Abstract
AbstractWe study closures of conjugacy classes in the symmetric matrices of the orthogonal group and we determine which one are normal varieties. In contrast to the result for the symplectic group where all classes have normal closure, there is only a relatively small portion of classes with normal closure. We perform a combinatorial computation on top of the same methods used by Kraft-Procesi and Ohta.
Highlights
In a fundamental paper of Kostant [4], the adjoint action on a reductive Lie algebra g defined over an algebrically closed field k of characteristic 0 is studied in detail
In his paper Kostant showed that if A is a regular nilpotent element of g, so that CA is the nilpotent cone of g, the normality is always the case
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Summary
In a fundamental paper of Kostant [4], the adjoint action on a reductive Lie algebra g defined over an algebrically closed field k of characteristic 0 is studied in detail. The method of the auxiliary variety Z developed by Kraft and Procesi in [5–7] was adapted by Ohta in [10] to the study of the singularities of the orbits in the orthogonal and symplectic symmetric spaces. In particular he proved that CA is always normal in the case of symplectic symmetric space. The main purpose of this paper is to give a necessary and sufficient condition on the partition corresponding to the orbit of a nilpotent symmetric element A in order to have the normality of CA. A special thank goes to my PhD advisor Giovanni Cerulli Irelli for many discussions about this problem and his precious help in editing this paper
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