According to a
theorem of Brieskorn and Slodowy,
the intersection of
the nilpotent cone of a simple Lie algebra with a transverse slice to
the subregular nilpotent orbit is a simple surface singularity. At
the opposite extremity of
the poset of nilpotent orbits,
the closure of
the minimal nilpotent orbit is also an isolated symplectic singularity, called a minimal singularity. For
classical Lie algebras, Kraft and Procesi showed that
these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures: specifically, any such singularity is ei
ther a simple surface singularity, a minimal singularity, or a union of two simple surface singularities of type
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311811&_mathId=si1.gif&_user=111111111&_pii=S0001870816311811&_rdoc=1&_issn=00018708&md5=6b6e4d21bc8f4104d86f57b8dbc2dbb6" title="Click to view the MathML source">A2k−1class="mathContainer hidden">class="mathCode">. In
the present paper, we complete
the picture by determining
the generic singularities of all nilpotent orbit closures in
exceptional Lie algebras (up to normalization in a few cases). We summarize
the results in some graphs at
the end of
the paper.
In most cases, we also obtain simple surface singularities or minimal singularities, though often with more complicated branching than occurs in the classical types. There are, however, six singularities that do not occur in the classical types. Three of these are unibranch non-normal singularities: an class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311811&_mathId=si1096.gif&_user=111111111&_pii=S0001870816311811&_rdoc=1&_issn=00018708&md5=a422179ec9cd78512ff8bd83eb976bb5" title="Click to view the MathML source">SL2(C)class="mathContainer hidden">class="mathCode">-variety whose normalization is class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311811&_mathId=si3.gif&_user=111111111&_pii=S0001870816311811&_rdoc=1&_issn=00018708&md5=8f0b4286c80131b1d4952f5e789ccf64" title="Click to view the MathML source">A2class="mathContainer hidden">class="mathCode">, an class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311811&_mathId=si151.gif&_user=111111111&_pii=S0001870816311811&_rdoc=1&_issn=00018708&md5=c10425f22db25af9fd876c94610b094e" title="Click to view the MathML source">Sp4(C)class="mathContainer hidden">class="mathCode">-variety whose normalization is class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311811&_mathId=si5.gif&_user=111111111&_pii=S0001870816311811&_rdoc=1&_issn=00018708&md5=bfa576f2fa56742f4c3f670f27846a37" title="Click to view the MathML source">A4class="mathContainer hidden">class="mathCode">, and a two-dimensional variety whose normalization is the simple surface singularity class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311811&_mathId=si6.gif&_user=111111111&_pii=S0001870816311811&_rdoc=1&_issn=00018708&md5=76245941e5801d8cf653ccda9a151af8" title="Click to view the MathML source">A3class="mathContainer hidden">class="mathCode">. In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, extending Slodowy's work for the singularity of the nilpotent cone at a point in the subregular orbit.