NOTE: I am very much not an expert on affine Kac-Moody groups or ind-varieties, so the following may be vague.


The first question is: Has anyone developed a theory of Frobenius splitting for ind-varieties in positive characteristic? (This might be very easy for all I know). Following from that, my main question is: If there is a reasonable theory of Frobenius splitting of ind-varieties, has anyone thought about whether affine flag varieties or affine Grassmannians corresponding to semisimple algebraic groups in positive characteristic are Frobenius split?



Frobenius splitting of affine flag varieties
If you have a strict ind-scheme (inductive limit of schemes with maps closed imersions), for example affine flag varieties as a limit of schubert cells, then you can ask that all these varieties are compatably Frobenius split, so the notion of a Frobenius splitting makes sense.


Then you can ask if the affine flag variety admits a Frobenius splitting. The asnwer is yes. I learnt this under mild conditions from a course of Xinwen Zhu, from which the original reference appears to be Faltings' paper "Algebraic loop groups and moduli spaces of bundles". Here the affine flag variety is defined in a particular way, and of course if you want to define it in a different way (eg using Kac-Moody) groups, then you must do some work to show that definitions agree.


This result is harder than in finite type since you don't have the smooth Schubert cell associated to the long word as a crutch to get you started.


