On uniqueness of submaximally symmetric
parabolic geometries

Let $G$ be a real or complex semisimple Lie group, $P\subset G$ a parabolic subgroup, and ${\mathfrak{p}}\subset{\mathfrak{g}}$ be their Lie algebras. Then ${\mathfrak{g}}$ admits a natural $P$-invariant (decreasing) filtration ${\mathfrak{g}}={\mathfrak{g}}^{-\nu}\supset...\supset{\mathfrak{g}}^{\nu}$ (we put ${\mathfrak{g}}^{i}={\mathfrak{g}}$ for $i<-\nu$, ${\mathfrak{g}}^{i}=0$ for $i>\nu$), ${\mathfrak{g}}^{1}={\mathfrak{p}}_{+}$ is the nilradical of ${\mathfrak{g}}^{0}={\mathfrak{p}}$, and $[{\mathfrak{g}}^{i},{\mathfrak{g}}^{j}]\subset{\mathfrak{g}}^{i+j}$ for all $i,j\in{\mathbb{Z}}$. There always exists grading element $\mathsf{Z}\in{\mathfrak{g}}$ whose ${\rm ad}_{\mathsf{Z}}$-eigenvalues $\forall j\in{\mathbb{Z}}$ (degrees) and eigenspaces ${\mathfrak{g}}_{j}:=\{x\in{\mathfrak{g}}:{\rm ad}_{\mathsf{Z}}(x)=jx\}$ ($\forall j\in{\mathbb{Z}}$) endow ${\mathfrak{g}}$ with the structure of a graded Lie algebra ${\mathfrak{g}}={\mathfrak{g}}_{-\nu}\oplus\ldots\oplus{\mathfrak{g}}_{\nu}$ compatible with the filtration, i.e. $[{\mathfrak{g}}_{i},{\mathfrak{g}}_{j}]\subset{\mathfrak{g}}_{i+j}$ and ${\mathfrak{g}}^{i}\cong\bigoplus_{j=i}^{\nu}{\mathfrak{g}}_{j}$. The associated-graded Lie algebra $\mathrm{gr}({\mathfrak{g}})$ is defined by $\mathrm{gr}_{i}({\mathfrak{g}}):={\mathfrak{g}}^{i}/{\mathfrak{g}}^{i+1}$. Given $\mathsf{Z}$ as above, we identify $\mathrm{gr}_{i}({\mathfrak{g}})\cong{\mathfrak{g}}_{i}$ as ${\mathfrak{g}}_{0}$-modules, and if $x\in{\mathfrak{g}}^{i}$, we denote by $\mathrm{gr}_{i}(x)\in{\mathfrak{g}}_{i}$ the projection to its leading part. We have $\mathsf{Z}\in{\mathfrak{z}}({\mathfrak{g}}_{0})$ (centre of ${\mathfrak{g}}_{0}$), ${\mathfrak{p}}={\mathfrak{g}}_{0}\oplus{\mathfrak{p}}_{+}$, and the Killing form on ${\mathfrak{g}}$ identifies $({\mathfrak{g}}/{\mathfrak{p}})^{*}\cong{\mathfrak{p}}_{+}$ as $P$-modules. Finally, letting $G_{0}=\{g\in P:{\rm Ad}_{g}({\mathfrak{g}}_{i})\subset{\mathfrak{g}}_{i},\,\forall i\}$ be the Levi subgroup (with Lie algebra ${\mathfrak{g}}_{0}$), and $P_{+}=\exp({\mathfrak{p}}_{+})\leq P$, we have $P\cong G_{0}\ltimes P_{+}$.

A parabolic geometry is a Cartan geometry $({\mathcal{G}}\to M,\omega)$ of type $(G,P)$, i.e. a (right) principal $P$-bundle ${\mathcal{G}}\to M$ with a Cartan connection $\omega\in\Omega^{1}({\mathcal{G}},{\mathfrak{g}})$:

1. (i)

   $\omega_{u}:T_{u}{\mathcal{G}}\to{\mathfrak{g}}$ is a linear isomorphism $\forall u\in{\mathcal{G}}$;

2. (ii)

   $\omega$ is $P$-equivariant: $R_{p}^{*}\omega={\rm Ad}_{p^{-1}}\circ\omega$, $\forall p\in P$;

3. (iii)

   $\omega(\zeta_{A})=A$, $\forall A\in{\mathfrak{p}}$, where $\zeta_{A}$ is the fundamental vertical vector field corresponding to $A$.

The curvature of $\omega$ is $K=d\omega+\frac{1}{2}[\omega,\omega]\in\Omega^{2}({\mathcal{G}},{\mathfrak{g}})$ (which is $P$-equivariant and horizontal, i.e. $K(\zeta_{A},\cdot)=0$), or equivalently we have the curvature function $\kappa:{\mathcal{G}}\to\bigwedge^{2}({\mathfrak{g}}/{\mathfrak{p}})^{*}\otimes{\mathfrak{g}}$ given by $\kappa(x,y)=K(\omega^{-1}(x),\omega^{-1}(y))$. The geometry is flat if $K=0$, which characterizes local equivalence to the flat model $(G\to G/P,\omega_{G})$, where $\omega_{G}$ is the (left-invariant) Maurer–Cartan form on $G$. Via the Killing form, the codomain of $\kappa$ identifies (as a $P$-module) with $C_{2}({\mathfrak{p}}_{+},{\mathfrak{g}}):=\bigwedge^{2}{\mathfrak{p}}_{+}\otimes{\mathfrak{g}}$. These are 2-chains in the complex $(C_{\bullet}({\mathfrak{p}}_{+},{\mathfrak{g}}),\partial^{*})$ with $\partial^{*}$ the Lie algebra homology differential. We say that $({\mathcal{G}}\to M,\omega)$ is normal if $\partial^{*}\kappa=0$ and it is regular if $\kappa({\mathfrak{g}}^{i},{\mathfrak{g}}^{j})\subset{\mathfrak{g}}^{i+j+1}$ for any $i,j$. Equivalently, if we naturally extend the filtration on ${\mathfrak{g}}$ to a filtration on $\bigwedge^{2}{\mathfrak{p}}_{+}\otimes{\mathfrak{g}}$, then we have $\kappa\in\ker(\partial^{*})^{1}$. This is the subspace of $\ker(\partial^{*})\subset\bigwedge^{2}{\mathfrak{p}}_{+}\otimes{\mathfrak{g}}$ on which $\mathsf{Z}$ acts with positive eigenvalues (degrees). There is a well-known equivalence of categories between regular, normal parabolic geometries and underlying geometric structures on $M$ (see [6] for details).

For any regular, normal parabolic geometry, a key invariant is its harmonic curvature $\kappa_{H}:{\mathcal{G}}\to H_{2}({\mathfrak{p}}_{+},{\mathfrak{g}}):=\frac{\ker(\partial^{*})}{\mathrm{im}(\partial^{*})}$, given by $\kappa_{H}=\kappa\,\,\,{\rm mod}\ \mathrm{im}(\partial^{*})$, and this $P$-equivariant function is a complete obstruction to flatness. Moreover, $H_{2}({\mathfrak{p}}_{+},{\mathfrak{g}})$ is a completely reducible ${\mathfrak{p}}$-representation, i.e. ${\mathfrak{p}}_{+}$-acts trivially. As ${\mathfrak{g}}_{0}$-modules, ${\mathfrak{g}}/{\mathfrak{p}}\cong{\mathfrak{g}}_{-}$, and $C^{k}({\mathfrak{g}}_{-},{\mathfrak{g}}):=\bigwedge^{k}{\mathfrak{g}}_{-}^{*}\otimes{\mathfrak{g}}$ yields a complex $(C^{\bullet}({\mathfrak{g}}_{-},{\mathfrak{g}}),\partial)$ with respect to the standard Lie algebra cohomology differential $\partial$, for which we have the (${\mathfrak{g}}_{0}$-invariant) algebraic Hodge decomposition:

where $\Box=\partial\partial^{*}+\partial^{*}\partial$ is the (${\mathfrak{g}}_{0}$-equivariant) algebraic Laplacian, with $\ker(\Box)=\ker(\partial)\cap\ker(\partial^{*})$. Then $H_{2}({\mathfrak{p}}_{+},{\mathfrak{g}})=\frac{\ker(\partial^{*})}{\mathrm{im}(\partial^{*})}\cong\ker(\Box)\cong\frac{\ker(\partial)}{\mathrm{im}(\partial)}\cong H^{2}({\mathfrak{g}}_{-},{\mathfrak{g}})$ as ${\mathfrak{g}}_{0}$-modules, which may be efficiently computed via Kostant’s theorem (§3.1). By regularity, $\kappa_{H}$ has image in the subspace $H_{2}({\mathfrak{p}}_{+},{\mathfrak{g}})^{1}\subseteq H_{2}({\mathfrak{p}}_{+},{\mathfrak{g}})$ on which $\mathsf{Z}$ acts with positive eigenvalues. This corresponds to some ${\mathfrak{g}}_{0}$-submodule $H^{2}_{+}({\mathfrak{g}}_{-},{\mathfrak{g}})\subseteq H^{2}({\mathfrak{g}}_{-},{\mathfrak{g}})$ under the above identification.

Finally, by [6, Thm.3.1.12], if $\kappa$ has lowest non-trivial degree $s>0$, then its leading part $\mathrm{gr}_{s}(\kappa)$ is harmonic and coincides with the degree $s$ component of $\kappa_{H}\neq 0$. In particular, $\kappa_{H}$ being a complete obstruction to flatness follows from this.

Two (regular, normal) parabolic geometries of type $(G,P)$ are equivalent if there is a principal bundle isomorphism that pulls back one Cartan connection to the other, and an automorphism is a self-equivalence. A Cartan geometry $({\mathcal{G}}\stackrel{{\scriptstyle\pi}}{{\to}}M,\omega)$ is (locally) homogeneous if there is a Lie group acting by (local) automorphisms whose projection to $M$ yields a (locally) transitive action on $M$. Infinitesimally, the symmetry algebra is

where ${\mathfrak{X}}({\mathcal{G}})^{P}$ are the $P$-invariant vector fields on ${\mathcal{G}}$.

Let us now summarize how to equivalently view $\mathfrak{inf}({\mathcal{G}},\omega)$ in a more algebraic manner [2, 5, 16]. Fix any $u\in{\mathcal{G}}$. Then $\omega_{u}:T_{u}{\mathcal{G}}\to{\mathfrak{g}}$ restricts to a linear injection on $\inf({\mathcal{G}},\omega)$. Letting ${\mathfrak{f}}={\mathfrak{f}}(u):=\omega_{u}(\inf({\mathcal{G}},\omega))$, the Lie bracket on $\inf({\mathcal{G}},\omega)$ transfers to the bracket on ${\mathfrak{f}}$ given by

The $P$-invariant filtration on ${\mathfrak{g}}$ induces a filtration on ${\mathfrak{f}}$ via ${\mathfrak{f}}^{i}:={\mathfrak{f}}\cap{\mathfrak{g}}^{i}$. By regularity, $\kappa({\mathfrak{g}}^{i},{\mathfrak{g}}^{j})\subset{\mathfrak{g}}^{i+j+1}$, so $[{\mathfrak{f}}^{i},{\mathfrak{f}}^{j}]_{\mathfrak{f}}\subset{\mathfrak{f}}\cap{\mathfrak{g}}^{i+j}={\mathfrak{f}}^{i+j}$, and $({\mathfrak{f}},[\cdot,\cdot]_{\mathfrak{f}})$ becomes a filtered Lie algebra (generally not a Lie subalgebra of ${\mathfrak{g}}$). By regularity, the associated-graded ${\mathfrak{s}}:=\mathrm{gr}({\mathfrak{f}})$, defined by ${\mathfrak{s}}_{i}:={\mathfrak{f}}^{i}/{\mathfrak{f}}^{i+1}$ is identified as a graded subalgebra of ${\mathfrak{g}}$ (via ${\mathfrak{s}}_{i}\hookrightarrow{\mathfrak{f}}^{i}/{\mathfrak{g}}^{i+1}\subseteq{\mathfrak{g}}^{i}/{\mathfrak{g}}^{i+1}\cong{\mathfrak{g}}_{i}$). The filtrand ${\mathfrak{f}}^{0}\subseteq{\mathfrak{g}}^{0}={\mathfrak{p}}$ satisfies the important algebraic condition ${\mathfrak{f}}^{0}\cdot\kappa=0$, which implies ${\mathfrak{f}}^{0}\cdot\kappa_{H}=0$. Since ${\mathfrak{p}}_{+}$ acts trivially on $H_{2}({\mathfrak{p}}_{+},{\mathfrak{g}})$, then ${\mathfrak{f}}^{1}\cdot\kappa_{H}=0$ always, so ${\mathfrak{s}}_{0}\cdot\kappa_{H}=0$, i.e. ${\mathfrak{s}}_{0}$ is contained in the annihilator ${\mathfrak{a}}_{0}:=\mathfrak{ann}(\kappa_{H}(u))\subseteq{\mathfrak{g}}_{0}$.

Now define the following (extrinsic) Tanaka prolongation algebra ${\mathfrak{a}}^{\phi}$ as in [16]:

The constraint ${\mathfrak{s}}_{0}\subseteq{\mathfrak{a}}_{0}$ propagates via Tanaka prolongation to the higher levels. More precisely, the following important inclusion holds:

Otherwise put, the symmetry algebra ${\mathfrak{f}}$ is a constrained filtered sub-deformation of ${\mathfrak{a}}^{\kappa_{H}}$, i.e.

1. (i)

   ${\mathfrak{f}}$ is a filtered deformation of the graded subspace ${\mathfrak{s}}(u)\subseteq{\mathfrak{a}}^{\kappa_{H}(u)}$, and

2. (ii)

   ${\mathfrak{f}}$ is constrained: e.g. it is a filtered subspace of ${\mathfrak{g}}$ and satisfies (2.3) for some $\kappa$.

The inclusion (2.4) was established in [16, Thm.2.4.6] on the open dense set of so-called regular points, i.e. those $u\in{\mathcal{G}}$ on which ${\rm dim}\,{\mathfrak{s}}_{i}(u)$ are locally constant functions $\forall i$, and was generalized to all points in [17, Thm.3.3]. If the given geometry is not flat, then $\kappa_{H}(u)\neq 0$ at some $u\in{\mathcal{G}}$, so $\mathsf{Z}\not\in\mathfrak{ann}(\kappa_{H}(u))$ by the regularity assumption on $\kappa$, and hence ${\rm dim}(\inf({\mathcal{G}},\omega))={\rm dim}({\mathfrak{s}})<{\rm dim}({\mathfrak{g}})$. Thus, the flat model is locally the unique maximally symmetric geometry. Defining

equation (2.4) immediately implies

A (regular, normal) geometry with ${\rm dim}(\inf({\mathcal{G}},\omega))={\mathfrak{S}}$ is submaximally symmetric. A priori, it should not be assumed that these are locally homogeneous, particularly if ${\mathfrak{S}}<{\mathfrak{U}}$.

Let $({\mathcal{G}}\to M,\omega)$ be homogeneous with respect to the Lie group $F$. Fix $u\in{\mathcal{G}}$, and let $F^{0}\subset F$ be the stabilizer of $o=\pi(u)\in M$. Given any $f_{0}\in F^{0}$, we have $f_{0}\cdot u=u\cdot\iota(f_{0})$ for some Lie group homomorphism $\iota:F^{0}\to P$. This defines a right $F^{0}$-action on $F\times P$ via $(f,p)\cdot f_{0}=(ff_{0},\iota(f_{0}^{-1})p)$ and we let $F\times_{F^{0}}P$ be the collection of all $F^{0}$-orbits $\overline{(f,p)}$. Letting ${\mathfrak{f}}$ and ${\mathfrak{f}}^{0}$ be the Lie algebras of $F$ and $F^{0}$ respectively, we have [6, Prop.1.5.15]:

1. (1)

   ${\mathcal{G}}\to M$ is equivalent to the associated bundle $F\times_{F^{0}}P\to F/F^{0}$.

2. (2)

   Any $F$-invariant Cartan connection $\omega\in\Omega^{1}(F\times_{F^{0}}P,{\mathfrak{g}})$ of type $({\mathfrak{g}},P)$ is completely determined by the following:

Indeed, given $\varpi$ as above, we obtain $\omega$ by factoring $\hat{\omega}^{1}\in\Omega^{1}(F\times P,{\mathfrak{g}})$ given by

The basepoint change $u\mapsto f\cdot u$ leaves $(\iota,\varpi)$ unchanged, but a fibrewise change $u\mapsto u\cdot p$ induces $(\iota,\varpi)\mapsto({\rm Ad}_{p^{-1}}\circ\iota,{\rm Ad}_{p^{-1}}\circ\varpi)$.

Define $\tilde{\kappa}(x,y):=[\varpi(x),\varpi(y)]-\varpi([x,y]_{\mathfrak{f}})$, so $\tilde{\kappa}\in\bigwedge^{2}({\mathfrak{f}}/{\mathfrak{f}}^{0})^{*}\otimes{\mathfrak{g}}$ by (C2’). The curvature of $\omega$ corresponds to $\kappa\in\bigwedge^{2}({\mathfrak{g}}/{\mathfrak{p}})^{*}\otimes{\mathfrak{g}}$ given by $\kappa(x,y)=\tilde{\kappa}(\varpi^{-1}(x),\varpi^{-1}(y))$. The notions of regularity and normality of $\kappa$ are immediately specialized to this algebraic setting, as is the quotient object $\kappa_{H}=\kappa\,\,{\rm mod}\ \mathrm{im}(\partial^{*})\in H_{2}({\mathfrak{p}}_{+},{\mathfrak{g}})^{1}$.

Note that $({\rm C}3)$ and (C2’) forces $\ker(\varpi)\subset{\mathfrak{f}}^{0}$ to be an ideal in ${\mathfrak{f}}$. The $F$-action on $F/F^{0}$ can always assumed to be infinitesimally effective, i.e. ${\mathfrak{f}}^{0}$ does not contain any non-trivial ideals of ${\mathfrak{f}}$ (hence, $\ker(\varpi)=0$). (Otherwise, we may without loss of generality quotient both $F$ and $F^{0}$ by the corresponding normal subgroup.) Consequently, we assume that $\varpi:{\mathfrak{f}}\to{\mathfrak{g}}$ is injective and identify ${\mathfrak{f}}$ with its image in ${\mathfrak{g}}$. This motivates the following definition.

The result below immediately follows from the general theory recalled in §2.2, but it is instructive to give proofs directly following from Definition 2.5 above.

Importantly, we note that the set of algebraic models of type $({\mathfrak{g}},{\mathfrak{p}})$:

• •

  admits a $P$-action via ${\mathfrak{f}}\mapsto{\rm Ad}_{p}{\mathfrak{f}}$ for any $p\in P$. All algebraic models in the same $P$-orbit are to be regarded as equivalent, so we must always account for this redundancy.

• •

  is partially ordered: declare that ${\mathfrak{f}}\leq{\mathfrak{f}}^{\prime}$ if and only if ${\mathfrak{f}}$ is a Lie subalgebra of ${\mathfrak{f}}^{\prime}$. We will focus on maximal elements ${\mathfrak{f}}$. (We view non-maximal elements as non-optimal descriptions of the same geometric structure.)

Since $\mathrm{gr}({\mathfrak{f}})={\mathfrak{s}}$, we may view ${\mathfrak{f}}\subseteq{\mathfrak{g}}$ as a graph over ${\mathfrak{s}}\subseteq{\mathfrak{g}}$. Namely, choosing some graded subspace ${\mathfrak{s}}^{\perp}\subseteq{\mathfrak{g}}$ so that ${\mathfrak{g}}={\mathfrak{s}}\oplus{\mathfrak{s}}^{\perp}$ (in fact, ${\mathfrak{s}}^{\perp}\subseteq{\mathfrak{p}}$ since ${\mathfrak{g}}_{-}\subseteq{\mathfrak{s}}$ by hypothesis), we can write

for some unique linear map ${\mathfrak{d}}:{\mathfrak{s}}\to{\mathfrak{s}}^{\perp}$ of positive degree, i.e. if $x\in{\mathfrak{s}}_{i}$, then ${\mathfrak{d}}(x)\in{\mathfrak{s}}^{\perp}\cap{\mathfrak{g}}^{i+1}$. We refer to ${\mathfrak{d}}$ as the deformation map and ${\mathfrak{d}}(x)$ as the tail of $x$.