Polarization (Lie algebra) explained

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

G be a Lie group, ak{g} the corresponding Lie algebra and ak{g} ^{*} its dual. Let \langle f,X\rangle denote the value of the linear form (covector) f\in ak{g} ^{*} on a vector X\in ak{g} . The subalgebra ak{h} of the algebra ak g is called subordinate of f\in ak{g} ^{*} if the condition\forall X, Y\in ak{h} \langle f,[X,Y]\rangle = 0 ,or, alternatively,\langle f,[ ak{h}, ak{h}]\rangle = 0 is satisfied. Further, let the group G act on the space ak{g} ^{*} via coadjoint representation Ad ^{*} . Let l{O} _{f} be the orbit of such action which passes through the point f and let ak{g} ^{f} be the Lie algebra of the stabilizer Stab (f) of the point f . A subalgebra ak{h}\subset ak{g} subordinate of f is called a polarization of the algebra ak{g} with respect to f , or, more concisely, polarization of the covector f , if it has maximal possible dimensionality, namely\dim ak{h} = (1)/(2) \left(\dim ak{g} + \dim ak{g} ^{f\right)} = \dim ak{g} - (1)/(2) \dim l{O} _{f} .

Pukanszky condition

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

Let

ak{h} be the polarization of algebra ak{g} with respect to covector f and ak{h} ^{\perp} be its annihilator: ak{h} ^{\perp} : = \{ λ \in ak{g} ^{*|\langleλ,ak{h}\rangle=} 0 \} . The polarization ak{h} is said to satisfy the Pukanszky condition iff + ak{h} ^{\perp\in l{O}} _{f.} 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

\langle f,[ ⋅ , ⋅ ]\rangle on the Lie algebra ak{g} .

• For some pairs

( ak{g},f) polarization may not exist.

• If the polarization does exist for the covector

f , then it exists for every point of the orbit l{O} _{f} as well, and if ak{h} is the polarization for f , then Ad _{gak{h}} is the polarization for Ad ^{*} _{g} f . Thus, the existence of the polarization is the property of the orbit as a whole.

• If the Lie algebra

ak{g} is completely solvable, it admits the polarization for any point f\in ak{g} ^{*} .

• If

l{O} is the orbit of general position (i. e. has maximal dimensionality), for every point f\in l{O} there exists solvable polarization.

Notes and References

1. Corwin . Lawrence . GreenLeaf . Frderick P. . Rationally varying polarizing subalgebras in nilpotent Lie algebras . Proceedings of the American Mathematical Society . 81 . 1 . 27–32 . American Mathematical Society . Berlin . 25 January 1981 . 1088-6826 . 10.2307/2043981 . 0477.17001 . free .
2. Dixmier. Jacques . Duflo. Michel . Hajnal. Andras . Kadison. Richard . Korányi. Adam . Rosenberg. Jonathan . Vergne. Michele . Lajos Pukánszky (1928 – 1996) . Notices of the American Mathematical Society . 45 . 4 . 492–499 . American Mathematical Society . April 1998 . 1088-9477 .
3. Pukanszky . Lajos . On the theory of exponential groups . Transactions of the American Mathematical Society . 126 . 487–507 . American Mathematical Society . March 1967 . 1088-6850 . 10.1090/S0002-9947-1967-0209403-7 . 0209403 . 0207.33605 . free .