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.

In this paper, we prove that the finite simple groups $\text{PSp}_{6}(q)$, ${\rm\Omega}_{7}(q)$ and $\text{PSU}_{7}(q^{2})$ are $(2,3)$-generated for all $q$. In particular, this result completes the classification of the $(2,3)$-generated finite classical simple groups up to dimension 7.

We study branching multiplicity spaces of complex classical groups in terms of ${\mathrm{GL} }_{2} $ representations. In particular, we show how combinatorics of ${\mathrm{GL} }_{2} $ representations are intertwined to make branching rules under the restriction of ${\mathrm{GL} }_{n} $ to ${\mathrm{GL} }_{n- 2} $. We also discuss analogous results for the symplectic and orthogonal groups.

We identify the groups ${\text{PSU} }_{6} (2)$, ${\text{PSU} }_{6} (2){: }2$, ${\text{PSU} }_{6} (2){: }3$ and $\text{Aut} ({\text{PSU} }_{6} (2))$ from the structure of the centralizer of an element of order three.

In the previous part of this paper, we constructed a large family of Hecke algebras on some classical groups G defined over p-adic fields in order to understand their admissible representations. Each Hecke algebra is associated to a pair (JΣ, ρΣ) of an open compact subgroup JΣ and its irreducible representation ρΣ which is constructed from given data Σ = (Γ, P′0, ϱ). Here, Γ is a semisimple element in the Lie algebra of G, P′0 is a parahoric subgroup in the centralizer of Γ in G, and ϱ is a cuspidal representation on the finite reductive quotient of P′0. In this paper, we explicitly describe those Hecke algebras when P′0 is a minimal parahoric subgroup, ϱ is trivial and ρΣ is a character.

This paper is concerned with rational representations of reductive algebraic groups over fields of positive characteristic $p$. Let $G$ be a simple algebraic group of rank $\ell$. It is shown that a rational representation of $G$ is semisimple provided that its dimension does not exceed $\ell p$. Furthermore, this result is improved by introducing a certain quantity $\mathcal{C}$ which is a quadratic function of $\ell$. Roughly speaking, it is shown that any rational $G$ module of dimension less than $\mathcal{C} p$ is either semisimple or involves a subquotient from a finite list of exceptional modules.

Suppose that $L_1$ and $L_2$ are irreducible representations of $G$. The essential problem is to study the possible extensions between $L_1$ and $L_2$ provided $\dim L_1 + \dim L_2$ is smaller than $\mathcal{C} p$. In this paper, all relevant simple modules $L_i$ are characterized, the restricted Lie algebra cohomology with coefficients in $L_i$ is determined, and the decomposition of the corresponding Weyl modules is analysed. These data are then exploited to obtain the needed control of the extension theory.

1991 Mathematics Subject Classification: 20G05.