Matrix coefficients and coadjoint orbits of compact Lie groups

Let be a compact Lie group. We use Weyl functional calculus (Anderson, 1969) and symplectic convexity theorems to determine the support and singular support of the operator-valued Fourier transform of the product of the -function and the pull-back of an arbitrary unitary irreducible representation of to the Lie algebra, strengthening and generalizing the results of Cazzaniga, 1992. We obtain as a consequence a new demonstration of the Kirillov correspondence for compact Lie groups.

     

Spectral synthesis for coadjoint orbits of nilpotent Lie groups

We determine the space of primary ideals in the group algebra \(L^{1}(G) \) of a connected nilpotent Lie group by identifying for every \(\pi \in \widehat{G} \) the family \(\mathcal I^\pi \) of primary ideals with hull \(\{\pi \} \) with the family of invariant subspaces of a certain finite dimensional sub-space \(\mathcal P_Q^\pi \) of the space of polynomials \(\mathcal P(G) \) on G.     

Generalized Littlewood–Richardson rule and sum of coadjoint orbits of compact Lie groups

Let G be a compact connected semisimple Lie group. We extend to all irreducible finite-dimensional representations of G a result of Heckman which provides a relation between the generalized Littlewood–Richardson rule and the sum of G-coadjoint orbits. As an application of our result, we describe the eigenvalues of a sum of two real skew-symmetric matrices.     

coadjoint orbits of compact Lie groups and generalized stereographic projection

We generalize the notion of stereographic projection to the case of an arbitrary compact Lie group and find the explicit form of the local complex parametrization of an orbit of the corresponding group. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 12, pp. 1714–1718, December, 1999.     

Nilpotent Lie algebras with 2-dimensional commutator ideals

We classify all (finitely dimensional) nilpotent Lie k-algebras h with 2-dimensional commutator ideals h^′, extending a known result to the case where h^′ is non-central and k is an arbitrary field. It turns out that, while the structure of h depends on the field k if h^′ is central, it is independent of k if h^′ is non-central and is uniquely determined by the dimension of h. In the case where k is algebraically or real closed, we also list all nilpotent Lie k-algebras h with 2-dimensional central commutator ideals h^′ and dim_kh?11.     

Deformations of adjoint orbits for semisimple Lie algebras and Lagrangian submanifolds

We give a coadjoint orbit's diffeomorphic deformation between the classical semisimple case and the semi-direct product given by a Cartan decomposition. The two structures admit the Hermitian symplectic form defined in a semisimple complex Lie algebra. We provide some applications such as the constructions of Lagrangian submanifolds.     

Harmonic morphisms from the classical non-compact semisimple Lie groups

We construct the first known complex-valued harmonic morphisms from the non-compact Lie groups SLn(R), SU^∗(2n) and Sp(n,R) equipped with their standard Riemannian metrics. We then introduce the notion of a bi-eigenfamily and employ this to construct the first known solutions on the non-compact Riemannian SO^∗(2n), SO(p,q), SU(p,q) and Sp(p,q). Applying a duality principle we then show how to manufacture the first known complex-valued harmonic morphisms from the compact Lie groups SO(n), SU(n) and Sp(n) equipped with semi-Riemannian metrics.     

Constructing semisimple subalgebras of semisimple Lie algebras

Algorithms are described that help with obtaining a classification of the semisimple subalgebras of a given semisimple Lie algebra, up to linear equivalence. The algorithms have been used to obtain classifications of the semisimple subalgebras of the simple Lie algebras of ranks ?8. These have been made available as a database inside the SLA package of GAP4. The subalgebras in this database are explicitly given, as well as the inclusion relations among them.     

Morse linear functionals on orbits of adjoint action of semisimple Lie groups

Linear functionals on the Lie algebra of an arbitrary semisimple compact Lie group with restrictions of these functionals onto an arbitrary orbit of the adjoint action are considered. Criteria for the criticality and non-degenerate criticality of a point on the orbit are formulated and proved for a given functional, a necessary and sufficient condition for a linear functional to be a Morse function on the orbit is also proved. A method calculating the indices of critical points and its applications in the study of topological properties of orbits are indicated.     

Poisson structures transverse to coadjoint orbits

We show that the Poisson structure transverse to a coadjoint orbit in the dual of a semisimple Lie algebra has a polynomial structure matrix, as conjectured by Damianou.     

Invariants of semisimple Lie algebras acting on associative algebras

If is a Lie algebra of derivations of an associative algebra , then the subalgebra of invariants is the set In this paper, we study the relationship between the structure of and the structure of , where is a finite dimensional semisimple Lie algebra over a field of characteristic zero acting finitely on , when is semiprime.

     

Invariant convex cones and causality in semisimple Lie algebras and groups

The convex cones in a simple Lie algebra $\text{G}$ invariant under the adjoint group G of $\text{G}$ are studied. Using a earlier abstract classification of such cones, we find explicit algebraic presentations of such cones in all the classical hermitian symmetric Lie algebras. (Nontrivial such cones exist only in these cases.) The G-orbits in such cones are listed. The notion of a temporal action of a Lie group with an invariant causal orientation upon a causally oriented manifold is defined. The canonical actions of such classical groups G as above on the S?hilov boundaries of the associated (tube-type) hermitian symmetric spaces are shown to be temporal actions. Corollaries are (1) the existence of nontrivial (Lie) semigroups S in the infinite-sheeted coverings $\text{G}\text{?}$ of G, which are invariant under conjugation by $\text{G}\text{?}$ and satisfy S ∩ S^?1 = {e}, and (2) the global causality (i.e. no “closed time-like curves”) of such covering groups $\text{G}\text{?}$.