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Function theory related to the group \(\mathrm{PSL}_2(\mathbb R)\). (English) Zbl 1263.22007

Farkas, Hershel M. (ed.) et al., From Fourier analysis and number theory to Radon transforms and geometry. In Memory of Leon Ehrenpreis. Berlin: Springer (ISBN 978-1-4614-4074-1/hbk; 978-1-4614-4075-8/ebook). Developments in Mathematics 28, 107-201 (2013).
The authors’ aim is to discuss some of the analytic aspects of the group \(G=\mathrm{PSL}_2(\mathbb R)\) acting on the hyperbolic plane and its boundary. Everything the authors do is related in some way to the spherical principal series representations of the group \(G\).
The authors review the standard models used to realize these representations and then describe a number of new properties and new models. Some of these are surprising and interesting in their own right, while others have already proved to be useful in connection with the study of cohomological applications of automorphic forms and may potentially have other applications in the future.
The principal series representations of \(G\) are indexed by a complex number \(s\), called the spectral parameter, which is assumed to have real part between 0 and 1. There are two basic realizations. One is the space \(\mathcal{V}_s\) of functions on \(\mathbb R\) with the (right) action of \(G\) given by \[ (\varphi|g)(t)=|ct+d|^{-2s}\varphi\left(\frac{at+b}{ct+d}\right),\quad t\in\mathbb R,\quad g=\left[\begin{matrix} a&b\\c&d\end{matrix}\right]\in G. \] The other is the space \(\mathcal E_s\) of functions \(u\) on \(\mathfrak H\) (complex upper half-plane) satisfying \[ \Delta u(z)=s(1-s)u(s),\quad z\in\mathfrak H, \] where \(\Delta=-y^2(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2})\), \(z=x+iy\in\mathfrak H\), is the hyperbolic Laplace operator, with the action \(u\mapsto u\circ g\). They are related by Helgason’s Poisson transform \[ \varphi(t)\mapsto (\operatorname{P}_s\varphi)(z)=\frac{1}{\pi}\int_{-\infty}^\infty \varphi(t)R(t;z)^{1-s}\,dt, \] where \(R(t;z)=\frac{y}{(z-t)(\overline z-t)}\) for \(z=x+iy\in\mathfrak H\) and \(t\in\mathbb C\).
The principal series representations can be realized in various ways. One of the aims of Section 2, Principal series representation \(\mathcal{V}_s\), is to gain insight by combining several of these models. Six standard models for the continuous vectors in the principal series representation are given. The larger space of hyperfunction vectors in some of these models is presented, and the isomorphism (for \(0<\operatorname{Re}s<1\)) between the principal series representations with the values \(s\) and \(1-s\) of the spectral parameter is discussed.
In Section 3, Laplace eigenfunctions and the Poisson transformation, the principal series representations are realized as the space of eigenfunctions of the Laplace operator \(\varDelta\) in the hyperbolic plane \(\mathbb H\). This model has several advantages: the action of \(G\) involves no automorphy factor at all; the model does not give a preferential treatment to any point; all vectors correspond to actual functions, with no need to work with distributions or hyperfunctions; and the values \(s\) and \(1-s\) of the spectral parameter give the same space. The isomorphism from the models on the boundary used so far to the hyperbolic plane model is given by a simple integral transform (Poisson map). Before discussing this transformation, the authors consider eigenfunctions of the Laplace operator on hyperbolic space and discuss Green’s form. Finally, second-order eigenfunctions, i.e., functions on \(\mathbb H\) that are annihilated by \((\varDelta-s(1-s))^2\), are considered.
In Section 4, Hybrid models for the principal series representation, the authors introduce the canonical model of the principal series, discussed earlier, and define first two other models of \(\mathcal{V}_s\) in functions or hyperfunctions on \(\partial\mathbb H\times\mathbb H\), called hybrid models, since they mix the properties of the model of \(\mathcal{V}_s\) in eigenfunctions, as discussed in Section 3, with the models discussed in Section 2. The second of these, called the flabby hybrid model, contains the canonical model as a special subspace. The advantage of the canonical model becomes very clear when the authors give an explicit inverse for the Poisson transformation whose image coincides exactly with the canonical model. The canonical model is characterized as a space of functions on \((\mathbb P^1_{\mathbb C}\setminus\mathbb P^1_{\mathbb R})\times \mathfrak H\) satisfying a certain system of differential equations. These differential equations are used to define a sheaf \(\mathcal D_s\) on \(\mathbb P^1_{\mathbb C}\times \mathfrak H\), the sheaf of mixed eigenfunctions. The properties of this sheaf and of its sections over other natural subsets of \(\mathbb P^1_{\mathbb C}\times \mathfrak H\) are studied.
The space \(\mathcal E_s\) of \(\lambda_s\)-eigenfunctions of the Laplace operator embeds canonically into the larger space \(\mathcal F_s\) of germs of eigenfunctions near the boundary of \(\mathbb H\). In Section 5, Eigenfunctions near \(\partial\mathbb H\) and the transverse Poisson transform, the authors introduce the subspace \(\mathcal W_s^\omega\) of \(\mathcal F_s\) consisting of eigenfunction germs that have the behavior \(y^s\times(\text{analytic across }\mathbb R)\) near \(\mathbb R\), together with the corresponding property near \(\infty\in\mathbb P_\mathbb R^1\), and show that \(\mathcal F_s\) splits canonically as the direct sum of \(\mathcal E_s\) and \(\mathcal W_s^\omega\). The space \(\mathcal W_s^\omega\) is shown to be isomorphic to \(\mathcal{V}_s^\omega\) by integral transformations, one of which is called the transverse Poisson transformation because it is given by the same integral as the usual Poisson transformation \(\mathcal{V}_s^\omega\to\mathcal E_s\) but with the integral taken across rather than along \(\mathbb P_\mathbb R^1\). This transformation gives another model \(\mathcal W_s^\omega\) of the principal series representation \(\mathcal{V}_s^\omega\), which has proved to be extremely useful in the cohomological study of Maas forms. The duality of \(\mathcal{V}_s^\omega\) and \(\mathcal{V}_{1-s}^{-\omega}\) is described in terms of a pairing of the isomorphic spaces \(\mathcal W_s^\omega\) and \(\mathcal E_{1-s}\). By using jets of \(\lambda_s\)-eigenfunctions of the Laplace operator, a smooth version \(\mathcal W_s^\infty\) of \(\mathcal W_s^\omega\) isomorphic to \(\mathcal{V}_s^\infty\) is constructed.
In Section 6, Boundary behavior of mixed eigenfunctions, the authors combine ideas from Sections 4 and 5. Representatives \(u\) of elements of \(\mathcal W_{1-s}^\omega\) have the special property that \((1-|w|^2)^{s-1}u(w)\) (in the circle model) or \(y^{s-1}u(z)\) (in the line model) extends analytically across the boundary \(\partial\mathbb H\). If such an eigenfunction occurs in a section \((h,u)\) of the sheaf \(\mathcal D_s\) of mixed eigenfunctions, we may ask whether a suitable multiple of \(h\) also extends across the boundary. It is shown that this is true locally, but not globally. The authors use the differential equations satisfied by \(y^{-s}u\) for representatives \(u\) of elements of \(\mathcal W_{s}^\omega\) to define an extension \(\mathcal A_s\) of the sheaf \(\mathcal E_s\) from \(\mathfrak H\) to \(\mathbb P_\mathbb C^1\). The sheaf \(\mathcal D_s\) on \(\mathbb P^1_{\mathbb C}\times\mathfrak H\) is also extended to a sheaf \(\mathcal D^\ast_s\) on \(\mathbb P^1_{\mathbb C}\times\mathbb P^1_{\mathbb C}\) that has the same relation to \(\mathcal A_{1-s}\) as the relation of \(\mathcal D_s\) to \(\mathcal E_s=\mathcal E_{1-s}\). It is shown that the power series expansion of sections of \(\mathcal A_s\) leads in a natural way to sections of \(\mathcal D^\ast_{1-s}\). Finally, the section of \(\mathcal D_s\) near \(\mathbb P^1_{\mathbb R}\times\mathfrak H\) is considered.
The eigenfunctions often have the local form \(y^s\times (\text{analytic})+y^{1-s}\times(\text{analytic})\) near points of \(\mathbb R\). In Section 7, Boundary splitting of eigenfunctions, this phenomenon is systematically considered in both the analytical context and the differentiable context. This leads, in particular, to a description of both \(\mathcal E_s^\omega=\operatorname{P}_s(\mathcal{V}_s^\omega)\) and \(\mathcal E_s^\infty= \operatorname{ P}_s(\mathcal{V}_s^\infty)\) in terms of boundary behavior. Results concerning the boundary behavior of elements of \(\mathcal E_s\) are known (also for more general groups). However, the proposed approach is more elementary and also includes several formulas that do not seem to be in the literature and that are useful for certain applications.
The authors end by describing a number of examples of eigenfunctions of the Laplace operator, of Poisson transforms, of transverse Poisson transforms, and of explicit potentials of Green’s form \(\{u,v\}\) for various special choices of \(u\) and \(v\), as well as some formulas for the action of the Lie algebra of \(G\).
For the entire collection see [Zbl 1250.00009].

MSC:

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22E30 Analysis on real and complex Lie groups
22-06 Proceedings, conferences, collections, etc. pertaining to topological groups
32A45 Hyperfunctions
35J08 Green’s functions for elliptic equations
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
46G15 Functional analytic lifting theory
58C40 Spectral theory; eigenvalue problems on manifolds
22E50 Representations of Lie and linear algebraic groups over local fields
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