Understanding reflection subgroups of (possibly infinite) groups generated by reflections plays a key role in understanding the symmetries and combinatorial structure of root systems, as well as the underlying geometry and the structure of the associated Lie algebra. Affine reflection systems provide a unifying framework that includes finite and affine root systems, as well as the root systems of extended affine Lie algebras.

In this talk, we classify the maximal root subsystems of affine reflection systems. This classification yields, in particular, a description of maximal reflection subgroups of the associated reflection groups. We also discuss a duality theorem: for each maximal root subsystem, either it is closed or its dual is closed in the dual root system. This generalizes classical results due to Borel and de Siebenthal (1950s) in the finite case and work by Dyer et al. (2011) in the affine setting.