Let K be a compact Lie group acting by automorphisms on a nilpotent Lie group N . One calls (K,N) a Gelfand pair when the integrable K-invariant functions on N form a commutative algebra under convolution. We prove that in this case the coadjoint orbits for G := K nN which meet the annihilator k⊥ of the Lie algebra k of K do so in single K-orbits. This generalizes a result of the authors and R....

We construct examples of two-step and three-step nilpotent Lie groups whose automorphism groups are “small” in the sense of either not having a dense orbit for the action on the Lie group, or being nilpotent (the latter being stronger). From the results we also get new examples of compact manifolds covered by two-step simply connected nilpotent Lie groups which do not admit Anosov automorphisms...

Let K be a field and let A be a finitely generated prime K-algebra. We generalize a result of Smith and Zhang, showing that if A is not PI and does not have a locally nilpotent ideal, then the extended centre of A has transcendence degree at most GKdim(A) − 2 over K. As a consequence, we are able to show that if A is a prime K-algebra of quadratic growth, then either the extended centre is a fi...

Let M be a Galois cover of a nilpotent coadjoint orbit of a complex semisimple Lie group. We define the notion of a perfect Dixmier algebra for M and show how this produces a graded (non-local) equivariant star product on M with several very nice properties. This is part of a larger program we have been developing for working out the orbit method for nilpotent orbits.

Denote by (R, ·) the multiplicative semigroup of an associative algebra R over an infinite field, and let (R, ◦) represent R when viewed as a semigroup via the circle operation x ◦ y = x + y + xy. In this paper we characterize the existence of an identity in these semigroups in terms of the Lie structure of R. Namely, we prove that the following conditions on R are equivalent: the semigroup (R,...

An improvement of a bound of Yankosky (2003) is presented in this paper, thanks to a restriction which has been recently obtained by the authors on the Schur multiplier M(L) of a finite dimensional nilpotent Lie algebra L. It is also described the structure of all nilpotent Lie algebras such that the bound is attained. An important role is played by the presence of a derived subalgebra of maxim...

In this paper we investigate expanding maps on infra-nilmanifolds. Such manifolds are obtained as a quotient E\L, where L is a connected and simply connected nilpotent Lie group and E is a torsion-free uniform discrete subgroup of LoC, with C a compact subgroup of Aut(L). We show that if the Lie algebra of L is homogeneous (i.e., graded and generated by elements of degree 1), then the correspon...

Let G be a connected and simply connected nilpotent Lie group with Lie algebra g and unitary dual Ĝ. The moment map for π ∈ Ĝ sends smooth vectors in the representation space of π to g∗. The closure of the image of the moment map for π is called its moment set. N. Wildberger has proved that the moment set for π coincides with the closure of the convex hull of the corresponding coadjoint orbit. ...

To any connected and simply connected nilpotent Lie group N , one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N , via such affine transformations. We succeed in translating the existence question of such a simply transitive affine action to a corresponding question on ...