RMC - Vol. 2 Num. 1, Nicolas Andruskiewitsch’

Full paper in PDF:
$% N. Andruskiewitsch’, On the complicatedness of the pair $(g,K)$ , Rev. Mat. Univ. Complut. Madrid 2 (1989), no. 1, 13–28%$.

On the Complicatedness of the Pair $(g, K )$

Nicolás ANDRUSKIEWITSCH’

Facultad de Matemática, Astronomía y Física — IMAF
Valparaíso y R. Martínez
Ciudad Universitaria — 5000 Córdoba
República Argentina

Received: July 12, 1988

ABSTRACT

Let $g= f o+ p$ be the complexification of a Cartan decomposition of a real-semisimple Lie algebra $gR$ and let $K$ be the analytic subgroup of the adjoint group of $g$ with Lie algebra $adg(f)$ . Let $L$ be an algebraic connected linear reductive complex group acting on a finite dimensional vector space $V$ . In the study of the orbits of this sort of actions, there are some criteria of non complicatedness e.g., cofreeness (the ring of all polynomial functions on is a free module over the ring of all L-invariants), etc. From this viewpoint, we show that the pair is complicated, at least when $gR$ is not a product of copies of $so(n,1)$ or $su(n,1)$ .

1980 Mathematics Subject Classification (1985 revision): 14D25, 14L30.