Let $E/\mathbb{Q}$ be an imaginary quadratic extension and
suppose $G$ is the unitary group attached to hermitian space over $E$ of
signature $(2,n)$. The symmetric domain $X$ attached to $G$ is a
quaternionic Kahler manifold in the sense of differential geometry. In
the late nineties N. Wallach studied harmonic analysis on $X$ in the
context of this quaternionic structure. He established a multiplicity
one theorem for spaces of generalized Whittaker periods appearing in the
cohomology of certain quaternionic $G$-bundles on $X$.

We prove new cases of Wallach's multiplicity one statement for some
degenerate generalized Whittaker periods and give explicit formulas for
these periods in terms of modified K-Bessel functions. Our results can
be interpreted as giving a refined Fourier expansion for automorphic
forms on $G$ which are quaternionic discrete series at infinity. As an
application we study the cusp forms on $G$ which arise as theta lifts of
holomorphic modular forms on quasi-split $\mathrm{U}(1,1)$. We show that
these cusp forms can be normalized so that all their Fourier
coefficients are algebraic numbers. (joint with Anton Hilado and Pan Yan)