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[1] as well as in harmonic analysis on Lie groups and mathematical physics.

## Contents

* 1 Definition
* 2 Pukanszky condition
* 3 Properties
* 4 References

## Definition

Let G be a Lie group, 𝔤 the corresponding Lie algebra and 𝔤* its dual. Let ⟨f,X⟩ denote the value of the linear form (covector) f∈𝔤* on a vector X∈𝔤. The subalgebra 𝔥 of the algebra 𝔤 is called subordinate of f∈𝔤* if the condition

∀X,Y∈𝔥 ⟨f,[X,Y]⟩=0,

or, alternatively,

⟨f,[𝔥,𝔥]⟩=0

is satisfied. Further, let the group G act on the space 𝔤* via coadjoint representation Ad*. Let 𝒪f be the orbit of such action which passes through the point f and 𝔤f be the Lie algebra of the stabilizer Stab(f) of the point f. A subalgebra 𝔥⊂𝔤 subordinate of f is called a polarization of the algebra 𝔤 with respect to f, or, more concisely, polarization of the covector f, if it has maximal possible dimensionality, namely

dim⁡𝔥=12(dim⁡𝔤+dim⁡𝔤f)=dim⁡𝔤−12dim⁡𝒪f.

## Pukanszky condition

The following condition was obtained by L. Pukanszky:[2]

Let 𝔥 be the polarization of algebra 𝔤 with respect to covector f and 𝔥⊥ be its annihilator: 𝔥⊥:={λ∈𝔤*|⟨λ,𝔥⟩=0}. The polarization 𝔥 is said to satisfy the Pukanszky condition if

f+𝔥⊥∈𝒪f.

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 ⟨f,[⋅,⋅]⟩ on the Lie algebra 𝔤.[4]
* For some pairs (𝔤,f) polarization may not exist.[4]
* If the polarization does exist for the covector f, then it exists for every point of the orbit 𝒪f as well, and if 𝔥 is the polarization for f, then Adg𝔥 is the polarization for Adg*f. Thus, the existence of the polarization is the property of the orbit as a whole.[4]
* If the Lie algebra 𝔤 is completely solvable, it admits the polarization for any point f∈𝔤*.[5]
* If 𝒪 is the orbit of general position (i. e. has maximal dimensionality), for every point f∈𝒪 there exists solvable polarization.[5]

## References

1. ↑ Corwin, Lawrence; GreenLeaf, Frderick P. (25 January 1981). "Rationally varying polarizing subalgebras in nilpotent Lie algebras". Proceedings of the American Mathematical Society (Berlin: American Mathematical Society) 81 (1): 27–32. doi:10.2307/2043981. ISSN 1088-6826. https://www.ams.org/journals/proc/1981-081-01/S0002-9939-1981-0589131-1/.
2. ↑ Dixmier, Jacques; Duflo, Michel; Hajnal, Andras; Kadison, Richard; Korányi, Adam; Rosenberg, Jonathan; Vergne, Michele (April 1998). "Lajos Pukánszky (1928 – 1996)". Notices of the American Mathematical Society (American Mathematical Society) 45 (4): 492 — 499. ISSN 1088-9477. https://www.ams.org/journals/notices/199804/199804FullIssue.pdf.
3. ↑ Pukanszky, Lajos (March 1967). "On the theory of exponential groups". Transactions of the American Mathematical Society (American Mathematical Society) 126: 487 — 507. doi:10.1090/S0002-9947-1967-0209403-7. ISSN 1088-6850. https://www.ams.org/journals/tran/1967-126-03/S0002-9947-1967-0209403-7/S0002-9947-1967-0209403-7.pdf.
4. ↑ 4.0 4.1 4.2 Kirillov, A. A. (1976), Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, 220, Berlin, New York: Springer-Verlag, ISBN 978-0-387-07476-4
5. ↑ 5.0 5.1 Dixmier, Jacques (1996), Enveloping algebras, Graduate Studies in Mathematics, 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, https://books.google.com/books?isbn=0821805606

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