Coxeter groups preserving orthants

Abstract

In this short, elementary note we prove that if a faithful reflection representation of a Coxeter group preserves an orthant, then that Coxeter group is a product of symmetric groups acting on its natural permutation representation. We also prove an affine analogue of this statement, where an orthant is preserved modulo an invariant sublattice. As a consequence, the existence of two different versions of the quantum geometric Satake equivalence is a purely type A phenomenon.