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Analysis on the minimal representation of $$\mathrm O(p,q)$$. II: Branching laws. (English) Zbl 1049.22006
[For Part I, see ibid. 180, No. 2, 486–512 (2003; Zbl 1046.22004).]
Let $$G$$ be a Lie group and $$G'$$ its subgroup, $$\widehat G$$ the unitary dual of $$G$$. If $$\pi\in\widehat G$$, the restriction $$\pi_{\widehat G'}$$ is generally not irreducible and we have an irreducible decomposition $\pi_{G'}\cong\int_{\widehat G'}m_{\pi}(\tau)\tau \,d\mu(\tau).$ This is the so-called branching law. The present paper is the second in a series of papers devoted to the analysis of the minimal representation $$\widetilde\omega^{p,q}$$ of the indefinite orthogonal group $$G = O(p,q)$$, i.e., from the point of view of conformal geometry, the authors find the restriction of $$\widetilde\omega^{p,q}$$ with respect to the symmetric pair $$(G,G') = (O(p,q),O(p'q')\times O(p'',q''))$$: A canonical conclusion obtained by them is the branching law for $$O(pq)\downarrow O(p,q')\times O(q'')$$: If $$q''\geq 1$$ and $$q'+ q'' = q$$, the twisted pull-back $$\widetilde\Phi{*_1}$$ of the local conformal map between spheres and hyperboloids gives an explicit irreducible decomposition of the unitary representation $$\tilde\omega^{p,q}$$ when restricted to $$O(p,q'')\times O(q'')$$ $\widetilde\Phi*_1: \widetilde\omega^{p,q}|_{O(p,q')\times O(q'')} \to \sum_{l=0}^{\infty} \pi^{p,q'}_{+,l+q''/2-1}\otimes \mathcal H^l (R^{q''}),$ where the $$\mathcal H^l(R^q)$$ are the usual spherical harmonics for the compact orthogonal group $$O(q)$$ and the $$\pi^{p,q}_{+,\lambda}$$ are the representations of the non-compact orthogonal group $$O(p,q)$$, which can be regarded as discrete series representations on hyperboloids, or as cohomological induced representations from characters of certain $$\theta$$-stable parabolic subalgebras. The authors give also the Parseval-Plancherel formula for the above restriction: let $$\widetilde\bigtriangleup_M$$ be the Yamabe operator on $$M = S^{p-1}\times S^{q-1}$$ and $$F\in \operatorname{Ker} \widetilde\bigtriangleup_M$$ and develop $$F$$ as $$f =\sum_{l=0}^{\infty} F_l^{(1)}F_l^{(2)}$$ according to the above decomposition, then $\| F\|^2_{\widetilde\omega^{p,q}}= \sum_{l=0}^{\infty}\| F^{(1)}\|^2_{\pi^{p,q'}_{+,l+q''/2-1}} \| F^{(2)}\|^2_{L^2 (S^{q''}-1)}.$ Finally the authors use certain Sobolev estimates to construct infinitely many discrete spectra when both factors in $$G'$$ are non-compact, they also conjecture the form of the fall discrete spectrum.
Reviewer: Fuliu Zhu (Hubei)

##### MSC:
 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 22E46 Semisimple Lie groups and their representations 53A30 Conformal differential geometry (MSC2010)
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