Summary: CORRIGENDUM TO `ORBIT CLOSURES IN THE ENHANCED
NILPOTENT CONE', PUBLISHED IN ADV. MATH. 219 (2008)
PRAMOD N. ACHAR AND ANTHONY HENDERSON
Abstract. In this note, we point out an error in the proof of Theorem 4.7 of
[P. Achar and A. Henderson, Orbit closures in the enhanced nilpotent cone,
Adv. Math. 219 (2008), 2762], a statement about the existence of affine
pavings for fibres of a certain resolution of singularities of an enhanced nilpo-
tent orbit closure. We also give independent proofs of later results that depend
on that statement, so all other results of that paper remain valid.
The paper [AH] carries out the determination of orbit closures in the enhanced
nilpotent cone V × N and their local intersection cohomology. A key role is played
by a certain resolution of singularities of the orbit closure Oµ;, denoted µ;.
In [AH, Theorem 4.7], we asserted that each fibre -1
µ;(v, x) of this resolution
has an affine paving. Regrettably, our proof of this statement was wrong: the error
comes four lines after equation (4.9), where we assumed without justification that
. We have not found either a correct proof or a counterexample, so the
existence of an affine paving for -1
µ;(v, x) is an open problem in general.