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Jet-determination of symmetries of parabolic geometries

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Abstract

We establish 2-jet determinacy for the symmetry algebra of the underlying structure of any (complex or real) parabolic geometry. At non-flat points, we prove that the symmetry algebra is in fact 1-jet determined. Moreover, we prove 1-jet determinacy at any point for a variety of non-flat parabolic geometries—in particular torsion-free, parabolic contact, and several other classes.

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Notes

  1. In [5], the isotropy is identified with an element of \(T_x^*M\), but we will not use this identification.

  2. General 2-jet determinacy of symmetry algebras of parabolic geometries has been known earlier—see Remark 3.9.

  3. For complex Yamaguchi-nonrigid geometries: (i) torsion-free geometries are classified in Appendix B, (ii) (GP) with \({\mathbb P}({\mathfrak {g}}^\nu )\) a single P-orbit are classified in Proposition 5.16.

  4. While \({\mathfrak {f}}(u) \subset {\mathfrak {g}}\) is a linear subspace, it is in general not a Lie subalgebra.

  5. The exceptions admitting non-trivial positive prolongation are classified in [25, Table 4], and \(A_\ell / P_{1,2}\) geometry from Example 4.3 appears in this list.

  6. The duality involution is trivial except in the following cases: \(A_\ell \) for \(\ell \ge 2\), \(D_\ell \) for \(\ell \) odd, or \(E_6\). In these cases, it is the unique non-trivial automorphism of the Dynkin diagram.

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Acknowledgements

We thank J.M. Landsberg for discussions on minuscule varieties and sub-cominuscule representations. We are grateful to A. Isaev and I. Kossovskiy for information about the results on stability and linearization in CR-geometry. B.K. was supported by the University of Tromsø while visiting the Australian National University (where this work was initiated) and the University of Vienna. D.T. was supported by a Lise Meitner Fellowship (project M1884-N84) of the Austrian Science Fund (FWF).

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Correspondence to Dennis The.

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Communicated by Ngaiming Mok.

Appendices

Appendix A: Dynkin diagram recipes and Kostant’s theorem

Let \({\mathfrak {g}}\) be a complex semisimple Lie algebra with Borel subalgebra \({\mathfrak {b}}\), Cartan subalgebra \({\mathfrak {h}}\), simple roots \(\{ \alpha _i \} \subset {\mathfrak {h}}^*\), dual basis \(\{ \mathsf {Z}_i \} \subset {\mathfrak {h}}\), coroots \(\alpha _i^\vee = \frac{2\alpha _i}{\langle \alpha _i, \alpha _i \rangle }\), and fundamental weights \(\{ \lambda _i \} \subset {\mathfrak {h}}^*\). Let \(\langle \cdot , \cdot \rangle \) be the symmetric bilinear form on \({\mathfrak {h}}^*\) induced from the Killing form B on \({\mathfrak {g}}\). Let:

  • \({\mathfrak {D}}({\mathfrak {g}})\) be the corresponding Dynkin diagram.

  • Given a parabolic subalgebra \({\mathfrak {b}}\subseteq {\mathfrak {p}}\subsetneq {\mathfrak {g}}\) with corresponding index set \(I_{\mathfrak {p}}\), let \({\mathfrak {D}}({\mathfrak {g}},{\mathfrak {p}})\) be the marked diagram obtained by putting crosses on \({\mathfrak {D}}({\mathfrak {g}})\) corresponding to \(I_{\mathfrak {p}}\).

  • Given a weight \(\mu \in {\mathfrak {h}}^*\), inscribe the coefficient \(r_i = \langle \mu , \alpha _i^\vee \rangle \) on the i-th node of \({\mathfrak {D}}({\mathfrak {g}})\) or \({\mathfrak {D}}({\mathfrak {g}},{\mathfrak {p}})\). Denote this by \({\mathfrak {D}}({\mathfrak {g}},\mu )\) or \({\mathfrak {D}}({\mathfrak {g}},{\mathfrak {p}},\mu )\) respectively. Its support consists of all connected components containing at least one nonzero coefficient over a node. This will be denoted by using 0 as a superscript on the previous diagram.

Table 5 Highest roots in terms of fundamental weights and simple roots

Let \({\mathfrak {g}}= {\mathfrak {g}}_{-\nu } \oplus \cdots \oplus {\mathfrak {g}}_\nu \) be the grading induced by \(\mathsf {Z}= \sum _{i \in I_{\mathfrak {p}}} \mathsf {Z}_i\).

(R.1) :

Structure of \({\mathfrak {g}}_0\): Removing crossed nodes from \({\mathfrak {D}}({\mathfrak {g}},{\mathfrak {p}})\) yields \({\mathfrak {D}}({\mathfrak {g}}_0^{ss})\) and \(\mathrm{dim}({\mathfrak {z}}({\mathfrak {g}}_0)) = |I_{\mathfrak {p}}|\).

(R.2) :

\({\mathfrak {g}}_{-1}\) as a \({\mathfrak {g}}_0^{ss}\)-module: Fix a crossed node i. For any adjacent uncrossed node j, inscribe the multiplicity of the bond between i and j if the bond is directed from i to j. Otherwise, inscribe a 1. Do this for each crossed node i to get the decomposition of \({\mathfrak {g}}_{-1}\) into irreducibles.

(R.3) :

\({\mathfrak {g}}_1\) as a \({\mathfrak {g}}_0^{ss}\)-module: Since \(({\mathfrak {g}}_{-1})^* \cong {\mathfrak {g}}_1\), we apply the duality involutionFootnote 6 to the \({\mathfrak {g}}_0^{ss}\) irreps in \({\mathfrak {g}}_{-1}\) to obtain the \({\mathfrak {g}}_0^{ss}\)-decomposition for \({\mathfrak {g}}_1\).

Now suppose that \({\mathfrak {g}}\) is simple and \(\lambda \) is its highest weight, given in Table 5.

(R.4) :

Top-slot \({\mathfrak {g}}_\nu \) as a \({\mathfrak {g}}_0^{ss}\)-module: Remove crosses from \({\mathfrak {D}}({\mathfrak {g}},{\mathfrak {p}},\lambda )\) to get the diagram \({\mathfrak {T}}({\mathfrak {g}}_0^{ss},\lambda )\).

(R.5) :

Effective part of the \({\mathfrak {g}}_0^{ss}\)-action on \({\mathfrak {g}}_\nu \): Restrict to the support \({\mathfrak {T}}^0({\mathfrak {g}}_0^{ss},\lambda )\) of \({\mathfrak {T}}({\mathfrak {g}}_0^{ss},\lambda )\).

Example A.1

(\(D_6 / P_{1,4}\)) . Here, \(\nu = 3\) and \({\mathfrak {g}}_0^{ss} = A_2 \times A_1 \times A_1\), but \(A_1 \times A_1\) acts trivially on \({\mathfrak {g}}_\nu \). The effective part is given by the \(A_2\)-action with the above weight.

The extended Dynkin diagram \(\widetilde{\mathfrak {D}}({\mathfrak {g}})\) augments \({\mathfrak {D}}({\mathfrak {g}})\) by a single node corresponding to \(-\lambda \) and corresponding bonds. Refer to the node(s) adjacent to \(-\lambda \) as the contact node(s). Equivalently, if \(\lambda = \sum _i r_i \lambda _i\), these are nodes j for which \(r_j \ne 0\). (See Table  5.) Mark these on \({\mathfrak {D}}({\mathfrak {g}},\lambda )\) with a .

(R.6) :

Simple roots orthogonal to \(\lambda \): All nodes in \({\mathfrak {D}}({\mathfrak {g}})\) except those marked with .

(R.7) :

The subalgebra \({\mathfrak {g}}(\lambda ) \subset {\mathfrak {g}}\): Remove all from \({\mathfrak {D}}({\mathfrak {g}},\lambda )\) to obtain a diagram \({\mathfrak {D}}_\lambda ({\mathfrak {g}},{\mathfrak {p}})\). (Equivalently, if \({\mathfrak {q}}\subset {\mathfrak {g}}\) is the parabolic subalgebra corresponding to crossing all , then \({\mathfrak {g}}(\lambda )\) is the semisimple part of \({\mathfrak {q}}\).)

(R.8) :

The ideal \({\mathfrak {l}}_{\mathfrak {p}}(\lambda ) \subset {\mathfrak {g}}(\lambda )\): From \({\mathfrak {D}}_\lambda ({\mathfrak {g}},{\mathfrak {p}})\), remove all cross-free connected components to obtain \({\mathfrak {D}}_\lambda ^0({\mathfrak {g}},{\mathfrak {p}})\) corresponding to an ideal \({\mathfrak {l}}_{\mathfrak {p}}(\lambda ) \subset {\mathfrak {g}}(\lambda )\). (All other ideals of \({\mathfrak {g}}(\lambda )\) lie in \({\mathfrak {g}}_0\).) Let \({\mathfrak {p}}(\lambda ) = {\mathfrak {l}}_{\mathfrak {p}}(\lambda ) \cap {\mathfrak {p}}\).

Using Lemma 5.8, we obtain the following recipe (see also Example 5.10):

  • (R.9) Top-slot orthogonal cascade:

    1. (a)

      Start with \({\mathfrak {D}}({\mathfrak {g}},{\mathfrak {p}},\lambda )\), where \({\mathfrak {g}}\) is simple, \({\mathfrak {p}}\subsetneq {\mathfrak {g}}\) is parabolic, and \(\lambda = \max \Delta ({\mathfrak {g}})\).

      • Termination condition: \({\mathfrak {T}}^0({\mathfrak {g}}_0^{ss},\lambda ) = \emptyset \) or .

      Remove all . Remove all connected components without crosses. Result: \({\mathfrak {D}}({\mathfrak {l}}_{\mathfrak {p}}(\lambda ),{\mathfrak {p}}(\lambda ))\). Now iterate using \({\mathfrak {D}}({\mathfrak {l}}_{\mathfrak {p}}(\lambda ),{\mathfrak {p}}(\lambda ),\mu )\) for the new \({\mathfrak {D}}({\mathfrak {g}},{\mathfrak {p}},\lambda )\), where \(\mu = \max \Delta ({\mathfrak {l}}_{\mathfrak {p}}(\lambda ))\).

    2. (b)

      For each diagram of the sequence produced in (a), write the highest root (use Table 5). Write the corresponding root in the initial \({\mathfrak {g}}\) by putting a zero coefficient for all nodes that carried a in the previous steps.

By Kostant’s version of the Bott–Borel–Weil theorem [8, 20], the \({\mathfrak {g}}_0\)-irreps \({\mathbb U}_\mu \subset H^2({\mathfrak {g}}_-,{\mathfrak {g}})\) are parametrized by the length two elements \(w \in W^{\mathfrak {p}}(2)\) of the Hasse diagram \(W^{\mathfrak {p}}\), which is a distinguished subset of the Weyl group W of \({\mathfrak {g}}\). Let \(\sigma _i\) denote the simple reflections in W, defined by \(\sigma _i(\alpha ) = \alpha - \langle \alpha , \alpha _i^\vee \rangle \alpha _i\). Then all \(w \in W^{\mathfrak {p}}(2)\) are of the form \(w = (jk) := \sigma _j \circ \sigma _k\), where

  1. (P.1)

    \(j \in I_{\mathfrak {p}}\) (i.e. a crossed node), and

  2. (P.2)

    \(j \ne k\) with either \(k \in I_{\mathfrak {p}}\) or k is adjacent to j in the Dynkin diagram of \({\mathfrak {g}}\).

The submodule \({\mathbb U}_\mu \subset H^2({\mathfrak {g}}_-,{\mathfrak {g}})\) has lowest \({\mathfrak {g}}_0\)-weight \(\mu = -w\cdot \lambda \), where \(\cdot \) denotes the affine action of W. By [8, Prop. 3.2.14 (1)], \(w \cdot 0 = -\alpha _j - \sigma _j(\alpha _k)\). If \(\lambda = \sum _i r_i \lambda _i\) (so \(r_i = \langle \lambda , \alpha _i^\vee \rangle \)) then

$$\begin{aligned} \mu&= -w\cdot \lambda = -w(\lambda ) - w\cdot 0 = -\sigma _j(\lambda - r_k \alpha _k) + \alpha _j + \sigma _j(\alpha _k) \nonumber \\&= -\lambda + (r_j + 1) \alpha _j + (r_k+1)(\alpha _k - c_{kj} \alpha _j), \end{aligned}$$
(A.1)

where \(c_{kj} = \langle \alpha _k, \alpha _j^\vee \rangle \) are entries of the Cartan matrix.

Using the natural \({\mathfrak {g}}_0\)-module isomorphism between \(H^2({\mathfrak {g}}_-,{\mathfrak {g}})\) and \(\ker (\Box ) \subset \bigwedge ^2 {\mathfrak {g}}_-^* \otimes {\mathfrak {g}}\), where \(\Box \) is the Kostant Laplacian, Kostant identified the following lowest \({\mathfrak {g}}_0\)-weight vector for \({\mathbb U}_{-w \cdot \lambda }\):

$$\begin{aligned} e_{\alpha _j} \wedge e_{\sigma _j(\alpha _k)} \otimes e_{w(-\lambda )}, \end{aligned}$$
(A.2)

where \(e_\gamma \) denotes a root vector corresponding to the root \(\gamma \in \Delta \). Since \(\lambda \in \Delta \), then \(w(-\lambda ) \in \Delta \).

Appendix B: Classification of Yamaguchi-nonrigid, torsion-free parabolic geometries

For any regular, normal parabolic geometry of type (GP), the harmonic curvature \(\kappa _H\) takes values in \(H^2_+({\mathfrak {g}}_-,{\mathfrak {g}})\). The geometry is Yamaguchi-rigid if \(H^2_+({\mathfrak {g}}_-,{\mathfrak {g}}) = 0\). All such geometries are automatically flat. In the non-rigid case, the geometry is torsion-free if the curvature \(\kappa \) takes values in the P-submodule \(\bigwedge ^2 {\mathfrak {p}}_+ \otimes {\mathfrak {p}}\subset \bigwedge ^2 {\mathfrak {p}}_+ \otimes {\mathfrak {g}}\). We will prove the following classification result.

Theorem B.1

Let G be a complex simple Lie group. All Yamaguchi-nonrigid, torsion-free (regular, normal) parabolic geometries of type (GP) are given by:

G

Range

P

\(w \in W^{\mathfrak {p}}\)

Description of some real forms

\(A_\ell \)

\(\ell \ge 2\)

\(P_1\)

(12)

Projective structures

 

\(\ell \ge 3\)

\(P_2\)

(21)

\((2,\ell -1)\)-Segré structures

 

\(\ell \ge 2\)

\(P_{1,2}\)

(21)

Torsion-free second order ODE system in \((\ell -1)\)-dep. vars

 

\(\ell \ge 3\)

\(P_{1,\ell }\)

\((1\ell )\)

Integrable Legendrian contact structures

\(B_\ell \)

\(\ell \ge 2\)

\(P_1\)

(12)

Odd-dimensional conformal structures

\(B_2\)

\(P_{1,2}\)

(12)

Scalar 3rd order ODE with vanishing Wünschman invariant

\(B_3\)

\(P_3\)

(32)

(3,6)-distributions

\(C_\ell \)

\(\ell \ge 2\)

\(P_1\)

(12)

Contact projective structure

 

\(\ell \ge 3\)

\(P_2\)

(21)

Split quaternionic contact structure

 

\(\ell \ge 3\)

\(P_{1,2}\)

(21)

Contact path geometry

\(D_\ell \)

\(\ell \ge 4\)

\(P_1\)

(12)

Even-dimensional conformal structures

\(G_2\)

\(P_1\)

(12)

(2,3,5)-distributions

Let \(\lambda \) be the highest weight of \({\mathfrak {g}}\). From (A.2), if the geometry is torsion-free, then \(\kappa _H\) takes values in the direct sum of those \({\mathbb U}_{-w\cdot \lambda } \subset H^2_+({\mathfrak {g}}_-,{\mathfrak {g}})\), where \(w \in W^{\mathfrak {p}}(2)\) and the grading element \(\mathsf {Z}\) satisfy

$$\begin{aligned} \mathsf {Z}(w(-\lambda )) \ge 0. \end{aligned}$$
(B.1)

The component of \(\kappa \) of lowest homogeneity is harmonic (see [8, Thm. 3.1.12] for a precise statement), so if (B.1) holds, then the geometry is torsion-free. The following appeared in [25, Lemma 4.3.2].

Lemma B.2

Let \({\mathfrak {g}}\) be complex simple. If \(w \in W^{\mathfrak {p}}(2)\), and \(w(-\lambda ) \in \Delta ^+\), then G / P is one of \(A_2 / P_1\), \(A_2 / P_{1,2}\), \(B_2 / P_1\), or \(B_2 / P_{1,2}\).

Thus, it suffices to study the case \(\mathsf {Z}(w(-\lambda )) = 0\). The rank 2 cases are easily settled by hand: \(B_2 / P_2 = C_2 / P_1\) and \(G_2 / P_1\) are the only cases with \(\mathsf {Z}(w(-\lambda )) = 0\). So let \(\ell := \mathrm {rank}(G) \ge 3\).

Since \(\lambda = \sum _a r_a \lambda _a\) is a dominant integral weight, then \(r_a := \langle \lambda , \alpha _a^\vee \rangle \in {\mathbb Z}_{\ge 0}\). (Indeed \(r_a \in \{ 0,1,2 \}\) always.) Hence, for \(w = (jk) \in W^{\mathfrak {p}}(2)\),

$$\begin{aligned} w(\lambda )&= \sigma _j\sigma _k(\lambda ) = \sigma _j(\lambda - r_k \alpha _k) = \lambda - r_j \alpha _j - r_k (\alpha _k - c_{kj} \alpha _j)\\&= \lambda - (r_j - r_k c_{kj}) \alpha _j - r_k \alpha _k. \end{aligned}$$

Since \(j \in I_{\mathfrak {p}}\), then \(\mathsf {Z}(\alpha _j) = 1\), so \(\mathsf {Z}(w(-\lambda )) = 0\) if and only if

$$\begin{aligned} \mathsf {Z}(\lambda ) = r_j - r_k c_{kj} + r_k \mathsf {Z}(\alpha _k). \end{aligned}$$
(B.2)

Since \(I_{\mathfrak {p}}\ne \emptyset \), then \(\mathsf {Z}(\lambda ) \ge |I_{\mathfrak {p}}| \ge 1\), so \(r_j = r_k = 0\) is impossible. Thus, \(r_j \ge 1\) or \(r_k \ge 1\). If \(r_a \ge 1\), node a will be referred to as a contact node. An important property is:

  1. (P.3)

    If \({\mathfrak {g}}\) is not of type A, there is a unique contact node a and we have \(\mathsf {Z}_a(\lambda ) = 2\). If \({\mathfrak {g}}\) is of type C, then \(r_a = 2\), and otherwise \(r_a = 1\).

Using (P.1), (P.2) (from Appendix A), and (P.3), we extract the implications of (B.2).

  1. (1)
    figure g

    . (j is a contact node)

    • Type A: Since \(\lambda = \lambda _1 + \lambda _\ell \), then we may assume \(j=1\), so \(r_j = 1\) and \(\mathsf {Z}_j(\lambda ) = 1\).

      • If \(r_k \ge 1\), then \(k = \ell \ge 3\), \(r_k = 1\), \(c_{kj} = 0\), and \(k \in I_{\mathfrak {p}}\). Thus, (B.2) implies \(\mathsf {Z}(\lambda ) = 2\). Since \(\lambda = \alpha _1 + \cdots + \alpha _\ell \) with \(j,k \in I_{\mathfrak {p}}\), then \(G/P = A_\ell / P_{1,\ell }\).

      • Otherwise, if \(r_k = 0\), then (B.2) implies \(\mathsf {Z}(\lambda ) = 1\), so \(G/P = A_\ell / P_1\).

    • Other types: We must have \(r_k = 0\), so (B.2) implies \(\mathsf {Z}(\lambda ) = r_j\).

      • Type C: Since \(\lambda = 2\lambda _1 = 2\alpha _1 + \cdots + 2\alpha _{\ell -1} + \alpha _\ell \), then \(j=1\), \(r_j = 2\), and \(\mathsf {Z}(\lambda ) = \mathsf {Z}_j(\lambda ) = 2\). Thus, \(G/P = C_\ell / P_1\).

      • Not type A,C: \(r_j = 1\), so \(\mathsf {Z}(\lambda ) = 1\) forces \(|I_{\mathfrak {p}}| = 1\). However, since j is the contact node, then \(\mathsf {Z}_j(\lambda ) = 2\), so \(\mathsf {Z}(\lambda ) = r_j\) is impossible.

  2. (2)
    figure h

    . (k is a contact node)

    • Type A: We may assume \(k=1\), so \(r_k = 1\) and (B.2) implies \(\mathsf {Z}(\lambda ) = \mathsf {Z}(\alpha _k) - c_{kj}\).

      • \(c_{kj} = 0\): Then \(I_{\mathfrak {p}}\supset \{ j, k \}\), so \(2 \le \mathsf {Z}(\lambda ) = \mathsf {Z}(\alpha _k) \le 1\), a contradiction.

      • \(c_{kj} \ne 0\): \(2 = j \in I_{\mathfrak {p}}\) and \(c_{kj} = -1\). If \(k \not \in I_{\mathfrak {p}}\), then \(\mathsf {Z}(\alpha _k) = 0\) and \(G/P = A_\ell / P_2\). If \(k \in I_{\mathfrak {p}}\), then \(\mathsf {Z}(\alpha _k) = 1\) and \(G/P = A_\ell / P_{1,2}\).

    • Type C: \(k=1\), \(r_k = 2\), and (B.2) implies \(\mathsf {Z}(\lambda ) = 2 \mathsf {Z}(\alpha _1) - 2 c_{1j}\).

      • \(j\ne 2\): \(c_{1j} = 0\), so \(1 \le \mathsf {Z}(\lambda ) = 2\mathsf {Z}(\alpha _1)\) implies \(1 \in I_{\mathfrak {p}}\) and \(\mathsf {Z}(\lambda ) = 2\). Since \(\lambda = 2(\sum _{i=1}^{\ell -1} \alpha _i) + \alpha _\ell \), then \(I_{\mathfrak {p}}= \{ 1 \}\). But \(j \in I_{\mathfrak {p}}\), so \(j=k=1\), a contradiction.

      • \(j=2\): \(c_{1j} = -1\), so \(\mathsf {Z}(\lambda ) = 2 + 2\mathsf {Z}(\alpha _1)\). If \(1 \not \in I_{\mathfrak {p}}\), then \(G/P = C_\ell / P_2\). If \(1 \in I_{\mathfrak {p}}\), then \(G/P = C_\ell / P_{1,2}\).

    • Not type A,C: By (P.3), \(r_k = 1\) and \(\mathsf {Z}_k(\lambda ) = 2\), so (B.2) implies \(\mathsf {Z}(\lambda ) = - c_{kj} + \mathsf {Z}(\alpha _k)\).

      • \(k \in I_{\mathfrak {p}}\): Since \(j \in I_{\mathfrak {p}}\), \(\mathsf {Z}_j(\lambda ) \ge 1\), so \( 3 \le \mathsf {Z}_j(\lambda ) + \mathsf {Z}_k(\lambda ) \le \mathsf {Z}(\lambda ) = 1-c_{kj}\). Since \(\ell = \mathrm {rank}(G) \ge 3\), then G must be doubly-laced with \(c_{kj} = -2\) and \(\mathsf {Z}(\lambda ) = 3\).

        • Type B: \(r_k=1\) and \(c_{kj} = -2\) implies \(k=2\) and \(j=\ell =3\). But then \(\mathsf {Z}(\lambda ) \ge \mathsf {Z}_2(\lambda ) + \mathsf {Z}_3(\lambda ) \ge 4\) (since \(\lambda = \alpha _1 + 2\alpha _2 + 2\alpha _3\)), a contradiction.

        • Type F: \(r_k=1\) so \(k=1\), but then \(c_{kj} \ne -2\), a contradiction.

      • \(k \not \in I_{\mathfrak {p}}\): Here, (B.2) implies \(\mathsf {Z}(\lambda ) = - c_{kj}\) with k the unique contact node.

        • Type B: \(k=2\) and \(\lambda = \alpha _1 + 2(\sum _{i=2}^\ell \alpha _i)\). If \(c_{kj} = -1\), then \(\mathsf {Z}(\lambda ) = \mathsf {Z}_j(\lambda ) = 1\), so G / P is \(B_\ell / P_1\). Otherwise, if \(c_{kj} = -2\), then \(G/P = B_3 / P_3\).

        • Type D: \(k=2\) and \(\lambda = \alpha _1 + 2(\sum _{i=2}^{\ell -2} \alpha _i) + \alpha _{\ell -1} + \alpha _\ell \), so \(G/P = D_\ell / P_1\).

        • Type E: \(\mathsf {Z}(\lambda ) = -c_{kj} = 1\), so \(G \ne E_8\). For \(E_6\), \(k=2\) and \(j=1\), but then \(c_{kj} = 0\). For \(E_7\), \(k=1\) and \(j=7\), but then \(c_{kj} = 0\). Both are contradictions.

        • Type F: \(k=1\), so \(j=2\), and \(3 = \mathsf {Z}_2(\lambda ) \le \mathsf {Z}(\lambda ) = 1\), a contradiction.

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Kruglikov, B., The, D. Jet-determination of symmetries of parabolic geometries. Math. Ann. 371, 1575–1613 (2018). https://doi.org/10.1007/s00208-017-1545-z

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