The spectra of 1-forms on simply connected compact irreducible Riemannian symmetric spaces

AbstractIn this paper we introduce two new methods for constructing harmonic morphisms from solvable Lie groups. The first method yields global solutions from any simply connected nilpotent Lie group and from any Riemannian symmetric space of non-compact type and rank r ≥ 3. The second method provides us with global solutions from any Damek–Ricci space and many non-compact Riemannian symmetric spaces. We then give a continuous family of 3-dimensional solvable Lie groups not admitting any complex-valued harmonic morphisms, not even locally.

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An extension theorem for non-compact split embedded Riemannian symmetric spaces and an application to their universal property

Advances in Geometry ◽

10.1515/advgeom-2019-0028 ◽

2020 ◽

Vol 20 (4) ◽

pp. 499-506

Author(s):

Julius Grüning ◽

Ralf Köhl

Keyword(s):

Symmetric Space ◽

Central Extension ◽

Hyperbolic Plane ◽

Symmetric Spaces ◽

Extension Theorem ◽

Universal Property ◽

Riemannian Symmetric Space ◽

Differential Structure ◽

Riemannian Symmetric Spaces ◽

Simply Connected

AbstractBy [5] it is known that a geodesic γ in an abstract reflection space X (in the sense of Loos, without any assumption of differential structure) canonically admits an action of a 1-parameter subgroup of the group of transvections of X. In this article, we modify these arguments in order to prove an analog of this result stating that, if X contains an embedded hyperbolic plane 𝓗 ⊂ X, then this yields a canonical action of a subgroup of the transvection group of X isomorphic to a perfect central extension of PSL2(ℝ). This result can be further extended to arbitrary Riemannian symmetric spaces of non-compact split type Y lying in X and can be used to prove that a Riemannian symmetric space and, more generally, the Kac–Moody symmetric space G/K for an algebraically simply connected two-spherical split Kac–Moody group G, as defined in [5], satisfies a universal property similar to the universal property that the group G satisfies itself.

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Continuous Hahn Polynomials of Differential Operator Argument and Analysis on Riemannian Symmetric Spaces of Constant Curvature

Canadian Journal of Mathematics ◽

10.4153/cjm-1992-044-4 ◽

1992 ◽

Vol 44 (4) ◽

pp. 750-773 ◽

Cited By ~ 1

Author(s):

Erich Badertscher ◽

Tom H. Koornwinder

Keyword(s):

Differential Operator ◽

Orthogonal Polynomials ◽

Constant Curvature ◽

Symmetric Spaces ◽

Spherical Functions ◽

Spaces Of Constant Curvature ◽

Hahn Polynomials ◽

Riemannian Symmetric Spaces ◽

Simply Connected ◽

Continuous Hahn Polynomials

AbstractFor the three types of simply connected Riemannian spaces of constant curvature it is shown that the associated spherical functions can be obtained from the corresponding (zonal) spherical functions by application of a differential operator of the form p(i d/dt), where p belongs to a system of orthogonal polynomials: Gegenbauer polynomials, Hahn polynomials or continuous symmetric Hahn polynomials. We give a group theoretic explanation of this phenomenon and relate the properties of the polynomials p to the properties of the corresponding representation. The method is extended to the case of intertwining functions.

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