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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Singularities in the nilpotent scheme of a classical group


Author: Wim Hesselink
Journal: Trans. Amer. Math. Soc. 222 (1976), 1-32
MSC: Primary 14B05; Secondary 14L15
MathSciNet review: 0429875
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Abstract: If $ (X,x)$ is a pointed scheme over a ring k, we introduce a (generalized) partition $ {\text{ord}}(x,X/k)$. If G is a reductive group scheme over k, the existence of a nilpotent subscheme $ N(G)$ of $ {\text{Lie}}(G)$ is discussed. We prove that $ {\text{ord}}(x,N(G)/k)$ characterizes the orbits in $ N(G)$, their codimension and their adjacency structure, provided that G is $ G{l_n}$, or $ S{p_n}$ and $ 1/2 \in k$. For $ S{O_n}$ only partial results are obtained. We give presentations of some singularities of $ N(G)$. Tables for its orbit structure are added.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0429875-8
PII: S 0002-9947(1976)0429875-8
Keywords: Partitions, linear extension of a local ring, regular sequence, classical group scheme, orbit, cross section, nilpotent variety, Kleinian singularity
Article copyright: © Copyright 1976 American Mathematical Society