This article discusses the existence problem of a compact quotient of a symmetric space by a properly discontinuous group with emphasis on the non-Riemannian case. Discontinuous groups are not always abundant in a homogeneous space G/H if H is non-compact. The first half of the article elucidates general machinery to study discontinuous groups for G/H, followed by the most update and complete list of symmetric spaces with/without compact quotients. In the second half, as applications of general theory, we prove: (i) there exists a 15 dimensional compact pseudo-Riemannian manifold of signature (7, 8) with constant curvature, (ii) there exists a compact quotient of the complex sphere of dimension 1, 3 and 7, and (iii) there exists a compact quotient of the tangential space form of signature (p, q) if and only if p is smaller than the Hurwitz-Radon number of q.

We shall consider separable C *-dynamical systems (A, G, α) for which the induced action of the group G on the primitive ideal space Prim(A) of the C *-algebra A is free. We shall discuss how the representation theory of the associated crossed product C *-algebra A α G depends on the representation theory of A and the properties of the action of G on Prim(A) and the spectrumÂ. After surveying some earlier results, we shall describe some recent joint work with Astrid an Huef. The main tools are the notion of strength of convergence in orbit spaces and the notions of upper and lower multiplicities for irreducible representations. We apply these ideas to give necessary and sufficient conditions, in terms of A and the action of G, for A α G to be (i) a continuous trace C *-algebra, (ii) a Fell algebra and (iii) a bounded trace C *-algebra. For the case of amenable G, we can apply a result of Leung and Ng to determine when A α G is (iv) a liminal C *-algebra and (v) a Type I C *-algebra. The results in (i) and (iii)-(v) extend some earlier special cases in which the C *-algebra A was assumed to have the corresponding property.

For a nilpotent Lie group G and its closed subgroup L, Lipsman [13] conjectured that the L-action on some homogeneous space of G is proper in the sense of Palais if and only if the action is free. Nasrin [14] proved this conjecture assuming that G is a 2-step nilpotent Lie group. However, Lipsman's conjecture fails for the 4-step nilpotent case. This paper gives an affirmative solution to Lipsman's conjecture for the 3-step nilpotent case.

Exponential family plays an important role in information geometry. In [TY18], we introduced a method to construct an exponential family P = {p θ } θ∈Θ on a homogeneous space G/H from a pair (V, v0).Here V is a representation of G and v0 is an H-fixed vector in V . Then the following questions naturally arise: (Q1) when is the correspondence θ → p θ injective? (Q2) when do distinct pairs (V, v0) and (V ′ , v ′ 0 ) generate the same family? In this paper, we answer these two questions (Theorems 1 and 2). Moreover, in Section 3, we consider the case (G, H) = (R>0, {1}) with a certain representation on R 2 . Then we see the family obtained by our method is essentially generalized inverse Gaussian distribution (GIG).

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