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This is a follow-up to my question about nilpotent orbits here asked in connection with an earlier discussion of symplectic resolutions. Leaving aside the connections with algebraic geometry and representation theory, I'm curious about what has been written down explicitly for the exceptional Lie algebras $E_6, E_7, E_8$.

Start with a simple Lie algebra $\mathfrak{g}$ of rank $n$ over $\mathbb{C}$ (or the Lie algebra of a simple algebraic group $G$ over any algebraically closed field of good characteristic). The finitely many nilpotent orbits in $\mathfrak{g}$ (or unipotent classes in $G$) have been well studied, with expositions for example in the books by Carter (1985) and Collingwood-McGovern (1993). Carter provides detailed tables in 13.1 and partial order graphs in 13.4 (though the $E_7, E_8$ graphs have a few missing edges compared to the correct pictures in Spaltenstein's lecture notes). In particular, it's easy for exceptional types to compare the total number of orbits, labelled by Dynkin or Bala-Carter, with the number of special orbits in Lusztig's sense.

While the special orbits are defined only indirectly via several kinds of representation theory, all Richardson orbits are special. The converse starts to fail in type $D_4$, where there is one special but non-Richardson orbit. Richardson orbits belong to one or more of the $2^n$ conjugacy classes of parabolic subalgebras in $\mathfrak{g}$; for instance, the subregular nilpotent orbit intersects the nilradical of each minimal parabolic $\mathfrak{p}$ in its dense orbit under the corresponding adjoint subgroup $P$. In type $F_4$ there are 16 orbits, 11 special and 9 of these Richardson.

Has anyone computed the full list of Richardson orbits in types $E_6, E_7, E_8$?

As noted earlier, Hirai's paper here gives "generators and relations" for producing such a list, along with explicit information about Richardson orbits with "non-even" Dynkin labelling. But for example, what is the list of Richardson orbits coming from the 256 classes of parabolics for $E_8$? One knows that there are 70 orbits, 46 of which are special; which of the latter are Richardson?

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Isn't this question a subset of the induction question (i.e. if we induce an orbit from a Levi subalgebra up to ${\mathfrak g}$, then what orbit do we get)? This was solved by Elashvili (for exceptional types) in 1979, and his computations were verified in a 2009 joint Elashvili-de Graaf paper ("Induced nilpotent orbits of the simple Lie algebras of exceptional type", in the Georgian Mathematical Journal but also on the arXiv).

By a standard result about Lusztig-Spaltenstein induction, two parabolic subalgebras with the same Levi subalgebra give the same induced orbit, so in fact we get a better bound than 2^n on the number of Richardson orbits: the number of Levi subalgebras up to conjugacy.

In type F4 the Levi subalgebras are of type: maximal torus, $A_1$, $\tilde{A}_1$, $A_1+\tilde{A}_1$, $A_2$, $B_2$, $\tilde{A}_2$, $A_2+\tilde{A}_1$, $A_1+\tilde{A}_2$, $B_3$, $C_3$, $F_4$. The corresponding Richardson orbits are: regular, subregular, subregular, $F_4(a_2)$, $C_3$, $F_4(a_3)$, $B_3$, $F_4(a_3)$, $F_4(a_3)$, $\tilde{A}_2$, $A_2$, $0$. The special orbits which aren't Richardson are $A_1+\tilde{A}_1$ and $\tilde{A}_1$.

In type $E_6$, the following orbits with non-even weighted Dynkin diagrams are Richardson: $D_5(a_1)$, $A_4+A_1$, $A_3$, $A_2+2A_1$, $2A_1$. These are induced respectively from the zero orbit in Levi subalgebras of type $A_2+A_1$, $A_2+2A_1$, $A_3+A_1$, $A_3+2A_1$ and $D_5$. So we get (I think) 15 Richardson orbits, and the two special orbits which aren't Richardson are $A_1$ and $A_2+A_1$.

I think that the number of Richardson orbits in type $E_7$ (resp. $E_8$) is 29 (resp. 34), but I could easily have made a counting error.

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  • $\begingroup$ Thanks. Your count of Richardson orbits for types $F_4$ and $E_n$ does seem consistent with Hirai. As you say, the literature on induced orbits is more comprehensive, though it still takes some work to see how many distinct Richardson orbits emerge from the larger list of conjugacy classes of Levi subalgebras. It surprises me that the results aren't made more explicit in the literature, but I'll look again at Elashvili's papers. $\endgroup$ – Jim Humphreys Jul 3 '14 at 14:13

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