##### Department of Mathematics,

University of California San Diego

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### Number Theory Seminar - Math 209

## Finn McGlade

#### UCSD

## 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 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)

### November 3, 2022

### 2:00 PM

APM 6402 and Zoom; see https://www.math.ucsd.edu/~nts

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