Even-degree Hermitian Ikeda lifts via Fourier-Jacobi descent

Abstract

Let $E/F$ be a CM extension and let $m\geq2$ be an even integer. Starting from the odd-degree Hermitian Ikeda lift of degree $m+1$ constructed by Yamana, we construct a Hermitian Ikeda lift of degree $m$ from the theta components of its first Fourier-Jacobi coefficient. We determine its global Arthur parameter and its local constituents at every finite place. Using the global multiplicity formula for unitary groups, we obtain an explicit multiplicity-free decomposition of the resulting representation of $\mathrm{U}_{m.m}(\mathbb{A}_{F,\mathbf{h}})$ and show that its isomorphism class is independent of the Fourier-Jacobi index. When $F=\mathbb{Q}$, we compare Fourier coefficients and identify the construction with Ikeda's even-degree Hermitian lift.