Vol. 12, No. 1, 2011

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Maximal Levi subgroups acting on the Euclidean building of $\mathrm{GL}_n(F)$

Jonathan Needleman

Vol. 12 (2011), No. 1, 7–19

In this paper we give a complete invariant of the action of GLn(F) × GLm(F) on the Euclidean building BGLn+m(F), where F is a discrete valuation field. We then use this invariant to give a natural metric on the resulting quotient space. In the special case of the torus acting on the tree BGL2(F), we obtain an algorithm for calculating the distance of any vertex in the tree to any fixed apartment.

affine building, Euclidean building, Levi subgroup, group action
Mathematical Subject Classification 2010
Primary: 20E42, 20G25
Received: 26 August 2009
Accepted: 19 May 2011
Jonathan Needleman