We obtain stochastic duality functions for specific Markov processes using representation theory of Lie algebras. The duality functions come from the kernel of a unitary intertwiner between $*$-representations, which provides (generalized) orthogonality relations for the duality functions. In particular, we consider representations of the Heisenberg algebra and $\mathfrak{su}(1,1)$. Both cases lead to orthogonal (self-)duality functions in terms of hypergeometric functions for specific interacting particle processes and interacting diffusion processes.