# The Gross–Prasad conjecture and local theta correspondence

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## Abstract

We establish the Fourier–Jacobi case of the local Gross–Prasad conjecture for unitary groups, by using local theta correspondence to relate the Fourier–Jacobi case with the Bessel case established by Beuzart-Plessis. To achieve this, we prove two conjectures of Prasad on the precise description of the local theta correspondence for (almost) equal rank unitary dual pairs in terms of the local Langlands correspondence. The proof uses Arthur’s multiplicity formula and thus is one of the first examples of a concrete application of this “global reciprocity law”.

## Mathematics Subject Classification

11F70 22E50## Notes

### Acknowledgments

We would like to thank Tasho Kaletha for useful discussions. W. T. Gan is partially supported by a Singapore government MOE Tier 2 Grant R-146-000-175-112. A. Ichino is partially supported by JSPS Grant-in-Aid for Scientific Research (B) 26287003. This material is based upon work supported by the National Science Foundation under Grant No. 0932078 000 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2014 semester.

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