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Abstract view

# Parabolic Subgroups with Abelian Unipotent Radical as a Testing Site for Invariant Theory

Let $L$ be a simple algebraic group and $P$ a parabolic subgroup with Abelian unipotent radical $P^u$. Many familiar varieties (determinantal varieties, their symmetric and skew-symmetric analogues) arise as closures of $P$-orbits in $P^u$. We give a unified invariant-theoretic treatment of various properties of these orbit closures. We also describe the closures of the conormal bundles of these orbits as the irreducible components of some commuting variety and show that the polynomial algebra $k[P^u]$ is a free module over the algebra of covariants.