The main problem that is solved in this paper has the following simple formulation (which is not used in its solution). The group $K = \mathrm{O}_p ({\bf C}) \times \mathrm{O}_q ({\bf C})$ acts on the space $M_{p,q}$ of $p\times q$ complex matrices by $(a,b) \cdot x = axb^{-1}$, and so does its identity component $K^0 = \SO_p ({\bf C}) \times \SO_q ({\bf C})$. A $K$-orbit (or $K^0$-orbit) in $M_{p,q}$ is said to be nilpotent if its closure contains the zero matrix. The closure, $\overline{\mathcal{O}}$, of a nilpotent $K$-orbit (resp.\ $K^0$-orbit) ${\mathcal{O}}$ in $M_{p,q}$ is a union of ${\mathcal{O}}$ and some nilpotent $K$-orbits (resp.\ $K^0$-orbits) of smaller dimensions. The description of the closure of nilpotent $K$-orbits has been known for some time, but not so for the nilpotent $K^0$-orbits. A conjecture describing the closure of nilpotent $K^0$-orbits was proposed in \cite{DLS} and verified when $\min(p,q) \le 7$. In this paper we prove the conjecture. The proof is based on a study of two prehomogeneous vector spaces attached to $\mathcal{O}$ and determination of the basic relative invariants of these spaces. The above problem is equivalent to the problem of describing the closure of nilpotent orbits in the real Lie algebra $\mathfrak{so} (p,q)$ under the adjoint action of the identity component of the real orthogonal group $\mathrm{O}(p,q)$.

A unified formulation is given to various generalizations of the classical numerical range including the $c$-numerical range, congruence numerical range, $q$-numerical range and von Neumann range. Attention is given to those cases having connections with classical simple real Lie algebras. Convexity and inclusion relation involving those generalized numerical ranges are investigated. The underlying geometry is emphasized.

Nous montrons que l'invariant de Hasse-Witt de la forme de Killing d'une alg{\`e}bre de Lie semi-simple $L$ s'exprime {\`a} l'aide de l'invariant de Tits de la repr{\'e}sentation irr{\'e}ductible de $L$ de poids dominant $\rho=\frac{1}{2}$ (somme des racines positives), et des invariants associ{\'e}s au groupe des sym{\'e}tries du diagramme de Dynkin de $L$.

Lie algebras graded by finite reduced root systems have been classified up to isomorphism. In this paper we describe the derivation algebras of these Lie algebras and determine when they possess invariant bilinear forms. The results which we develop to do this are much more general and apply to Lie algebras that are completely reducible with respect to the adjoint action of a finite-dimensional subalgebra.