterns
analogous to those observed by Friedan, Qiu and Shenker in the Virasoro
algebra's representation theory.
