L-Functoriality for Local Theta Correspondence of Supercuspidal Representations with Unipotent Reduction

The preservation principle of local theta correspondences of reductive dual pairs over a p-adic field predicts the existence of a sequence of irreducible supercuspidal representations of classical groups. Adams and Harris-Kudla-Sweet have a conjecture about the Langlands parameters for the sequence of supercuspidal representations. In this paper we prove modified versions of their conjectures for the case of supercuspidal representations with unipotent reduction.

Refined global Gan–Gross–Prasad conjecture for Fourier–Jacobi periods on symplectic groups

In this paper, we propose a conjectural identity between the Fourier–Jacobi periods on symplectic groups and the central value of certain Rankin–Selberg $L$-functions. This identity can be viewed as a refinement to the global Gan–Gross–Prasad conjecture for $\text{Sp}(2n)\times \text{Mp}(2m)$. To support this conjectural identity, we show that when $n=m$ and $n=m\pm 1$, it can be deduced from the Ichino–Ikeda conjecture in some cases via theta correspondences. As a corollary, the conjectural identity holds when $n=m=1$ or when $n=2$, $m=1$ and the automorphic representation on the bigger group is endoscopic.

Dichotomy for generic supercuspidal representations ofG2

The local Langlands conjectures imply that to every generic supercuspidal irreducible representation ofG2over ap-adic field, one can associate a generic supercuspidal irreducible representation of either PGSp6or PGL3. We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations ofG2and other representations of PGSp6and PGL3. This correspondence arises from theta correspondences inE6andE7, analysis of Shalika functionals, and spin L-functions. Our main result reduces the conjectural Langlands parameterization of generic supercuspidal irreducible representations ofG2to a single conjecture about the parameterization for PGSp6.