For a connected reductive group G over a nonarchimedean local field F of positive characteristic, Genestier-Lafforgue and Fargues-Scholze have attached a semisimple parameter ${\mathcal {L}}^{ss}(\pi )$ to each irreducible representation $\pi $. Our first result shows that the Genestier-Lafforgue parameter of a tempered $\pi $ can be uniquely refined to a tempered L-parameter ${\mathcal {L}}(\pi )$, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of ${\mathcal {L}}^{ss}(\pi )$ for unramified G and supercuspidal $\pi $ constructed by induction from an open compact (modulo center) subgroup. If ${\mathcal {L}}^{ss}(\pi )$ is pure in an appropriate sense, we show that ${\mathcal {L}}^{ss}(\pi )$ is ramified (unless G is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show $\mathcal {L}^{ss}(\pi )$ is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is ${\mathbb {P}}^1$ and a simple application of Deligne’s Weil II.

In this paper, we investigate the twisted GGP conjecture for certain tempered representations using the theta correspondence and establish some special cases, namely when the L-parameter of the unitary group is the sum of conjugate-dual characters of the appropriate sign.

We prove the local Langlands conjecture for the exceptional group $G_2(F)$ where F is a non-archimedean local field of characteristic zero.

In a series of three earlier papers, we considered a family of restriction problems for classical groups (over local and global fields) and proposed precise answers to these problems using the local and global Langlands correspondence. These restriction problems were formulated in terms of a pair $W \subset V$ of orthogonal, Hermitian, symplectic, or skew-Hermitian spaces. In this paper, we consider a twisted variant of these conjectures in one particular case: that of a pair of skew-Hermitian spaces $W = V$.

This paper generalizes the Gan–Gross–Prasad (GGP) conjectures that were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the non-generic L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of Arthur parameters that governs the branching laws for ${{\rm GL}}_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in Gan et al. [Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I, Astérisque 346 (2012), 1–109] including Bessel models and Fourier–Jacobi models.

We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of $\text{PGSp}_{6}$ discovered by Pollack, we prove that a cuspidal representation of $\text{PGSp}_{6}$ is a (weak) functorial lift from the exceptional group $G_{2}$ if its (partial) Spin L-function has a pole at $s=1$.

We develop the theory of the doubling zeta integral of Piatetski-Shapiro and Rallis for metaplectic groups Mp2n , and we use it to give precise definitions of the local γ-factors, L-factors, and ε-factors for irreducible representations of Mp2n × GL1, following the footsteps of Lapid and Rallis.

Using theta correspondence, we classify the irreducible representations of Mp2n in terms of the irreducible representations of SO2n+1 and determine many properties of this classification. This is a local Shimura correspondence which extends the well-known results of Waldspurger for n=1.

We prove an explicit formula for periods of certain automorphic forms on SO5 × SO4 along the diagonal subgroup SO4 in terms of L-values. Our formula also involves a quantity from the theory of endoscopy, as predicted by the refined Gross–Prasad conjecture.

We determine essentially completely the theta correspondence arising from the dual pair ${\it PGL}_3 \times G_2 \subset E_6$ over a p-adic field. Our first result determines the theta lift of any non-supercuspidal representation of PGL3 and shows that the lifting respects Langlands functoriality. Our second result shows that the theta lift $\theta(\pi)$ of a (non-self-dual) supercuspidal representation $\pi$ of PGL3 is an irreducible generic supercuspidal representation of G2; we also determine $\theta(\pi)$ explicitly when $\pi$ has depth zero.

We study an equidistribution problem of Linnik using Hecke operators.

We construct an automorphic realization of the global minimal representation of quaternionic exceptional groups, using the theory of Eisenstein series, and use this for the study of theta correspondences.

We consider the restriction of the reflection representation to various reductive dual pairs in exceptional groups, and determine the correspondence of generic representations.

We study the correspondence of representations arising by restricting the minimal representation of the linear group of type ${{E}_{7}}$ and relative rank 4. The main tool is computations of the Jacquet modules of the minimal representation with respect to maximal parabolic subgroups of ${{G}_{2}}$ and $\text{P}{{\text{U}}_{3}}\left( D \right)$ .