Triple parabolic cosets have unique Bruhat minima

Source abstract

A folklore result says that any parabolic double coset $W_{I} w W_{J}$ in a Coxeter group $W$ is an interval of the Bruhat order, provided that $W_{I}$ and $W_{J}$ are finite. We include a self-contained proof, without assuming that $W$ is finite. Furthermore, we prove that each "triple coset" $W_{I} u W_{J} v W_{K}$ has a Bruhat minimum and -- if $W_{I}, W_{J}, W_{K}$ are finite -- a Bruhat maximum, but (in general) is not a Bruhat interval. This proves and extends a conjecture on MathOverflow #472072.