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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board Member: Bannai, Eiichi / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Pasini, Antonio / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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A rank 3 tangent complex of PSp4(q), q odd

John Sarli / Philipp McClurg

Citation Information: Advances in Geometry. Volume 1, Issue 4, Pages 365–371, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advg.2001.022, January 2006

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Let G=PSp4(q), q=pk odd. We show that the geometry of root subgroups of G is the tangent envelope of a system of conics that comprise the (q,q)generalized quadrangle associated with G. The flags of this geometry form a rank 3 chamber complex in the sense of Tits [9], as one would expect from the theory of symmetric spaces for Lie groups. By way of application, we give an intrinsic interpretation of symplectic 2-transvections. We then show that the subgroup generated by a pair of shortroot subgroups not contained in a pSylow is determined by the geometry. In particular, we describe the incidence conditions under which such pairs are contained in the maximal subgroups of G corresponding to the pluspoint and minuspoint stabilizers in the orthogonal construction of G ([3], xii).

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