For a locally compact metrisable group G, we study the action of ${\rm Aut}(G)$ on ${\rm Sub}_G$ , the set of closed subgroups of G endowed with the Chabauty topology. Given an automorphism T of G, we relate the distality of the T-action on ${\rm Sub}_G$ with that of the T-action on G under a certain condition. If G is a connected Lie group, we characterise the distality of the T-action on ${\rm Sub}_G$ in terms of compactness of the closed subgroup generated by T in ${\rm Aut}(G)$ under certain conditions on the center of G or on T as follows: G has no compact central subgroup of positive dimension or T is unipotent or T is contained in the connected component of the identity in ${\rm Aut}(G)$ . Moreover, we also show that a connected Lie group G acts distally on ${\rm Sub}_G$ if and only if G is either compact or it is isomorphic to a direct product of a compact group and a vector group. All the results on the Lie groups mentioned above hold for the action on ${\rm Sub}^a_G$ , a subset of ${\rm Sub}_G$ consisting of closed abelian subgroups of G.

It is shown that Jamison sequences, introduced in 2007 by Badea and Grivaux, arise naturally in the study of topological groups with no small subgroups, of Banach or normed algebra elements whose powers are close to identity along subsequences, and in characterizations of (self-adjoint) positive operators by the accretiveness of some of their powers. The common core of these results is a description of those sequences for which non-identity elements in Lie groups or normed algebras escape an arbitrary small neighborhood of the identity in a number of steps belonging to the given sequence. Several spectral characterizations of Jamison sequences are given, and other related results are proved.

Let $W$ be a Weyl group, $\sum $ a set of simple reflections in $W$ related to a basis $\Delta $ for the root system $\Phi $ associated with $W$ and $\theta $ an involution such that $\theta (\Delta )\,=\,\Delta $. We show that the set of $\theta $- twisted involutions in $W$, ${{\mathcal{J}}_{\theta }}\,=\,\{w\,\in \,W\,|\,\theta (w)\,=\,{{w}^{-1}}\}$ is in one to one correspondence with the set of regular involutions ${{\mathcal{J}}_{\text{ID}}}$. The elements of ${{\mathcal{J}}_{\theta }}$ are characterized by sequences in $\sum $ which induce an ordering called the Richardson–Springer Poset. In particular, for $\Phi $ irreducible, the ascending Richardson–Springer Poset of ${{\mathcal{J}}_{\theta }}$, for nontrivial $\theta $ is identical to the descending Richardson–Springer Poset of ${{\mathcal{J}}_{\text{ID}}}$.

By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an extension of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups preserves the cohomology.

Let $G$ be a real Lie group and $X$ a real analytic manifold. Suppose that $G$ acts analytically on $X$ with finitely many orbits. Then the orbits are subanalytic in $X$. As a consequence we show that the micro-support of a $G$-equivariant sheaf on $X$ is contained in the conormal variety of the $G$-action.

In this article, making use of the second author’s criterion for exponentiality of a connected solvable Lie group, we give a rather simple necessary and sufficient condition for the semidirect product of a torus acting on certain connected solvable Lie groups to be exponential.

If a Lie group acts faithfully as a transitive group of contact transformations of a compact manifold it is either compact with centre of dimension at most 1 or non-compact simple. The latter case is described

We prove a homogenization formula for a sub-Laplacian are left invariant Hörmander vector fields) on a connected Lie group Gof polynomial growth. Then using a rescaling argument inspired from M. Avellanedaand F. H. Lin [2], we prove Harnack inequalities for the positive solutions of the equation (∂/∂t+ L)u= 0. Using these inequalities and further exploiting the algebraic structure of Gwe prove that the Riesz transforms , are bounded on Lq,1 < q <+∞ and from L1to weak-L1.

The action of Lie groups as transitive groups of restricted contact transformations of compact manifolds are classified.

Let G be a Lie group with adjoint orbits θi and corresponding conjugacy classes Ci = exp θi, i = 1, 2, 3. We show that if G is nilpotent or compact, there is a neighbourhood U of 0 in g such that if θi ∊ U then θ3 ⊂ θ1 + θ2 if and only if C3 ⊂ C1C2.

Given a finite dimensional Euclidean vector space V, ( , ) and an involution τ of V, one can form the bilinear function ( , )τ defined by (x, y)τ = (τ(x), y), x,y ∊ V.

Let O(τ) = {ϕ ∊ GL(V)|(ϕx, ϕy)τ = (x, y)τ}.

If t is self-adjoint the structure of O(t) is well known. The purpose of this paper is to detemine the structure of O(t) in the general case. This structure is also determined in the complex and quaternionic case, as well as the case when the condition on t is replaced by τ2 = ∊ι, ∊ ∈ ℝ.