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Journal of Lie Theory 22 (2012), No. 1, 155--161
Copyright Heldermann Verlag 2012

Some Transitive Linear Actions of Real Simple Lie Groups

Laura Geatti
Dip. di Matematica, UniversitÓ di Roma 2 "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy

Martin Moskowitz
CUNY Graduate Center, Ph.D. Program in Mathematics, 365 Fifth Avenue, New York, NY 10016, U.S.A.


\def\C{{\mathbb{C}}} \def\H{{\mathbb{H}}} \def\R{{\mathbb{R}}} In a recent paper of M. Moskowitz and R. Sacksteder [An extension of the Minkowski-Hlawka theorem, Mathematika 56 (2010) 203-216], essential use was made of the fact that in its natural linear action the real symplectic group, Sp$(n,\R)$, acts transitively on $\R^{2n}\setminus\{0\}$ (similarly for the theorem of Hlawka itself, SL$(n,\R)$ acts transitively on $\R^n\setminus\{0\}$). This raises the natural question as to whether there are {\it proper connected} Lie subgroups of either of these groups which also act transitively on $\R^{2n}\setminus\{0\}$, (resp. $\R^n\setminus\{0\}$). Here we determine all the minimal ones. These are Sp$(n,\R)\subseteq {\rm SL}(2n,\R)$ and SL$(n,\C) \subseteq{\rm SL}(2n,\R)$ acting on $\R^{2n}\setminus \{0\}$; on $\R^{4n}\setminus \{0\}$, they are Sp$(2n,\R)\subseteq{\rm SL}(4n,\R)$ and SL$(n,\H) (={\rm SU}^*(2n)) \subseteq{\rm SL}(4n,\R)$.

Keywords: Transitive linear action, reductive group, actions of compact groups on spheres, special linear and real symplectic groups.

MSC: 22E46, 22F30, 54H15, 57S15

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