Abstract
In this paper we study the \(\mathfrak {sp}(2m)\)-invariant Dirac operator \(D_s\) which acts on symplectic spinors, from an orthogonal point of view. By this we mean that we will focus on the subalgebra \(\mathfrak {so}(m) \subset \mathfrak {sp}(2m)\), as this will allow us to derive branching rules for the space of 1-homogeneous polynomial solutions for the operator \(D_s\) (hence generalising the classical Fischer decomposition in harmonic analysis for a vector variable in \({\mathbb {R}}^m\)). To arrive at this result we use techniques from representation theory, including the transvector algebra \({\mathcal {Z}}(\mathfrak {sp}(4),\mathfrak {so}(4))\) and tensor products of Verma modules.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bergeron, N., Millson, J.J., Ralston, J.: The relative Lie algebra cohomology of the Weil representation of \(\text{SO}(n, 1)\). arXiv preprint (2014)
Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Research Notes in Mathematics, vol. 76. Pitman, London (1982)
Brackx, F., Schepper, H.D., Eelbode, D., Lávička, R., Souček, V.: Fischer decomposition for \(\mathfrak{osp}(4|2)\)-monogenics in quaternionic Clifford analysis. Math. Methods Appl. Sci. 39(16), 4874–4891 (2016)
Crumeyrolle, A.: Symplectic Clifford Algebras. Clifford Algebras and Their Applications in Mathematical Physics, Springer, Dordrecht (1986)
De Bie, H., Somberg, P., Souček, V.: The metaplectic Howe duality and polynomial solutions for the symplectic Dirac operator. J. Geom. Phys. 75, 120–128 (2014)
De Bie, H., Eelbode, D., Roels, M.: The harmonic transvector algebra in two vector variables. J. Algebra 473, 247–282 (2017)
De Bie, H., Holíková, M., Somberg, P.: Basic aspects of symplectic Clifford analysis for the symplectic Dirac operator. Adv. Appl. Clifford Algebras 27(2), 1103–1132 (2017)
Delanghe, R., Sommen, F., Souček, S.: Clifford Algebra and Spinor-Valued Functions. Springer, Berlin (1992)
Frahm, J., Weiske, C.: Symmetry breaking operators for real reductive groups of rank one. J. Funct. Anal. 279(5) (2020)
Gilbert, J., Murray, M.: Clifford algebras and Dirac operators in harmonic analysis (No. 26). Cambridge University Press, Cambridge (1991)
Goodman, R.: Multiplicity-free spaces and Schur–Weyl–Howe duality. Representations of Real and p-adic Groups 2, 305–415 (2004)
Goodman, R., Wallach, N.R.: Symmetry, Representations, and Invariants. Springer, Dordrecht (2009)
Habermann, K., Habermann, L.: Introduction to symplectic Dirac operators. Springer, Heidelberg (2006)
Holíková, M: Symplectic spin geometry. Doctoral dissertation, Charles University, Prague, Czech Republic (2016)
Howe, R., Tan, E.C., Willenbring, J.: Stable branching rules for classical symmetric pairs. Trans. Am. Math. Soc. 357(4), 1601–1626 (2005)
Klimyk, A.U.: Infinitesimal operators for representations of complex Lie groups and Clebsch–Gordan coefficients for compact groups. J. Phys. A Math. Gen. 15(10), 3009 (1982)
Kostant, B.: Symplectic spinors. Rome Symp. XIV, 139–152 (1974)
Krýsl, S.: Howe duality for the metaplectic group acting on symplectic spinor valued forms. J. Lie theory 22(4), 1049–1063 (2012)
Lávička, R.: Separation of Variables in the Semistable Range. Topics in Clifford Analysis Topics in Clifford Analysis Topics in Clifford Analysis, Birkhäuser, Cham (2019)
Reuter, M.: Symplectic Dirac–Kähler fields. J. Math. Phys. 40(11), 5593–5640 (1999)
Van Lancker, P., Sommen, F., Constales, D.: Models for irreducible representations of \({\sf Spin} (m)\). Adv. Appl. Clifford Algebras 11(1), 271–289 (2001)
Zhelobenko, D.P.: Extremal projectors and generalized Mickelsson algebras over reductive Lie algebras. Math. USSR-Izvestiya 33(1), 85–100 (1989)
Acknowledgements
The author G. Muarem was supported by the FWO-EoS project G0H4518N.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Uwe Kaehler.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Eelbode, D., Muarem, G. The Orthogonal Branching Problem for Symplectic Monogenics. Adv. Appl. Clifford Algebras 32, 32 (2022). https://doi.org/10.1007/s00006-022-01215-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00006-022-01215-1