Intersection and Incidence Distances Between Parabolic Subgroups of a Reductive Group

Abstract

Let Γ be a reductive algebraic group, and let P,Q ⊂ Γ be a pair of parabolic subgroups. Some properties of the intersection and incidence distances $$ \begin{array}{c}\hfill {\mathrm{d}}_{\mathrm{in}}\left(P,Q\right)= \max \left\{ \dim P, \dim Q\right\}- \dim \left(P\cap Q\right),\hfill \\ {}\hfill {\mathrm{d}}_{\mathrm{in}\mathrm{c}}\left(P,Q\right)=\mathrm{mix}\left\{ \dim P, \dim Q\right\}- \dim \left(P\cap Q\right)\hfill \end{array} $$ are considered (if P,Q are Borel subgroups, both numbers coincide with the Tits distance dist(P,Q) in the building Δ(Γ) of all parabolic subgroups of Γ). In particular, if Γ = GL(V ) and P = Pv,Q = Pu are stabilizers in GL(V ) of linear subspaces v, u ⊂ V , we obtain the formula din(P, Q) = − d 2 + a 1 d + a 2, where d = din(v, u) = max{dim v, dim u} − dim(v ∩ u) is the intersection distance between the subspaces v and u and where a1 and a2 are integers expressed in terms of dim V , dim v, and dim u. Bibliography: 7 titles.