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generalized Verma modules; conformal geometry in dimension $5$; exceptional Lie algebra ${\mathrm{Lie~}G_2}$; F-method; branching problem
The branching problem for a couple of non-compatible Lie algebras and their parabolic subalgebras applied to generalized Verma modules was recently discussed in [15]. In the present article, we employ the recently developed F-method, [10], [11] to the couple of non-compatible Lie algebras $\mathrm{Lie~}G_2\stackrel{i}{\hookrightarrow }{\operatorname{so}(7)}$, and generalized conformal ${\operatorname{so}(7)}$-Verma modules of scalar type. As a result, we classify the $i({\mathrm{Lie~}G_2}) \cap {\mathfrak{p}}$-singular vectors for this class of $\operatorname{so}(7)$-modules.
[1] Čap, A., Slovák, J.: Parabolic geometries, I: Background and General Theory. Mathematical Surveys and Monographs, American Mathematical Society, 2009. MR 2532439 | Zbl 1183.53002
[2] Dixmier, J.: Algebres Enveloppantes. Gauthier-Villars Editeur, Paris–Bruxelles–Montreal, 1974. Zbl 0308.17007
[3] Eastwood, M. G., Graham, C. R.: Invariants of conformal densities. Duke Math. J. 63 (1991), 633–671. DOI 10.1215/S0012-7094-91-06327-1 | Zbl 0745.53007
[4] Graham, R. C., Willse, T.: Parallel tractor extension and ambient metrics of holonomy split $G_2$. Zbl 1268.53075
[5] Humphreys, J. E., Jr., : Representations of Semisimple Lie Algebras in the BGG Category $ {\mathcal{O}}$. Graduate Studies in Mathematics, vol. 94, American Mathematical Society, 2008. MR 2428237
[6] Juhl, A.: Families of conformally covariant differential operators, Q–curvature and holography. Progress in Math., Birkhäuser, 2009. MR 2521913 | Zbl 1177.53001
[7] Kobayashi, T.: Discrete decomposability of the restriction of $A_{\mathfrak{q}}( \lambda )$ with respect to reductive subgroups and its applications. Invent. Math. 117 (1994), 181–205, Part II, Ann. of Math. (2) 147 (1998), 709–729; Part III, Invent. Math. 131 (1998), 229–256. DOI 10.1007/BF01232239
[8] Kobayashi, T.: Multiplicity–free theorems of the restriction of unitary highest weight modules with respect to reductive symmetric pairs. Progress in Math, vol. 280, Birkhäuser, 2007, pp. 45–109. MR 2369496
[9] Kobayashi, T.: Restrictions of generalized Verma modules to symmetric pairs. Transform. Groups 17 (2012), 523–546. DOI 10.1007/s00031-012-9180-y | MR 2921076 | Zbl 1257.22014
[10] Kobayashi, T., Ørsted, B., Somberg, P., Souček, V.: Branching laws for Verma modules and applications in parabolic geometry, I. preprint.
[11] Kobayashi, T., Ørsted, B., Somberg, P., Souček, V.: Branching laws for Verma modules and applications in parabolic geometry, II. preprint.
[12] Kostant, B.: Verma modules and the existence of quasi–invariant differential operators. Lecture Notes in Math., Springer Verlag, 1974, pp. 101–129.
[13] Lepowsky, J.: A generalization of the Bernstein–Gelfand–Gelfand resolution. J. Algebra 49 (1977), 496–511. DOI 10.1016/0021-8693(77)90254-X | Zbl 0381.17006
[14] Matumoto, H.: The homomorphisms between scalar generalized Verma modules associated to maximal parabolic subalgebras. Duke Math. J. 131 (2006), 75–118. DOI 10.1215/S0012-7094-05-13113-1 | MR 2219237 | Zbl 1129.17008
[15] Milev, T., Somberg, P.: The branching problem for generalized Verma modules, with application to the pair $(\operatorname{so}(7), \operatorname{Lie}\, G_2)$.
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