Completely reducible subcomplexes of spherical buildings

TL;DR

This paper extends Serre's result by proving that convex subcomplexes of irreducible spherical buildings are completely reducible if each vertex of a certain type has an opposite.

Contribution

It generalizes Serre's theorem to a broader class of convex subcomplexes in spherical buildings, establishing a new criterion for complete reducibility.

Findings

Convex subcomplexes with opposite vertices are completely reducible.

Extension of Serre's theorem to new classes of spherical building subcomplexes.

Provides a criterion for complete reducibility based on vertex opposites.

Abstract

We generalize a result of Serre's to show that if every vertex of some fixed type of a convex subcomplex of an irreducible spherical building has an opposite, then the subcomplex is completely reducible.