There is a result about the real representations of a simple Lie group $G$ which is known as Soergel conjecture or Vogan duality. We'll focus on the formulation that for a fixed character of $G$ $\cup_{\sigma} \mathfrak D\mathrm{-mod}(K_\sigma\backslash G/B_\chi) = \mathrm{same\ for\ {}^\vee\ }$ where the union goes over maximal compact subgroups.
Now I've seen a way to draw this statement using the homotopy maps of an interval to the classifying space of $G$: if you take the maps of interval drawn as B----(G)----B, you get $B\backslash G/ B$, when if you take half-interval drawn as B----(G)----K, you get $B\backslash G/K$ (by definition such a map is divided by whatever subgroups are drawn on the vertices).
In other words, if we introduce the notation $\chi(G, I/2)$ for the union (over max compact subgroups) of spaces of D-modules on the space of maps of half-interval to $\mathrm{pt}/G$, one expects to have a statement that $\chi(G; I/2) = \chi(G^{\vee}; I/2)$. This is nice to draw.
Those, not surprisingly, were contents of a talk. Now comes the cryptic paragraph from my notes:
This comes from 4d theory who takes maps from a cylinder to $\mathrm{pt}/G$, with $A = \mathfrak D\mathrm{-mod}[I\backslash \mathcal L G/I]$ and $B = \mathcal O\mathrm{-mod}[\mathrm{Steinberg}_{G^\vee}/ G^{\vee}]$ and $A=B$ by either
S-duality or Bezrukavnikov.
Do you know a good reference for this 4d-theory and/or an explanation for how it helps to prove the above? By the way, the math part must be equivalent to the discussion in Loop Spaces and Langlands Parameters.
