Abstract
The orbit method of Kirillov is an important tool in representation theory of Lie groups, which links together harmonic analysis, symplectic geometry and mathematical physics. The method gets its name from a study of orbits of the coadjoint representation, which turn to be simply the matrix conjugation in the important case of matrix groups. The orbit methods began from the success for nilpotent Lie groups and steadily evolved to tackle more difficult cases, e.g. non-compact semisimple Lie groups.
An interesting modification of the orbit method can be obtained if we consider hypercomplex extensions of the coadjoint space. Geometrically it corresponds to an action on the space of quadrics rather than points. In the case of SL(2,R) this lead to a better description of complementary series, which is difficult to deal within the traditional orbit method.
Related Links
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