# Kazhdan-Lusztig graph for the Springer fiber of the minimal special unipotent class?

This graph was determined in the case of simply-laced root systems by Igor Dolgachev and Norman Goldstein here. For other root systems the original question should be modified, leading to a precise conjecture which remains unproved. It takes some preparation to state.

Briefly, the fibers in the Springer resolution of singularities of a unipotent class in a simple algebraic group (say over $\mathbb{C}$ or a field of good prime characteristic) are projective varieties, arising from the fixed points of a unipotent element on the flag variety $G/B$. (It's equivalent to study nilpotent orbits of the Lie algebra.) The varieties, as well as their irreducible components (all of equal dimension), have known dimensions (Springer, Steinberg, Spaltenstein), but it remains challenging to describe geometrically the fibers or the patterns of component intersections. The best understood nontrivial case involves the subregular class, where the fiber is a "Dynkin curve" having copies of $\mathbb{P}^1$ as components. The incidence graph in types $A,D,E$ is just the corresponding Dynkin diagram, which coincides in the subregular case with the associated "Kazhdan-Lusztig graph". The latter is defined at the end of the 1979 KL Inventiones paper and has one vertex for each component, with vertices joined by an edge just when the components intersect in codimension one.

Dolgachev and Goldstein studied the opposite extreme of the minimal (nontrivial) unipotent class and worked out its KL graph for types $A,D,E$. Their theorem is that the graph is the same as that for the subregular class. Their study of the minimal class for $G_2$ yields a different graph. Their consultation with Spaltenstein reported in the paper suggests substitution of the minimal special class. (I asked Dolgachev about that a year ago when he was here, but he hadn't seen it pursued further.) The special classes came up in Lusztig's work on Springer representations of Weyl groups, but have no geometric characterization so far. Anyway, Lusztig-Spaltenstein duality for the partial ordering of special classes (generalizing transpose duality for partitions in type $A$) leads in types $B, C, F_4$ to a minimal special class defined by the short root groups, but in type $G_2$ to the subregular class. Comparing data for the Langlands dual types $B, C$ suggests strongly this formulation:

CONJECTURE: In all types, the KL graph for the subregular unipotent class agrees with the graph for the Langlands dual minimal special unipotent class.

A bit of numerical evidence in favor: the number of components of the Springer fiber for the minimal special class in Langlands self-dual type $F_4$ is 6, as for the subregular class of $F_4$, whereas for $B_3$ and $C_3$ the relevant numbers 4, 5 get switched and similarly for types $B_n, C_n$ in general via Springer theory. (For other pairs of classes related in this way, more than a graph would be needed to get a good comparison. Another problem.) Aside from wanting more geometric evidence related to the stated conjecture, my question here is:

Is there a good reason for Langlands duality to play any role in this essentially geometric question?

[EDIT: The paper is now freely available online; link added. But I haven't yet found an answer to my question.]

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This is not quite an answer but an observation which perhaps may clarify the situation. In the simply-laced case the coincidence of Kazhdan-Lusztig graphs for the minimal and the subregular orbit is known, as pointed out in the posting, so we want to reduce the general case of your conjecture to the simply-laced one. Let $\mathfrak{g}$ be a nonsimplylaced simple Lie algebra. I claim that (as can be observed e.g. from the case by case treatment in Slodowy's LNM volume) the Springer fiber for the subregular element in $\mathfrak{g}$ coincides with the subregular Springer fiber in the unfolding of the Langlands dual Lie algebra. [One point this is related to is the action of the affine braid group of the dual Lie algebra on the derived category of coherent sheaves on the resolution of the slice -- since we have an embedding of the affine braid group for $G$ to that of its unfolding, the above statement is consistent with that action]. In particular, it doesn't come from any direct geometric map between the Lie algebras or related spaces. Now one can expect a geometric relation between the minimal special Springer fiber for $\mathfrak{g}$ and the minimal Springer fiber for its unfolding (no Langlands duality this time), or at least a relation between their KL graphs: this would provide a proof of the Conjecture in view of the result by Dolgachev and Goldstein. In fact, Paul Levy's answer to: Uniform proof of dimension formula for minimal special nilpotent orbit? seems to point in the direction of such a relation.