Yakimova, Oksana: Gelfand pairs. - Bonn, 2005. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5N-05134
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5N-05134,
author = {{Oksana Yakimova}},
title = {Gelfand pairs},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2005,
note = {Let K be a compact subgroup of a real Lie group G. The homogeneous space G/K is called commutative or the pair (G, K) is called a Gelfand pair if the algebra of G-invariant differential operators on G/K is commutative. Symmetric Riemannian homogeneous spaces,introduced by Élie Cartan, and weakly symmetric homogeneous spaces, introduced by Selberg in his celebrated work on the trace formula, are commutative. In this Dissertation we prove an effective commutativity criterion and obtain the complete classification of Gelfand pairs.
As was proved by Vinberg, if G/K is commutative, then, up to a local isomorphism, G is a semi-direct product G=N⊂L, where N is the nilpotent radical of G, L is reductive with K⊂L, and N is either 2-step nilpotent or abelian. In Chapter 1 we impose on G/K two technical constrains: principality and Sp1-saturation. These conditions describe the behaviour of the connected centres Z(L)⊂L, Z(K)⊂K and normal subgroups of K and L isomorphic to Sp1. Under these constraints, the classification problem is reduced to the reductive case (G=L) and the Heisenberg case (L=K). In Chapter 1, we describe principal commutative homogeneous spaces such that there is a simple non-commutative normal subgroup Li of L which is not contained K and is not locally isomorphic to SU2.
In Chapter 2, G is supposed to be reductive. In this case the notions of commutative and weakly symmetric homogeneous spaces are equivalent; moreover, as was proved by Akhiezer and Vinberg, weakly symmetric spaces are real forms of complex affine spherical homogeneous spaces. Spherical affine homogeneous spaces are classified by Krämer (in case G is simple), and by Brion and Mikityuk, independently (G is semisimple). Classifications of Brion and Mikityuk are not complete. They describe only principal spherical homogeneous spaces. In Chapter 2, we fill in the gaps in these classifications and explicitly describe commutative homogeneous spaces of reductive groups.This chapter also contains a classification of weakly symmetric structures on G/K. We obtain many new examples of weakly symmetric Riemannian manifolds. Most of them are not symmetric under some particular choice of a G-invariant Riemannian metric.
In Chapter 3, we complete classification of principal Sp1-saturated commutative spaces of Heisenberg type, started by Benson-Ratcliff and Vinberg.
In Chapter 4, constraints of principality and Sp1-saturation are removed. Thus, all Gelfand pairs are classified.
In Chapter 5, we classify principal maximal Sp1-saturated weakly symmetric homogeneous spaces. The question whether each commutative homogeneous space is weakly symmetric was posed by Selberg. It was answered a few years ago in a negative way by Lauret. On the other hand, Akhiezer and Vinberg showed that commutative homogeneous spaces of reductive groups are weakly symmetric. We prove that if G/K is commutative, L=K, and N is a product of several Heisenberg groups, then G/K is weakly symmetric. Several new examples of commutative, but not weakly symmetric homogeneous spaces are obtained.},

url = {http://hdl.handle.net/20.500.11811/2142}

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