Intersection and incidence distances between parabolic subgroups of a reductive group

N. Gordeev ; U. Rehmann

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 27, Tome 430 (2014), pp. 103-113

Résumé

Let $\Gamma$ be a reductive algebraic group and let $P,Q\subset\Gamma$ be a pair of parabolic subgroups. We consider here some properties of intersection and incident distances \begin{gather*} d_\mathrm{in}(P,Q)=\max\{\dim P,\dim Q\}-\dim (P\cap Q),\\ d_\mathrm{inc}(P,Q)=\min\{\dim P,\dim Q\}-\dim (P\cap Q) \end{gather*} (if $P,Q$ are Borel subgroups, both numbers coincide with the Tits distance $\operatorname{dist}(P,Q)$ in the building $\Delta(\Gamma)$ of all parabolic subgroups of $\Gamma$). In particular, if $\Gamma=\mathrm{GL}(V)$ and $P=P_v$, $Q=P_u$ are stabilizers in $\mathrm{GL}(V)$ of linear subspaces $v,u\subset V$ we obtain the formula $$ d_\mathrm{in}(P,Q)=-d^{\,2}+a_1d+a_2 $$ where $d=d_\mathrm{in}(v,u)=\max\{\dim v,\dim u\}-\dim(v\cap u)$ is the intersection distance between the subspaces $v,u$, and where $a_1, a_2$ are integers expressed in terms of $\dim V,\dim v,\dim u$.