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.

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

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

or, alternatively,

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

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

${\displaystyle \dim {\mathfrak {h}}={\frac {1}{2}}\left(\dim \,{\mathfrak {g}}+\dim \,{\mathfrak {g}}^{f}\right)=\dim \,{\mathfrak {g}}-{\frac {1}{2}}\dim \,{\mathcal {O}}_{f}}$.

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

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

${\displaystyle f+{\mathfrak {h}}^{\perp }\in {\mathcal {O}}_{f}.}$

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]

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