What are situations where one can conclude that all entries of an intersection form are divisible by a fixed integer?

More precisely: $F \subset S$ is a proper connected (usually reducible) half-dimensional subvariety of a smooth variety $S$. In this situation one define the "refined intersection form" on the top degree homology of $F$. The top homology has a basis over $\mathbb{Z}$ given by the fundamental classes of components of $F$ of maximal dimension, amd the intersection form gives a pairing on this space.

(To define this intersection form one uses the isomorphism of $H^{top}(F) = H^{n}(S, S \setminus F)$ and the product on relative cohomology. Informally one takes two components of $F$, moves them inside $S$ until they are transverse, and counts intersection points. For example, if $F$ is smooth one can take $S = T^*F$ and the self-intersection of $[F]$ is the Euler characteristic of $F$, up to $\pm 1$.)

For reasons coming from representation theory we have many examples where all entries in the intersection form are divisible by a integer, which happens to be the order of a finite group $W(e)$.

Is this a familiar phenomenon somewhere else? What geometric tools are used to prove such divisibility?

(In our situation the finite group $W(e)$ acts on the whole set-up, and my naive guess was that we should consider the quotient by $W(e)$ and proceed as in this question. However there are many finite groups which act, and whose orders have nothing to do with $|W(e)|$. So if this is the explanation, then $W(e)$ should be special in some way.)

For those with some background in geometric representation theory here is a more precise description of my situation:

  • $\mathcal{N}$ is the nilpotent cone in a complex semi-simple Lie algebra;
  • $e \in \mathcal{N}$ and $S_e$ the intersection of $\mathcal{N}$ with a Slodowy slice through $e$;
  • $\pi : T^*(G/B) \to \mathcal{N}$ is the Springer resolution;
  • $\pi : S \to S_e$ is the corresponding resolution of $S_e$;
  • $F = \pi^{-1}(e)$ is the Springer fibre inside $S$.

We can show in type $A$ that all entries of the intersection form are divisible by $W(e)$, the Weyl group of the reductive part of the centraliser of $e$. However the proof is entirely non-geometric. We do not know if this is true in other types.



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