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.
Definition
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,
![{\displaystyle \langle f,\,[{\mathfrak {h}},\,{\mathfrak {h}}]\rangle =0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
.
Pukanszky condition
The following condition was obtained by L. Pukanszky:[2]
Let
be the polarization of algebra
with respect to covector
and
be its annihilator:
. The polarization
is said to satisfy the Pukanszky condition if
![{\displaystyle f+{\mathfrak {h}}^{\perp }\in {\mathcal {O}}_{f}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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.[3]
Properties
- 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]
References
- ^ Corwin, Lawrence; GreenLeaf, Frderick P. (25 January 1981). "Rationally varying polarizing subalgebras in nilpotent Lie algebras". Proceedings of the American Mathematical Society. 81 (1). Berlin: American Mathematical Society: 27–32. doi:10.2307/2043981. ISSN 1088-6826. Zbl 0477.17001.
- ^ 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.
- ^ Pukanszky, Lajos (March 1967). "On the theory of exponential groups" (PDF). Transactions of the American Mathematical Society. 126. American Mathematical Society: 487–507. doi:10.1090/S0002-9947-1967-0209403-7. ISSN 1088-6850. MR 0209403. Zbl 0207.33605.
- ^ a b c Kirillov, A. A. (1976) [1972], Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, vol. 220, Berlin, New York: Springer-Verlag, ISBN 978-0-387-07476-4, MR 0412321
- ^ a b Dixmier, Jacques (1996) [1974], Enveloping algebras, Graduate Studies in Mathematics, vol. 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, MR 0498740