Abstract
We study analytic properties of the action of \(\mathrm{{PSL}}_{2}(\mathbb{R})\) on spaces of functions on the hyperbolic plane. The central role is played by principal series representations. We describe and study a number of different models of the principal series, some old and some new. Although these models are isomorphic, they arise as the spaces of global sections of completely different equivariant sheaves and thus bring out different underlying properties of the principal series.
The two standard models of the principal series are the space of eigenfunctions of the hyperbolic Laplace operator in the hyperbolic plane (upper half-plane or disk) and the space of hyperfunctions on the boundary of the hyperbolic plane. They are related by a well-known integral transformation called the Poisson transformation. We give an explicit integral formula for its inverse.
The Poisson transformation and several other properties of the principal series become extremely simple in a new model that is defined as the space of solutions of a certain two-by-two system of first-order differential equations. We call this the canonical model because it gives canonical representatives for the hyperfunctions defining one of the standard models.
Another model, which has proved useful for establishing the relation between Maass forms and cohomology, is in spaces of germs of eigenfunctions of the Laplace operator near the boundary of the hyperbolic plane. We describe the properties of this model, relate it by explicit integral transformations to the spaces of analytic vectors in the standard models of the principal series, and use it to give an explicit description of the space of \({C}^{\infty }\)-vectors.
Mathematics Subject Classification (2010): 22E50,22E30,22E45,32A45,35J08,43A65,46F15,58C40
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Notes
- 1.
More generally, for any differentiable functions u and v on \(\mathbb{H}\) we have
$$\mathrm{d}\{u,v\}\; =\; 2\mathrm{i}\,\mathrm{d}[u,v]\; =\;{\Bigl ( (\Delta u)\,v - u\,(\Delta v)\Bigr )}\,\mathrm{d}\mu,$$where \(\mathrm{d}\mu \) (\(\,=\, {y}^{-2}\,\mathrm{d}x\,\mathrm{d}y\) in the upper half-plane model) is the invariant measure in \(\mathbb{H}\).
- 2.
Here one has the choice to impose any desired regularity conditions (C 0, \({C}^{\infty }\), C ω, …) in the second variable or in both variables jointly. We do not fix any such choice since none of our considerations depend on which choice is made and since in any case the most interesting elements of this space, like the canonical representative introduced below, are analytic in both variables.
- 3.
The Pochhammer symbol (x) k is defined for k < 0 as \({(x - 1)}^{-1}\cdots {(x -\vert k\vert )}^{-1}\), so that \({(x)}_{k} = \Gamma (x + k)/\Gamma (x)\) in all cases.
References
E. P. van den Ban, H. Schlichtkrull, Asymptotic expansions and boundary values of eigenfunctions on Riemannian spaces. J. reine angew. Math. 380 (1987), 108–165.
R. Bruggeman, J. Lewis and Preprint.
A. Erdélyi et al., Higher transcendental functions, Vol. I. McGraw-Hill, 1953.
R. Hartshorne, Algebraic geometry. Springer-Verlag, 1977.
S. Helgason, Topics in harmonic analysis on homogeneous spaces. Progr. in Math. 13, Birkhäuser, 1981.
S. Helgason, Groups and geometric analysis. Math. Surv. and Monogr. 83, AMS, 1984.
M. Kashiwara, A. Kowata, K. Minekura, K. Okamoto, T. Oshima, M. Tanaka, Eigenfunctions of invariant differential operators on a symmetric space. Ann. Math. 107 (1978), 1–39.
A. W. Knapp, Representation theory of semisimple Lie groups, an overview based on examples. Princeton University Press, Princeton, 1986.
J. B. Lewis, Eigenfunctions on symmetric spaces with distribution-valued boundary forms. J. Funct. Anal. 29 (1978), 287–307.
J. Lewis, D. Zagier, Period functions for Maass wave forms. I. Ann. Math. 153 (2001), 191–258.
H. Schlichtkrull, Hyperfunctions and harmonic analysis on symmetric spaces. Progr. in Math. 49, Birkhäuser, 1984.
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Bruggeman, R., Lewis, J., Zagier, D. (2013). Function Theory Related to the Group PSL2(ℝ). In: Farkas, H., Gunning, R., Knopp, M., Taylor, B. (eds) From Fourier Analysis and Number Theory to Radon Transforms and Geometry. Developments in Mathematics, vol 28. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4075-8_7
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