# How to decide if two surfaces occurring in Springer theory are isomorphic?

In the study of a simple algebraic group (say over $\mathbb{C}$) and related geometry of its flag variety associated with the Springer correspondence, one encounters pairs of surfaces which have some crucial data in common. It would be interesting to know that they are actually isomorphic, but the data typically come indirectly from a study of the algebraic group structure along with the behavior of Weyl group representations in Springer's theory. This looks much the same over any algebraically closed field of good characteristic $p \geq 0$. The surfaces themselves are Springer fibers attached to nilpotent elements (or orbits) in the Lie algebra and live in the cotangent bundle of the flag variety $\mathcal{B}$ (realized as $G/B$ when a Borel subgroup $B$ is fixed).

So far the geometric structure of Springer fibers is poorly understood in most cases, but the general theory reveals their dimensions (inversely proportional to dimensions of nilpotent orbits) as well as the equidimensionality of their irreducible components. In the first nontrivial case, the fiber has dimension 1 (a Dynkin curve) corresponding to the subregular nilpotent orbit and has components determined by the Dynkin diagram. Here the variety is a union of intersecting projective lines and is the same for Lie types $G_2, C_3, D_4$ even though in each case the centralizer of an associated nilpotent element in $G$ modulo its identity component will vary: $S_3, S_2, S_1 = 1$.

The structure of Springer fibers is already difficult to study in rank 2 (or higher), with some partial results for simply-connected Lie types. But in the cases just mentioned (and presumably others), there are 2-dimensional components for adjacent nilpotent orbits having some known data that agree. For example, the pair $G_2, C_3$ both lead to surfaces with two irreducible components and with total cohomology dimension 6. (Here the cohomology can be classical or etale.)

Are there algebraic geometry criteria which can ensure isomorphism of such surfaces based on limited data?

The interest of all this becomes apparent when working in characteristic $p$ (say larger than the Coxeter number), where the number of simple modules in a "regular" block of the reduced enveloping algebra for the Lie algebra corresponding to a given nilpotent element turns out to be equal to the total dimension of the cohomology of the associated Springer fiber (Bezrukavnikov-Mirkovic-Rumynin). Empirical study of special cases suggests strongly that such modules "deform" from a nilpotent orbit to an orbit in its closure in a way dependent only on the cohomology or K-theory of the fiber. In particular, this seems to be true for the indicated pair when passing from the subregular orbit to one of dimension one less. (Similarly for the pair $C_3, D_4$, where for the 2-dimensional Springer fibers there are three components and total cohomology dimension 8.)

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