Summary: NORMALITY OF ORBIT CLOSURES IN THE ENHANCED
PRAMOD N. ACHAR, ANTHONY HENDERSON, AND BENJAMIN F. JONES
Abstract. We continue the study of the closures of GL(V )-orbits in the en-
hanced nilpotent cone V × N begun by the first two authors. We prove that
each closure is an invariant-theoretic quotient of a suitably-defined enhanced
quiver variety. We conjecture, and prove in special cases, that these enhanced
quiver varieties are normal complete intersections, implying that the enhanced
nilpotent orbit closures are also normal.
The geometry of nilpotent orbits in complex semisimple Lie algebras is a topic of
central importance in numerous branches of representation theory. A fundamental
question on this topic is: Are the closures of nilpotent orbits normal varieties?
This question was answered in the affirmative for nilpotent orbits in type A by
KraftProcesi [KP1] in 1979. In other types, the answer turns out to be "not
always": an explicit determination of the nilpotent orbits with normal closures was
carried out in types B and C by KraftProcesi [KP2], and in types G2, F4, and
E6 by Kraft [Kr], Broer [B], and Sommers [S1], respectively. The case of type D
was partially resolved by KraftProcesi [KP2] and completed by Sommers [S2]. A
complete answer is not yet known in types E7 and E8.