Journal of Lie Theory

no. 1
Some Transitive Linear Actions of Real Simple Lie Groups
Laura Geatti1; Martin Moskowitz2
1Dip. di Matematica, Università di Roma 2 "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy
2CUNY Graduate Center, Ph.D. Program in Mathematics, 365 Fifth Avenue, New York, NY 10016, U.S.A.
Journal of Lie Theory, Volume 22 (2012) no. 1, pp. 155-161
\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$(n,\H) (={\rm SU}^*(2n)) \subseteq{\rm SL}(4n,\R)$.
DOI: 10.5802/jolt.664
Classification: 22E46, 22F30, 54H15, 57S15
Keywords: Transitive linear action, reductive group, actions of compact groups on spheres, special linear and real symplectic groups

Laura Geatti  1 ; Martin Moskowitz  2

1 Dip. di Matematica, Università di Roma 2 "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy
2 CUNY Graduate Center, Ph.D. Program in Mathematics, 365 Fifth Avenue, New York, NY 10016, U.S.A. 
Laura Geatti; Martin Moskowitz. Some Transitive Linear Actions of Real Simple Lie Groups. Journal of Lie Theory, Volume 22 (2012) no. 1, pp. 155-161. doi: 10.5802/jolt.664
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