Speaker: 陈飞，清华大学丘成桐数学科学中心

Inviter: 邹佳良

Title: On local intertwining periods

Language: Chinese

Time & Venue: 2026.10.09 10:30-11:30 南楼N820

Abstract: Let F be a non-Archimedean local field with char(F)≠2 and let G be a reductive group over F. Let θ be an involution of G and let H=Gᵅ be the corresponding symmetric subgroup. Given a smooth character χ: H → ℂˣ, one of the central problems of the relative Langlands program is to understand when a representation is (H,χ)-distinguished. That is, for which π ∈ Rep(G), Hom_H(π,χ)≠0 holds? In this talk, we focus on the case where π=I_Pᴳ(σ) is an induced representation. Here P is a parabolic subgroup of G and M is a Levi of P that is θ-stable. For any ℓ ∈ Hom_{H∩M}(σ, δ_{P∩M} δ_P⁻¹ᐟ²χ), f ∈ I_Pᴳ(σ) and λ a θ anti-invariant unramified character of M, one can formally write down an integral ∫_{H∩P\H} χ⁻¹(h) ℓ(f_λ(h)) dh, known as the local intertwining period. When the integral converges absolutely, it defines a non-zero element in Hom_H(I_Pᴳ(σ⊗λ),χ). Previously, it was known in several special cases or under certain modulus assumption that the above integral converges absolutely for λ in an open subset and admits a meromorphic continuation. We study the local intertwining periods from two aspects. On the one hand, we develop an algebraic theory which works for any algebraically closed coefficient field of characteristic different from the residue characteristic of F. On the other hand, when the coefficient field is ℂ, we establish the absolute convergence and meromorphic continuation in full generality. This is based on joint work with Nadir Matringe.