On the Decomposition of a Representation of SO_n When Restricted to SO_n-1

Abstract

Let k be a local field, with char(k) ≠ 2. A quadratic space V over k is a finite dimensional vector space together with a non-degenerate quadratic form Q: V → k.The special orthogonal group SO(V) consists of all linear maps T: V → V which satisfy:Q(Tv) = Q(v) for all ν and det T = 1.

The Gross–Prasad conjecture and local theta correspondence

“…In this appendix, we prove a conjecture of Gross-Prasad and Rallis [23,Conjecture 2.6] under a certain working hypothesis. B.1.…”

“…In [23], [24], [15], [16], a restriction problem in the representation theory of classical groups was studied and a precise conjecture was formulated for this restriction problem. This so-called Gross-Prasad (GP) conjecture has generated much interest in recent years.…”

On the Periods of Automorphic Forms on Special Orthogonal Groups and the Gross–Prasad Conjecture

“…Let π 1 ≅ ⊗ v π 1,v and π 0 ≅ ⊗ v π 0,v be irreducible tempered cuspidal automorphic representations of G 1 (A) and G 0 (A), respectively. Assume that Hom G 0 (kv) (π 1,v ⊗π 0,v , C) ̸ = {0} for any place v of k. Then the global Gross-Prasad conjecture [12] asserts that In this paper, we would like to formulate a conjecture, which expresses the period 〈ϕ 1 …”

“…In early 90's, Gross and Prasad [12], [13] gave a series of fascinating conjectures on the restriction of automorphic representation of a special orthogonal group to a smaller special orthogonal subgroup. We now recall their global conjecture.…”

The Local Langlands Conjecture for Sp(4)

“…To deal with the case p = 2, it is probably easier to work with the parameter ϕ. One has the following general statement [GP,Corollary 6.6]:…”