Descents of unipotent cuspidal representations of finite classical groups (Q2663057)

The local Gan-Gross-Prasad conjecture concerns the dimension \(m(\pi,\pi')\) of Hom spaces between irreducible representations \(\pi\) and \(\pi'\) of two classical groups of degrees \(m\) and \(n\), after restriction to a certain common subgroup. Related to this is the study of so-called local descents of admissible representations \(\pi\); roughly, this means finding a `minimal' \(\pi'\) such that \(m(\pi,\pi')\) is non-zero; such results can be viewed as a refinement of the conjecture. The present paper describes the descents of unipotent cuspidal representations of finite symplectic and orthogonal groups, complementing the authors' corresponding study for unitary groups in [\textit{D. Liu} and \textit{Z. Wang}, Trans. Am. Math. Soc. 373, No. 6, 4223--4253 (2020; Zbl 1496.20016)]. The study has a natural dichotomy; depending on whether \(n - m\) is even or odd (the Fourier-Jacobi case and the Bessel case, respectively). The main tool is the theta correspondence (or Howe correspondence) along with a see-saw argument which allows one to transfer results between related reductive dual pairs. After setting up the necessary theory of Harish-Chandra series, describing the theta correspondence and see-saw argument, and deriving preliminary lemmas in Sections 2 to 4, Section 5 proves the Bessel case of the main theorem and Section 6 proves the Fourier-Jacobi case.