Fourier coefficients on quaternionic U(2,n)

Abstract: 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 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 $G$-bundles on $X$.

We prove an analogous 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 give a refinement of the Fourier expansion of certain automorphic forms on $G$ which are quaternionic discrete series at infinity. As an application, we study the Fourier expansion of 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)

number theory

Audience: researchers in the topic

Comments: pre-talk at 1:20pm

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UCSD number theory seminar

Series comments: Most talks are preceded by a pre-talk for graduate students and postdocs. The pre-talks start 40 minutes prior to the posted time (usually at 1:20pm Pacific) and last about 30 minutes.

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