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[1] as well as in harmonic analysis on Lie groups and mathematical physics.
Let be a Lie group, the corresponding Lie algebra and its dual. Let denote the value of the linear form (covector) on a vector . The subalgebra of the algebra is called subordinate of if the condition
- ,
or, alternatively,
is satisfied. Further, let the group act on the space via coadjoint representation . Let be the orbit of such action which passes through the point and let be the Lie algebra of the stabilizer of the point . A subalgebra subordinate of is called a polarization of the algebra with respect to , or, more concisely, polarization of the covector , if it has maximal possible dimensionality, namely
- .
- Polarization is the maximal totally isotropic subspace of the bilinear form on the Lie algebra .[4]
- For some pairs polarization may not exist.[4]
- If the polarization does exist for the covector , then it exists for every point of the orbit as well, and if is the polarization for , then is the polarization for . Thus, the existence of the polarization is the property of the orbit as a whole.[4]
- If the Lie algebra is completely solvable, it admits the polarization for any point .[5]
- If is the orbit of general position (i. e. has maximal dimensionality), for every point there exists solvable polarization.[5]
Dixmier, Jacques; Duflo, Michel; Hajnal, Andras; Kadison, Richard; Korányi, Adam; Rosenberg, Jonathan; Vergne, Michele (April 1998). "Lajos Pukánszky (1928 – 1996)" (PDF). Notices of the American Mathematical Society. 45 (4). American Mathematical Society: 492–499. ISSN 1088-9477.