The dimension formula for affine Springer fibers of regular symmetric-space elements

Let $F$ be the Laurent series field with valuation $v$, let $(G,G_0)$ be the symmetric pair considered in the paper, and let

$$\gamma=\begin{pmatrix}0&X\Y&0\end{pmatrix}\in\mathfrak{g}_1(F)$$

be regular semisimple. Regard $XY$ as an element of $\mathfrak{gl}_n$, and let $[XY,-]$ denote the commutator map restricted to the Lie algebra of the upper triangular Borel subgroup. Dimension formula conjecture.

$$\dim_k X(G_0,\gamma)=v\bigl(\det([XY,-])\bigr).$$

This conjecture would extend the dimension formula established in the paper for the Cartan subspaces treated there to arbitrary regular semisimple elements. The authors describe it as a conjecture they hope to resolve in a future paper; its status is therefore open.