Polarization at a coadjoint functional

Let $\mathfrak g$ be a finite-dimensional real Lie algebra and $\ell\in\mathfrak g^*$ . Set

B_\ell(X,Y)=\ell([X,Y]),\qquad \mathfrak g_\ell=\{X:B_\ell(X,\mathfrak g)=0\}.

A real polarization at $\ell$ is a Lie subalgebra $\mathfrak p\subseteq\mathfrak g$ such that $\ell([\mathfrak p,\mathfrak p])=0$ and

\dim\mathfrak p=\tfrac12(\dim\mathfrak g+\dim\mathfrak g_\ell).

Thus $\mathfrak p$ is a maximal-dimensional isotropic subspace for $B_\ell$ that is also closed under the Lie bracket. A complex polarization is defined analogously as a complex Lie subalgebra of $\mathfrak g_{\mathbb C}$ , subordinate to the complex-linear extension of $\ell$ , with the corresponding complex dimension.

Relation to the coadjoint orbit

The radical $\mathfrak g_\ell$ is the stabilizer Lie algebra of $\ell$ , while $B_\ell$ descends to the nondegenerate Kirillov form on $\mathfrak g/\mathfrak g_\ell$ , the tangent space to the coadjoint orbit. The dimension formula therefore says that $\mathfrak p/\mathfrak g_\ell$ is Lagrangian in that symplectic quotient. The bracket-closure condition is additional: a merely maximal isotropic subspace need not integrate to a subgroup and is not a polarization.

From a polarization to a representation

Suppose $G$ is connected and simply connected nilpotent with Lie algebra $\mathfrak g$ , and let $P=\exp(\mathfrak p)$ . Subordination makes

\chi_\ell(\exp X)=e^{i\ell(X)}

a unitary character of $P$ . The unitarily induced representation $\operatorname{Ind}_P^G\chi_\ell$ is irreducible, is independent up to unitary equivalence of the chosen polarization, and depends only on the coadjoint orbit of $\ell$ . These conclusions are special to the nilpotent orbit-method setting; the bare definition alone does not guarantee them for a general Lie group.

Heisenberg example

For the three-dimensional Heisenberg algebra with basis $X,Y,Z$ and $[X,Y]=Z$ , choose $\ell(Z)=\lambda\neq0$ . Then $\mathfrak g_\ell=\mathbb RZ$ , and $\mathfrak p=\operatorname{span}\{Y,Z\}$ is a polarization: it is abelian and has dimension $2=(3+1)/2$ . The line $\mathbb RY$ is subordinate but not a polarization because it is too small. Choosing $\operatorname{span}\{X,Z\}$ instead leads to an equivalent irreducible representation.

Conventions and scope

Some sources call any maximal subordinate subalgebra a polarization. For nilpotent Lie algebras this is normally paired with the dimension condition in the core. For solvable groups, additional conditions such as the Pukánszky condition can be required. A complex polarization may also be required to be positive or invariant under a stabilizer; those are strengthened notions and are not included here.

References

A. A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics 64, American Mathematical Society, 2004. AMS record. Relevant: Chapters 2–3 on polarizations, the Heisenberg group, and nilpotent groups.
Lawrence J. Corwin and Frederick P. Greenleaf, Representations of Nilpotent Lie Groups and Their Applications, Part I: Basic Theory and Examples, Cambridge Studies in Advanced Mathematics 18, Cambridge University Press, 1990. Publisher front matter. Relevant: Chapter 2 on coadjoint orbits, polarizations, and induced representations.