On the Springer resolution of the minimal unipotent conjugacy class

Abstract

Let C be a simple ~~rn~~~x Lie roup, G, an roups of G. ft was shown b that the projection resolves the singularities of U, The fibre only on the conjuga~y class of an element u E U. The second ~r~j~~ti~n V-, d ident~~es this fibre with the subvariety of 8 of elements fixed under u, which acts on A? by conjugation. The projection, n, is an isomorphism over the conjuga uni~utent elements~ the unique class whose closure is the f is the fibre of m Greg , of regular variety U. On the G, the only other fibre whose structure has been n~wn is the fibre U over a sub ular unipotent element u. Th cmjugacy class, Csregr of these ss which 'goes next after' Greg i-e. U= C&J eisreg. ft was shown by J. Tits and R. Steinberg (cf. f&4, p. 147]) that u is isomorphic to a union of nonsingu~ar rational curves, w~~ich form a configuration descri diagram of a sirn~~e grout C' of type A, I) or 43; C = G' if C is If ~8 is identified with the quotient variety c/B0 for a fixed rel subgroup I30 of en each irreduci le component, C, of ith the subvariety where Pi is one of the ~=r~~~~ minim ing BO.

Springer Fibers for the Minimal and the Minimal Special Nilpotent Orbits

“…type A, D, E. Here we summarize some properties of these orbits and corresponding Springer fibers. These can be read from subsequent sections of this paper, [DG84], [Car93], [CM93], etc.…”

“…Also Irr(B N ) = {X 1 , X 2 , · · · , X n−1 }. For i = j, codim BN X i ∩ X j = 1 if and only if |i − j| = 1 by [DG84]. Thus Γ N is described as follows.…”

“…Then Irr B N = {X 1 , · · · , X n−2 , X ′ n−1 , X ′′ n−1 }. For i, j ≤ n − 2, by [DG84] codim BN X i ∩ X j = 1 if and only if |i − j| = 1. Also, codim BN X j ∩ X ′ n−1 = 1 and/or codim BN X j ∩ X ′′ n−1 = 1 if and only if j = n − 2.…”

“…Here we have a similar issue as in the remark of Section 3.2, thus the argument here is also similar to it. Furthermore, in general X i are not smooth; for example, if G = SO 8 we may show that B N ≃ S(s(3124232)) ∪ S(s(4123121)) ∪ S(s(3124231)) ∪ S(s(3124121)) using the method of [DG84]. However, the Kazhdan-Lusztig polynomial P id,s(3124231) (q) = 2q + 1, thus S(s(3124231)) is not even rationally smooth.…”

“…(See also [Hum11] and [Hum16b].) In [DG84] Dolgachev and Goldstein observed a duality between the graphs attached to Springer fibers corresponding to the subregular and the minimal nilpotent orbit when G is of simply-laced type, but this duality breaks down when G is not simply-laced. Instead, Humphreys conjectured that one needs to consider the minimal special nilpotent orbit instead of the minimal one, and also that the Langlands dual should come into the picture.…”