Complex adjoint orbits in Lie theory and geometry

Reducible Spectral Curves and the Hyperkähler Geometry of Adjoint Orbits

We study the hyperkähler geometry of a regular semisimple adjoint orbit of SL(k, C) via the algebraic geometry of the corresponding reducible spectral curve. It is by now well-known that adjoint orbits of complex semisimple Lie groups admit hyperkähler structures. Among several constructions of such structures, it is the one given by Kronheimer [11], later extended by Biquard [5] and by Kovalev...

Star-shapedness and K-orbits in Complex Semisimple Lie Algebras

Given a complex semisimple Lie algebra g = k + ik (k is a compact real form of g), let π : g → h be the orthogonal projection (with respect to the Killing form) onto the Cartan subalgebra h := t + it, where t is a maximal abelian subalgebra of k. Given x ∈ g, we consider π(Ad(K)x), where K is the analytic subgroup G corresponding to k, and show that it is star-shaped. The result extends a resul...

Complex Geometry and Representations of Lie Groups

Let Z = G/Q be a complex flag manifold and let Go be a real form of G. Then the representation theory of the real reductive Lie group Go is intimately connected with the geometry of Go-orbits on Z. The open orbits correspond to the discrete series representations and their analytic continuations, closed orbits correspond to the principal series, and certain other orbits give the other series of...

Interactions between Lie Theory and Algebraic Geometry

Sums of Adjoint Orbits

We show that the sum of two adjoint orbits in the Lie algebra of an exponential Lie group coincides with the Campbell-Baker-Hausdorff product of these two orbits. Introduction N. Wildberger and others have recently investigated the structure of the hypergroup of the adjoint orbits in relation with the class hypergroup of compact Lie groups. A generalization of the notion of this type of hypergr...