Pieces of nilpotent cones for classical groups

Achar, Pramod; Henderson, Anthony; Sommers, Eric

doi:10.1090/S1088-4165-2011-00393-9

Pieces of nilpotent cones for classical groups
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by Pramod N. Achar, Anthony Henderson and Eric Sommers

Represent. Theory 15 (2011), 584-616

DOI: https://doi.org/10.1090/S1088-4165-2011-00393-9

Published electronically: August 22, 2011

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Abstract:

We compare orbits in the nilpotent cone of type $B_n$, that of type $C_n$, and Kato’s exotic nilpotent cone. We prove that the number of $\mathbb {F}_q$-points in each nilpotent orbit of type $B_n$ or $C_n$ equals that in a corresponding union of orbits, called a type-$B$ or type-$C$ piece, in the exotic nilpotent cone. This is a finer version of Lusztig’s result where corresponding special pieces in types $B_n$ and $C_n$ have the same number of $\mathbb {F}_q$-points. The proof requires studying the case of characteristic $2$, where more direct connections between the three nilpotent cones can be established. We also prove that the type-$B$ and type-$C$ pieces of the exotic nilpotent cone are smooth in any characteristic.

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