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Coadjoint orbits of compact Lie groups and generalized stereographic projection. (English. Ukrainian original) Zbl 0933.22020

Ukr. Math. J. 51, No. 12, 1939-1944 (1999); translation from Ukr. Mat. Zh. 51, No. 12, 1714-1718 (1999).
Let \(\mathfrak G\) be a compact Lie group with Lie algebra \(\mathfrak g\) and complexification \(\mathfrak G^C\). Denote by \(Z\) a contragredient subgroup for a nilpotent radical of the parabolic subgroup \(P\subset{\mathfrak G}^C\). The author defines the stereographic projection as a mapping \(\pi :\mathfrak g \mapsto Z\) the restriction of which to the orbit \(O_\Lambda \simeq K\setminus \mathfrak G\) is a coordinate diffeomorphism. Here \(K\) is the stabilizer of the element \(\Lambda\) which belongs to the Cartan subalgebra \(\mathfrak h\). In the case where \(\mathfrak G=\text{SU}(n)\) the author obtains an explicit formula for the stereographic projection \(\pi \) which turns out to be equivariant with respect to the \(\mathfrak G\)-action.

MSC:

22E46 Semisimple Lie groups and their representations
53C30 Differential geometry of homogeneous manifolds
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
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