In Moskowitz M., 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 \ {0} (similarly for the theorem of Hlawka itself, SL(n, R) acts transitively on \ {01). This raises the natural question as to whether there are proper connected Lie subgroups of either of these groups which also act transitively on R-2n \ {0}, (resp. R-n \ {01}. Here we determine all the minimal ones. These are Sp(n, R) subset of SL(2n, R) and SL(n, C) subset of SL(2n, R) acting on R-2n \ {0}; on R-4n \ {0}, they are Sp(2n, R) subset of SL(4n, R) and SL(n, H)(= SU* (2n)) subset of SL(4n, R).
Geatti, L., Moskowitz, M. (2012). Some Transitive Linear Actions of Real Simple Lie Groups. JOURNAL OF LIE THEORY, 22(1), 155-161.
Some Transitive Linear Actions of Real Simple Lie Groups
GEATTI, LAURA;
2012-01-01
Abstract
In Moskowitz M., 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 \ {0} (similarly for the theorem of Hlawka itself, SL(n, R) acts transitively on \ {01). This raises the natural question as to whether there are proper connected Lie subgroups of either of these groups which also act transitively on R-2n \ {0}, (resp. R-n \ {01}. Here we determine all the minimal ones. These are Sp(n, R) subset of SL(2n, R) and SL(n, C) subset of SL(2n, R) acting on R-2n \ {0}; on R-4n \ {0}, they are Sp(2n, R) subset of SL(4n, R) and SL(n, H)(= SU* (2n)) subset of SL(4n, R).File | Dimensione | Formato | |
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