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)