Restriction of Most Degenerate Representations of $O(1,N)$ with Respect to Symmetric Pairs

J. Math. Sci. Univ. Tokyo
Vol. 22 (2015), No. 1, Page 279–338.

Möllers, Jan ; Oshima, Yoshiki
Restriction of Most Degenerate Representations of $O(1,N)$ with Respect to Symmetric Pairs
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We find the complete branching law for the restriction of certain unitary representations of $O(1,n+1)$ to the subgroups $O(1,m+1)\times O(n-m)$, $0\leq m\leq n$. The unitary representations we consider are those induced from a character of a parabolic subgroup or its irreducible quotient. They belong either to the unitary spherical principal series, the spherical complementary series or discrete series for the hyperboloid. In the crucial case $0\lt m\lt n$ he decomposition consists of a continuous part and a discrete part. The continuous part is given by a direct integral of unitary principal series representations whereas the discrete part consists of finitely many representations which either belong to the complementary series or are discrete series for the hyperboloid. The explicit Plancherel formula is computed on the Fourier transformed side of the non-compact realization of the representations 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. To derive the spectral decomposition of the ordinary differential operator we use Kodaira's formula for the spectral decomposition of Schrödinger type operators.

Keywords: Unitary representation, complementary series, principal series, discrete series, branching law, Bessel operators, hypergeometric function, Kodaira--Titchmarsh formula.

Mathematics Subject Classification (2010): Primary 22E46; Secondary 33C05, 34B24.
Mathematical Reviews Number: MR3329198

Received: 2014-06-24