Restriction of complementary series representations of O(1,N) to symmetric subgroups

TY - UNPB

T1 - Restriction of complementary series representations of O(1,N) to symmetric subgroups

AU - Möllers, Jan

AU - Oshima, Yoshiki

PY - 2012

Y1 - 2012

N2 - We find the complete branching law for the restriction of complementary series representations of O(1, n+1) to the symmetric subgroup O(1,m+1)× O(n − m), 0 m < n. The decomposition consists of a continuous part and a discrete part which is trivial for some parameters. The continuous part is given by a direct integral of principal series representations whereas the discrete part consists of finitely many complementary series representations. The explicit Plancherel formula is computed on the Fourier transformed side of the non-compact realization of the complementary series by using the spectral decomposition of a certain hypergeometric type ordinary differential operator. The main tool connecting this differential operator with the representations are second order Bessel operators which describe the Lie algebra action in this realization.

AB - We find the complete branching law for the restriction of complementary series representations of O(1, n+1) to the symmetric subgroup O(1,m+1)× O(n − m), 0 m < n. The decomposition consists of a continuous part and a discrete part which is trivial for some parameters. The continuous part is given by a direct integral of principal series representations whereas the discrete part consists of finitely many complementary series representations. The explicit Plancherel formula is computed on the Fourier transformed side of the non-compact realization of the complementary series by using the spectral decomposition of a certain hypergeometric type ordinary differential operator. The main tool connecting this differential operator with the representations are second order Bessel operators which describe the Lie algebra action in this realization.

M3 - Working paper

T3 - Preprints

BT - Restriction of complementary series representations of O(1,N) to symmetric subgroups

PB - Department of Mathematics, Aarhus University

ER -