## Analysis on the minimal representation of O($$p,$$ $$q$$). III: Ultrahyperbolic equations on $$\mathbb{R}^{p-1,q-1}$$.(English)Zbl 1039.22005

A new construction of minimal unitary representation [B. Kostant, Prog. Math. 92, 85–124 (1990; Zbl 0739.22012), B. Binegar and R. Zierau, Commun. Math. Phys. 138, 245–258 (1991; Zbl 0748.22009)] of the noncompact orthogonal group $$O(p, q)$$ via Euclidean Fourier analysis is given, which is an extension of the $$q=2$$ case, where the representation is the mass zero, spin zero representation realized in the Hilbert space of solutions to the corresponding wave equation (1970, Todorov). It is taken into account that the group $$O(p,q)$$ acts as the Möbius group of conformal transformations on the real finite-dimensional vector space $$R^{p-1,q-1}$$, and preserves a space of solutions of the ultrahyperbolic Laplace equation on $$R^{p-1,q-1}$$. In the paper under review a Hilbert space of solutions is constructed in an intrinsical way, so that $$O(p,q)$$ becomes a continuous irreducible unitary representation in this Hilbert space. It is also proved that this representation is unitarily equivalent to the representation of $$L^2(C)$$, where $$C$$ is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the Lie algebra of $$O(p,q)$$. The paper is organized as follows:
1. Introduction. 2. Ultrahyperbolic equation on $$R^{p-1,q-1}$$ and conformal group. 3. Square integrable functions on the cone. 4. Green function and inner product. 5. Bessel function and an integral formula of spherical functions. 6. Explicit inner product on solutions $$\square_{R^{p-1,q-1}} f=0$$.

### MSC:

 2.2e+46 Representations of Lie and linear algebraic groups over real fields: analytic methods

### Citations:

Zbl 0739.22012; Zbl 0748.22009; Zbl 1049.22006; Zbl 1046.22004
Full Text:

### References:

 [1] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Reprint of the 1972 edition, Dover, New York, 1992. [2] Bailey, W.N., Some infinite integrals involving Bessel functions, Proc. London math. soc., 40, 37-48, (1935-36) · Zbl 0012.21005 [3] Binegar, B.; Zierau, R., Unitarization of a singular representation of SO(p,q), Comm. math. phys., 138, 245-258, (1991) · Zbl 0748.22009 [4] Dvorsky, A.; Sahi, S., Explicit Hilbert spaces for certain unipotent representations II, Invent. math., 138, 203-224, (1999) · Zbl 0937.22006 [5] A. Erdélyi, Higher Transcendental Functions, Vol. I, Graw-Hill, New York, 1953. [6] A. Erdélyi, Tables of Integral Transforms, Vol. II, McGraw-Hill, New York, 1954. [7] I.M. Gelfand, G.E. Shilov, Generalized Functions, Vol. I, Academic Press, New York, 1964. [8] V. Guillemin, S. Sternberg, Variations on a Theme by Kepler, Amer. Math. Soc. Colloq. Publ. Vol. 42, Amer. Math. Soc., Providence, 1990. · Zbl 0724.70006 [9] Hörmander, L., Asgeirsson’s Mean value theorem and related identities, J. func. anal., 184, 337-402, (2001) · Zbl 1019.35022 [10] Kobayashi, T., Discrete decomposability of the restriction of $$Aq(λ)$$ with respect to reductive subgroups and its applications, Invent. math., 117, 181-205, (1994), Part II, Ann. of Math. 147 (1998) 709-729; Part III, Invent. Math. 131 (1998) 229-256 · Zbl 0826.22015 [11] T. Kobayashi, B. Ørsted, Analysis on the minimal representation of O(p,q)—Part I. Realization via conformal geometry, to appear in Adv. in Math. · Zbl 1046.22004 [12] T. Kobayashi, B. Ørsted, Analysis on the minimal representation of O(p,q)—Part II. Branching laws, to appear in Adv. in Math. · Zbl 1049.22006 [13] B. Kostant, The vanishing scalar curvature and the minimal unitary representation of SO(4,4), in: A. Connes et al. (Eds.), Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Progress Math. 92 (1990) 85-124. · Zbl 0739.22012 [14] I.T. Todorov, Derivation and solution of an infinite-component wave equation for the relativistic Coulomb problem, Lecture Notes in Physics, Group Representations in Mathematics and Physics (Rencontres, Battelle Res. Inst., Seattle, 1969), Vol. 6, Springer, Berlin, 1970, pp. 254-278.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.