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
In [5], the isotropy is identified with an element of \(T_x^*M\), but we will not use this identification.
General 2-jet determinacy of symmetry algebras of parabolic geometries has been known earlier—see Remark 3.9.
While \({\mathfrak {f}}(u) \subset {\mathfrak {g}}\) is a linear subspace, it is in general not a Lie subalgebra.
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.
References
Beloshapka, V.K.: On the dimension of the group of automorphisms of an analytic hypersurface. Math. USSR Izvestiya 14, 223–245 (1980)
Bochner, S.: Compact groups of differentiable transformations. Ann. Math. 46(2), 372–381 (1945)
Branson, T., Čap, A., Eastwood, M.G., Gover, A.R.: Prolongations of geometric overdetermined systems of PDE. Int. J. Math. 17, 641–664 (2006)
Calderbank, D.M.J., Eastwood, M.G., Matveev, V.S., Neusser, K.: C-projective geometry. arXiv:1512.04516 (2015)
Čap, A., Melnick, K.: Essential Killing fields of parabolic geometries. Indiana Univ. Math. J. 62(6), 1917–1953 (2013)
Čap, A., Melnick, K.: Essential Killing fields of parabolic geometries: projective and conformal structures. Cent. Eur. J. Math. 11(12), 2053–2061 (2013)
Čap, A., Neusser, K.: On automorphism groups of some types of generic distributions. Differ. Geom. Appl. 27(6), 769–779 (2009)
Čap, A., Slovák, J.: Parabolic Geometries I: Background and General Theory, Mathematical Surveys and Monographs, vol. 154, American Mathematical Society (2009)
Cartan, É.: Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes. I. Ann. Mat. Pura Appl. 11, 17–90 (1932)
Cartan, H.: Sur les Groupes de Transformations Analytiques. Act. Sc. et Int, Hermann, Paris (1935)
Chern, S.S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974)
Ebenfelt, P., Lamel, B., Zaitsev, D.: Finite jet determination of local analytic CR automorphisms and their parametrization by 2-jets in the finite type case. Geom. Funct. Anal. 13, 546–573 (2003)
Ezhov, V.V.: On the linearization of automorphisms of a real analytic hypersurface. Math. USSR Izvestiya 27(1), 53–84 (1986)
Ezhov, V.V.: An example of a real-analytic hypersurface with a nonlinearizable stability group. Math. Notes 44, 824–828 (1988)
Frances, C.: Local dynamics of conformal vector fields. Geom. Dedic. 158, 35–59 (2012)
Frances, C., Melnick, K.: Normal forms for conformal vector fields. Bull. SMF 141(3), 377–421 (2013)
Isaev, A.V.: Proper actions of high-dimensional groups on complex manifolds. Bull. Math. Sci. (2015). doi:10.1007/s13373-015-0069-7
Knapp, A.W.: Lie Groups Beyond an Introduction. Progress in Mathematics, vol. 42, 2nd edn. Birkhäuser Boston Inc., Boston (2002)
Kobayashi, S.: Transformation Groups in Differential Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 70. Springer-Verlag, New York-Heidelberg (1972)
Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. Math. 74(2), 329–387 (1961)
Kostant, B.: The cascade of orthogonal roots and the coadjoint structure of the nilradical of a Borel subgroup of a semisimple Lie group. Mosc. Math. J. 12(3), 605–620, 669 (2012)
Kruglikov, B.: Point classification of second order ODEs: tresse classification revisited and beyond (with an appendix by Kruglikov and V.Lychagin). In: Abel Symp. 5, Differential Equations: Geometry, Symmetries and Integrability, pp. 199–221. Springer (2009)
Kruglikov, B.: Finite-dimensionality in Tanaka theory. Ann. Inst. Henri Poincar Anal. Non Linaire 28(1), 75–90 (2011)
Kruglikov, B., Matveev, V., The, D.: Submaximally symmetric \(c\)-projective structures. Int. J. Math. 27, 1650022 (2016). doi:10.1142/S0129167X16500221. 34 pages
Kruglikov, B., The, D.: The gap phenomenon in parabolic geometries. J. Reine Angew. Math. (Crelle’s Journal) 2017(723), 153–215 (2014). doi:10.1515/crelle-2014-0072
Kruzhilin, N.G., Loboda, A.V.: Linearization of local automorphisms of pseudoconvex surfaces. Dokl. Akad. Nauk SSSR 271, 280–282 (1983)
Loboda, A.V.: On local automorphisms of real-analytic hypersurfaces. Math. USSR-Izv. 18, 537–559 (1982)
Landsberg, J.M., Manivel, L.: On the projective geometry of rational homogeneous varieties. Comment. Math. Helv. 78(1), 65–100 (2003)
Melnick, K., Neusser, K.: Strongly essential flows on irreducible parabolic geometries. Trans. Am. Math. Soc. 368(11), 8079–8110 (2016)
Nagano, T., Ochiai, T.: On compact Riemannian manifolds admitting essential projective transformations. J. Fac. Sci. Univ. Tokyo Sect. IA Math 33(2), 233–246 (1986)
Neusser, K.: Prolongation on regular infinitesimal flag manifolds. Int. J. Math. 23(4), 1–41 (2012)
Schoen, R.: On the conformal and CR automorphism groups. Geom. Funct. Anal. 5(2), 464–481 (1995)
Tanaka, N.: On the pseudo-conformal geometry of hypersurfaces of the space of \(n\) complex variables. J. Math. Soc. Jpn. 14, 397–429 (1962)
Webster, S.: On the transformation group of a real hypersurface. Trans. Am. Math. Soc. 231(1), 179–190 (1977)
Yamaguchi, K.: Differential systems associated with simple graded Lie algebras, In: Progress in Differential Geometry, Adv. Stud. Pure Math. 22, pp. 413–494. Kinokuniya Company, Tokyo (1993)
Yamaguchi, K.: \(G_2\)-geometry of overdetermined systems of second order. In: Analysis and Geometry in Several Complex Variables (Katata, 1997), Trends Math., pp. 289–314. Birkhäuser Boston, Boston (1999)
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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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:
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\({\mathfrak {D}}({\mathfrak {g}})\) be the corresponding Dynkin diagram.
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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 \(I_{\mathfrak {p}}\).
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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.
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) :
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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) :
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\({\mathfrak {g}}_{-1}\) 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) :
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\({\mathfrak {g}}_1\) 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) :
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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) :
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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) :
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Simple roots orthogonal to \(\lambda \): All nodes in \({\mathfrak {D}}({\mathfrak {g}})\) except those marked with .
- (R.7) :
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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) :
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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):
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(R.9) Top-slot orthogonal cascade:
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(a)
Start with \({\mathfrak {D}}({\mathfrak {g}},{\mathfrak {p}},\lambda )\), where \({\mathfrak {g}}\) is simple, \({\mathfrak {p}}\subsetneq {\mathfrak {g}}\) is parabolic, and \(\lambda = \max \Delta ({\mathfrak {g}})\).
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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 \(\mu = \max \Delta ({\mathfrak {l}}_{\mathfrak {p}}(\lambda ))\).
-
-
(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.
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(a)
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
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(P.1)
\(j \in I_{\mathfrak {p}}\) (i.e. a crossed node), and
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(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
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 \({\mathbb U}_{-w \cdot \lambda }\):
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 (G, P), 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 (G, P) 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
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 \(w = (jk) \in W^{\mathfrak {p}}(2)\),
Since \(j \in I_{\mathfrak {p}}\), then \(\mathsf {Z}(\alpha _j) = 1\), so \(\mathsf {Z}(w(-\lambda )) = 0\) if and only if
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:
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(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).
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(1)
. (j is a contact node)
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Type A: Since \(\lambda = \lambda _1 + \lambda _\ell \), then we may assume \(j=1\), so \(r_j = 1\) and \(\mathsf {Z}_j(\lambda ) = 1\).
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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 \(G/P = A_\ell / P_{1,\ell }\).
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Otherwise, if \(r_k = 0\), then (B.2) implies \(\mathsf {Z}(\lambda ) = 1\), so \(G/P = A_\ell / P_1\).
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Other types: We must have \(r_k = 0\), so (B.2) implies \(\mathsf {Z}(\lambda ) = r_j\).
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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, \(G/P = C_\ell / P_1\).
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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)
. (k is a contact node)
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Type A: We may assume \(k=1\), so \(r_k = 1\) and (B.2) implies \(\mathsf {Z}(\lambda ) = \mathsf {Z}(\alpha _k) - c_{kj}\).
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\(c_{kj} = 0\): Then \(I_{\mathfrak {p}}\supset \{ j, k \}\), so \(2 \le \mathsf {Z}(\lambda ) = \mathsf {Z}(\alpha _k) \le 1\), a contradiction.
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\(c_{kj} \ne 0\): \(2 = j \in I_{\mathfrak {p}}\) 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 \(\mathsf {Z}(\lambda ) = 2 \mathsf {Z}(\alpha _1) - 2 c_{1j}\).
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\(j\ne 2\): \(c_{1j} = 0\), 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.
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\(j=2\): \(c_{1j} = -1\), 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 \(\mathsf {Z}(\lambda ) = - c_{kj} + \mathsf {Z}(\alpha _k)\).
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\(k \in I_{\mathfrak {p}}\): 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 \(\mathsf {Z}(\lambda ) = 3\).
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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.
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Type F: \(r_k=1\) so \(k=1\), but then \(c_{kj} \ne -2\), a contradiction.
-
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\(k \not \in I_{\mathfrak {p}}\): Here, (B.2) implies \(\mathsf {Z}(\lambda ) = - c_{kj}\) with k the unique contact node.
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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 \(G/P = B_3 / P_3\).
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Type D: \(k=2\) and \(\lambda = \alpha _1 + 2(\sum _{i=2}^{\ell -2} \alpha _i) + \alpha _{\ell -1} + \alpha _\ell \), so \(G/P = D_\ell / P_1\).
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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.
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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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DOI: https://doi.org/10.1007/s00208-017-1545-z