Geodesic
Intersection and incidence distances between parabolic subgroups of a reductive group
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 27, Tome 430 (2014), pp. 103-113 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\dim V,\dim v,\dim u$. 
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     title = {Intersection and incidence distances between parabolic subgroups of a reductive group},
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}
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N. Gordeev; U. Rehmann. Intersection and incidence distances between parabolic subgroups of a reductive group. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 27, Tome 430 (2014), pp. 103-113. http://geodesic.mathdoc.fr/item/ZNSL_2014_430_a8/

[1] A. Borel, Linear Algebraic groups, Graduate texts in Math., 126, 2nd enl. ed., Springer-Verlag, New York Inc., 1991 | DOI | MR | Zbl

[2] N. Bourbaki, Éléments de Mathématique. Groupes ey Algébres de Lie, Chap. IV, V, VI, 2ème édition, Masson, Paris, 1981 | MR

[3] R. W. Carter, Simple Groups of Lie Type, Reprint of the 1972 original, Wiley Classics Library, John Wiley and Sons, New York, 1989 | MR | Zbl

[4] R. W. Carter, Finite Groups of Lie Type. Conjugacy Classes and Complex Characters, John Wiley Sons, Chichester er al., 1985 | MR | Zbl

[5] Ph. Griffiths, J. Harris, Principles of algebraic geometry, John Wiley Sons, New York, 1978 | MR | Zbl

[6] N. Gordeev, U. Rehmann, “On linearly Kleiman groups”, Transf. Groups, 18:3 (2013), 685–709 | DOI | MR | Zbl

[7] J. Tits, Buildings of Spherical Type and and Finite BN-Pairs, Lect. Note Maths., 386, Springer-Verlag, 1974 | MR | Zbl