Symmetry breaking operators for strongly spherical reductive pairs

TY - JOUR

T1 - Symmetry breaking operators for strongly spherical reductive pairs

AU - Frahm, Jan

PY - 2023/10/11

Y1 - 2023/10/11

N2 - A real reductive pair (G, H) is called strongly spherical if the homogeneous space (G × H)/ diag(H) is real spherical. This geometric condition is equivalent to the representa-tion-theoretic property that dim Hom H(π| H, τ) < ∞ for all smooth admissible representations π of G and τ of H. In this paper we explicitly construct for all strongly spherical pairs (G, H) intertwining operators in Hom H(π| H, τ) for π and τ spherical principal series representations of G and H. These so-called symmetry breaking operators depend holomorphically on the induction parameters and we further show that they generically span the space Hom H(π| H, τ). In the special case of multiplicity one pairs we extend our construction to vector-valued principal series representations and obtain generic formulas for the multiplicities between arbitrary principal series. As an application, we prove an early version of the Gross–Prasad conjecture for complex orthogonal groups, and also provide lower bounds for the dimension of the space of Shintani functions.

AB - A real reductive pair (G, H) is called strongly spherical if the homogeneous space (G × H)/ diag(H) is real spherical. This geometric condition is equivalent to the representa-tion-theoretic property that dim Hom H(π| H, τ) < ∞ for all smooth admissible representations π of G and τ of H. In this paper we explicitly construct for all strongly spherical pairs (G, H) intertwining operators in Hom H(π| H, τ) for π and τ spherical principal series representations of G and H. These so-called symmetry breaking operators depend holomorphically on the induction parameters and we further show that they generically span the space Hom H(π| H, τ). In the special case of multiplicity one pairs we extend our construction to vector-valued principal series representations and obtain generic formulas for the multiplicities between arbitrary principal series. As an application, we prove an early version of the Gross–Prasad conjecture for complex orthogonal groups, and also provide lower bounds for the dimension of the space of Shintani functions.

KW - 22E46 (Primary), 11F70, 53C30 (Secondary)

KW - math.NT

KW - math.RT

UR - http://www.scopus.com/inward/record.url?scp=85174564489&partnerID=8YFLogxK

U2 - 10.4171/PRIMS/59-2-2

DO - 10.4171/PRIMS/59-2-2

M3 - Journal article

SN - 0034-5318

VL - 59

SP - 259

EP - 337

JO - Publications of the Research Institute for Mathematical Sciences

JF - Publications of the Research Institute for Mathematical Sciences

IS - 2

ER -