Intersection and Incidence Distances Between Parabolic Subgroups of a Reductive Group

• Published in: Journal of mathematical sciences (New York, N.Y.) N.Y.), 2016-12, Vol.219 (3), p.405-412
• Main Authors: Gordeev, N., Rehmann, U.
• Format: Article
• Language: English
• Subjects: Incidence; Integers; Mathematics; Mathematics and Statistics; Subgroups; Subspaces
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-016-3116-3

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 d in P Q = max dim P , dim Q − dim P ∩ Q , d i n c P Q = mix dim P , dim Q − dim P ∩ Q 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 = P v ,Q = P u are stabilizers in GL(V ) of linear subspaces v, u ⊂ V , we obtain the formula d in ( 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 a 1 and a 2 are integers expressed in terms of dim V , dim v, and dim u. Bibliography: 7 titles.