The Orbit Method and Gelfand Pairs Associated with Nilpotent Lie Groups

On Gelfand pairs associated with nilpotent Lie groups

Gelfand pairs associated with finite Heisenberg groups

A topological group G together with a compact subgroup K are said to form a Gelfand pair if the set L1(K\G/K) of K-bi-invariant integrable functions on G is a commutative algebra under convolution. The situation where G and K are Lie groups has been the focus of extensive and ongoing investigation. Riemannian symmetric spaces G/K furnish the most widely studied and best understood examples. ([H...

The Space of Bounded Spherical Functions on the Free Two Step Nilpotent Lie Group

Let N be a connected and simply connected 2-step nilpotent Lie group and K be a compact subgroup of Aut(N). We say that (K,N) is a Gelfand pair when the set of integrable K-invariant functions on N forms an abelian algebra under convolution. In this paper, we construct a one-to-one correspondence between the set ∆(K, N) of bounded spherical functions for such a Gelfand pair and a set A(K, N) of...

The structure of a pair of nilpotent Lie algebras

Assume that $(N,L)$, is a pair of finite dimensional nilpotent Lie algebras, in which $L$ is non-abelian and $N$ is an ideal in $L$ and also $mathcal{M}(N,L)$ is the Schur multiplier of the pair $(N,L)$. Motivated by characterization of the pairs $(N,L)$ of finite dimensional nilpotent Lie algebras by their Schur multipliers (Arabyani, et al. 2014) we prove some properties of a pair of nilpoten...

Some Two–Step and Three–Step Nilpotent Lie Groups with Small Automorphism Groups

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...