Polarization (Lie algebra)
In representation theory, polarization is the maximal totally isotropic subspace of a certain skew-symmetric bilinear form on a Lie algebra. The notion of polarization plays an important role in construction of irreducible unitary representations of some classes of Lie groups by means of the orbit method as well as in harmonic analysis on Lie groups and mathematical physics. Definition. Let formula_1 be a Lie group, formula_2 the corresponding Lie algebra and formula_3 its dual. Let formula_4 denote the value of the linear form (covector) formula_5 on a vector formula_6. The subalgebra formula_7 of the algebra formula_8  is called subordinate of formula_5 if the condition or, alternatively, is satisfied. Further, let the group formula_1 act on the space formula_3 via coadjoint representation formula_14. Let formula_15 be the orbit of such action which passes through the point formula_16 and formula_17 be the Lie algebra of the stabilizer formula_18 of the point formula_16. A subalgebra formula_20 subordinate of formula_16 is called a polarization of the algebra formula_2 with respect to formula_16, or, more concisely, polarization of the covector formula_16, if it has maximal possible dimensionality, namely Pukanszky condition. The following condition was obtained by L. Pukanszky: Let formula_7 be  the polarization of algebra formula_2 with respect to covector formula_16 and formula_29 be its annihilator: formula_30. The polarization formula_7 is said to satisfy the Pukanszky condition if L. Pukanszky has shown that this condition guaranties applicability of the Kirillov's orbit method initially constructed for nilpotent groups to more general case of solvable groups as well.