Stability phenomena for Kac-Moody groups
Nitu Kitchloo
TL;DR
The paper establishes $p$-local homological stability for families of Kac-Moody groups obtained by extending generalized Dynkin diagrams, with the prototypical family $E_n$ analyzed through classifying-space decompositions. It uses a homotopy-colimit framework over the poset of spherical subsets to describe $BK(A_I)$ and $BW(A_I)$, and proves that $BE_{9+n}$ stabilize to $BE$, supported by a collapses in a spectral sequence for primes $p>5$. The argument extends to general families $M_n$ with a distinguished node, yielding stability for the corresponding $BW(M_n)$. A notable emergent structure is an $A_ abla$-action by stable $SU$-bundles on the stabilized space, leading to a principal $BSU$-fibration and an interpretation that stabilized $E$-bundles correspond to $E/SU$-bundles, suggesting deeper geometric connections to topology and string-theory contexts.
Abstract
We show that a canonical procedure of extending generalized Dynkin diagrams gives rise to families of Kac-Moody groups that satisfy homological stability. We also briefly sketch some emergent structure that appears on stabilization. Our results are illustrated for the family {E_n} which is of interest in String theory. The techniques used involve homotopy decompositions of classifying spaces of Kac-Moody groups.