Irreducible components of exotic Springer fibres

Abstract

Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (the fibres of the resolution of the exotic nilpotent cone), and prove that they are naturally in bijection with standard bitableaux. As a result, we deduce the existence of an exotic Robinson-Schensted bijection, which is a variant of the type C Robinson-Schensted bijection between pairs of same-shape standard bitableaux and elements of the Weyl group; this bijection is described explicitly in the sequel to this paper. Note that this is in contrast with ordinary type C Springer fibres, where the parametrisation of irreducible components, and the resulting geometric Robinson-Schensted bijection, are more complicated. As an application, we explicitly describe the structure in the special cases where the irreducible components of the exotic Springer fibre have dimension 2, and show that in those cases one obtains Hirzebruch surfaces. Contents

Irreducible components of exotic Springer fibres II : The exotic Robinson–Schensted algorithm

“…In this paper, we examine the 'exotic Robinson-Schensted algorithm' -analogously obtained using the geometry of Kato's exotic nilpotent cone as a substitute for the ordinary nilpotent cone of type C. The resulting combinatorial algorithm is more tractable and is not related to other type B/C generalisations of the Robinson-Schensted algorithm appearing in the literature, such as the 'naive' extension first defined by Stanley in [Sta82] or the one involving domino tableaux that goes back to the work of Barbasch and Vogan [BV82]. This builds on our previous paper [NRS16], parametrising irreducible components of exotic Springer fibres. Note that this is different from the exotic Robinson-Schensted correspondence constructed by Henderson and Trapa in [HT12].…”

“…In this section we recall some fundamental properties of the exotic nilpotent cone and the relevant results from [NRS16] that will be needed to establish the exotic Robinson-Schensted correspondence. Readers only interested in the actual algorithm can skip this section.…”

“…Similarly a bitableau of shape (µ, ν) ∈ Q n is standard if every integer between 1 to n occurs exactly once, and the increasing condition is satisfied in each of the two tableaux. To match the conventions of [NRS16] and [AH08], we reverse the direction of the rows of the first tableau, so numbers are increasing along rows from right-to-left. We let T (µ, ν) denote the set of standard Young bitableaux of shape (µ, ν).…”

“…Let π : N −→ N be the exotic Springer resolution as defined in [Kat11]. In [NRS16], we showed that the irreducible components of the fibres of π are in bijection with standard Young bitableaux, using Spaltenstein's techniques from [Spa76]. This allowed us to define an analogous exotic Steinberg variety whose irreducible components are parametrised in two separate ways: one way by elements of the Weyl group W (C n ) = C 2 S n (signed permutations), and the other by pairs of standard Young bitableaux.…”

“…• In Section 4 we examine the exotic Springer fibres, understanding the restriction of exotic Jordan types of pairs of generic points of the fibre. • In Section 5 we use the results from Section 4 and [NRS16] to construct the reverse bumping algorithm from the geometry of exotic Springer fibres and thus prove the main theorem.…”