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 [math]\displaystyle{ G }[/math] be a Lie group, [math]\displaystyle{ \mathfrak{g} }[/math] the corresponding Lie algebra and [math]\displaystyle{ \mathfrak{g}^* }[/math] its dual. Let [math]\displaystyle{ \langle f,\,X\rangle }[/math] denote the value of the linear form (covector) [math]\displaystyle{ f\in\mathfrak{g}^* }[/math] on a vector [math]\displaystyle{ X\in\mathfrak{g} }[/math]. The subalgebra [math]\displaystyle{ \mathfrak{h} }[/math] of the algebra [math]\displaystyle{ \mathfrak g }[/math] is called subordinate of [math]\displaystyle{ f\in\mathfrak{g}^* }[/math] if the condition

[math]\displaystyle{ \forall X, Y\in\mathfrak{h}\ \langle f,\,[X,\,Y]\rangle = 0 }[/math],

or, alternatively,

[math]\displaystyle{ \langle f,\,[\mathfrak{h},\,\mathfrak{h}]\rangle = 0 }[/math]

is satisfied. Further, let the group [math]\displaystyle{ G }[/math] act on the space [math]\displaystyle{ \mathfrak{g}^* }[/math] via coadjoint representation [math]\displaystyle{ \mathrm{Ad}^* }[/math]. Let [math]\displaystyle{ \mathcal{O}_f }[/math] be the orbit of such action which passes through the point [math]\displaystyle{ f }[/math] and [math]\displaystyle{ \mathfrak{g}^f }[/math] be the Lie algebra of the stabilizer [math]\displaystyle{ \mathrm{Stab}(f) }[/math] of the point [math]\displaystyle{ f }[/math]. A subalgebra [math]\displaystyle{ \mathfrak{h}\subset\mathfrak{g} }[/math] subordinate of [math]\displaystyle{ f }[/math] is called a polarization of the algebra [math]\displaystyle{ \mathfrak{g} }[/math] with respect to [math]\displaystyle{ f }[/math], or, more concisely, polarization of the covector [math]\displaystyle{ f }[/math], if it has maximal possible dimensionality, namely

[math]\displaystyle{ \dim\mathfrak{h} = \frac{1}{2}\left(\dim\,\mathfrak{g} + \dim\,\mathfrak{g}^f\right) = \dim\,\mathfrak{g} - \frac{1}{2}\dim\,\mathcal{O}_f }[/math].

## Pukanszky condition

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

Let [math]\displaystyle{ \mathfrak{h} }[/math] be the polarization of algebra [math]\displaystyle{ \mathfrak{g} }[/math] with respect to covector [math]\displaystyle{ f }[/math] and [math]\displaystyle{ \mathfrak{h}^\perp }[/math] be its annihilator: [math]\displaystyle{ \mathfrak{h}^\perp := \\{\lambda\in\mathfrak{g}^*|\langle\lambda,\,\mathfrak{h}\rangle = 0\\} }[/math]. The polarization [math]\displaystyle{ \mathfrak{h} }[/math] is said to satisfy the Pukanszky condition if

[math]\displaystyle{ f + \mathfrak{h}^\perp\in\mathcal{O}_f. }[/math]

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 [math]\displaystyle{ \langle f,\,[\cdot,\,\cdot]\rangle }[/math] on the Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math].[4]
* For some pairs [math]\displaystyle{ (\mathfrak{g},\,f) }[/math] polarization may not exist.[4]
* If the polarization does exist for the covector [math]\displaystyle{ f }[/math], then it exists for every point of the orbit [math]\displaystyle{ \mathcal{O}_f }[/math] as well, and if [math]\displaystyle{ \mathfrak{h} }[/math] is the polarization for [math]\displaystyle{ f }[/math], then [math]\displaystyle{ \mathrm{Ad}_g\mathfrak{h} }[/math] is the polarization for [math]\displaystyle{ \mathrm{Ad}^*_g f }[/math]. Thus, the existence of the polarization is the property of the orbit as a whole.[4]
* If the Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] is completely solvable, it admits the polarization for any point [math]\displaystyle{ f\in\mathfrak{g}^* }[/math].[5]
* If [math]\displaystyle{ \mathcal{O} }[/math] is the orbit of general position (i. e. has maximal dimensionality), for every point [math]\displaystyle{ f\in\mathcal{O} }[/math] 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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