Gelfand pairs

Let K ⊂ G be a compact subgroup of a real Lie group G. Denote by D(X) thealgebra of G-invariant differential operators on the homogeneous space X = G/K. ThenX is called commutative or the pair (G,K) is called a Gelfand pair if the algebra D(X)is commutative. Symmetric Riemannian homogeneous spaces introduced by Élie Cartanand weakly symmetric homogeneous spaces introduced by Selberg in [41] are commutative.In this Dissertation we prove an effective commutativity criterion and obtain the completeclassification of Gelfand pairs.If X = G/K is commutative, then, up to a local isomorphism, G has a factorisationG = N h L, where N is either 2-step nilpotent or abelian and L is reductive with K ⊂ L,see [43]. In Chapter 1 we impose on X 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 reductive case (G = L) and Heisenberg case (L = K).In Chapter 1, we describe principal commutative homogeneous spaces such that there is asimple non-commutative ideal li 6= su2 of LieL which is not contained in LieK.In Chapter 2, G is supposed to be reductive. In this case the notions of commutative andweakly symmetric homogeneous spaces are equivalent; moreover, weakly symmetric spacesare real forms of complex affine spherical homogeneous spaces, see Akhiezer-Vinberg [1].Spherical affine homogeneous spaces are classified by Krämer [25] (G is simple), by Brion[10] and Mikityuk [30] (G is semisimple). Classifications of [10] and [30] are not complete.They describe only principal spherical homogeneous spaces. In Chapter 2, we fill in the gapsin these classifications and explicitly describe commutative homogeneous spaces of reductivegroups. 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 themare not symmetric under some particular choice of a G-invariant Riemannian metric.In Chapter 3, we complete classification of principal Sp1-saturated commutative spacesof Heisenberg type, started by Benson-Ratcliff [3] and Vinberg [43], [44].In Chapter 4, constraints of principality and Sp1-saturation are removed. Thus, allGelfand pairs are classified.In Chapter 5, we classify principal maximal Sp1-saturated weakly symmetric homoge-neous spaces. The question whether each commutative homogeneous space is weakly sym-metric was posed by Selberg [41]. It was answered a few years ago in a negative way byLauret [26]. On the other hand, commutative homogeneous spaces of reductive groups areweakly symmetric, see [1]. We prove that if X = (N h K)/K is commutative and N is aproduct of several Heisenberg groups, then X is weakly symmetric. Several new examplesof commutative, but not weakly symmetric homogeneous spaces are obtained.