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We study actions of Lie supergroups, in particular, the hitherto elusive notion of orbits through odd (or more general) points. Following categorical principles, we derive a conceptual framework for their treatment and therein prove general existence theorems for the isotropy (or stabiliser) supergroups and orbits through general points. In this setting, we show that the coadjoint orbits always admit a (relative) supersymplectic structure of Kirillov–Kostant–Souriau type. Applying a family version of Kirillov’s orbit method, we decompose the regular representation of an odd Abelian supergroup into an odd direct integral of characters and construct universal families of representations, parametrised by a supermanifold, for two different super variants of the Heisenberg group.



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Research supported by Deutsche Forschungsgemeinschaft (DFG), grant nos. SFB/TR 12 (all authors), the Heisenberg grant AL 698/3-1 (A.A.), the Leibniz prize to M. Zirnbauer ZI 513/2-1 (A.A.), SFB TRR 183 (A.A.), and the Institutional Strategy of the University of Cologne within the German Excellence Initiative (all authors).



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1. Aguilar, M., Gitler, S. and Prieto, C., Algebraic Topology from a Homotopical Viewpoint, Universitext (Springer, New York, 2002).
2. Alldridge, A., Fréchet globalisations of Harish–Chandra modules, Int. Math. Res. Not. IMRN (2016), arXiv:1403.4055.
3. Alldridge, A. and Hilgert, J., Invariant Berezin integration on homogeneous supermanifolds, J. Lie Theory 20(1) (2010), 6591.
4. Alldridge, A., Hilgert, J. and Laubinger, M., Harmonic analysis on Heisenberg–Clifford Lie supergroups, J. Lond. Math. Soc. (2) 87(2) (2013), 561585, doi:10.1112/jlms/jds058.
5. Alldridge, A., Hilgert, J. and Wurzbacher, T., Singular superspaces, Math. Z. 278 (2014), 441492, doi:10.1007/s00209-014-1323-5.
6. Alldridge, A. and Palzer, W., Asymptotics of spherical superfunctions on rank one Riemannian symmetric superspaces, Doc. Math. 19 (2014), 13171366.
7. Alldridge, A. and Shaikh, Z., Superbosonization via Riesz superdistributions, Forum Math. Sigma 2 (2014), e9, 64, doi:10.1017/fms.2014.5.
8. Almorox, A. L., Supergauge theories in graded manifolds, in Differential Geometric Methods in Mathematical Physics (Salamanca, 1985), Lecture Notes in Mathematics, Volume 1251, pp. 114136 (Springer, Berlin, 1987), doi:10.1007/BFb0077318.
9. Andler, M. and Sahi, S., Equivariant cohomology and tensor categories, Preprint, 2008, arXiv:0802.1038.
10. Balduzzi, L., Carmeli, C. and Cassinelli, G., Super G-spaces, in Symmetry in Mathematics and Physics, Contemporary Mathematics, Volume 490, pp. 159176 (American Mathematical Society, Providence, RI, 2009), doi:10.1090/conm/490/09594.
11. Bourbaki, N., Elements of Mathematics. General Topology. Part 1 (Hermann, Paris; Addison-Wesley Publishing Co., Reading, MA–London–Don Mills, Ont, 1966).
12. Boyer, C. P. and Sánchez-Valenzuela, O. A., Lie supergroup actions on supermanifolds, Trans. Amer. Math. Soc. 323(1) (1991), 151175, doi:10.2307/2001621.
13. Carmeli, C., Cassinelli, G., Toigo, A. and Varadarajan, V. S., Unitary representations of super Lie groups and applications to the classification and multiplet structure of super particles, Comm. Math. Phys. 263(1) (2006), 217258, doi:10.1007/s00220-005-1452-0.
14. Carmeli, C., Cassinelli, G., Toigo, A. and Varadarajan, V. S., Erratum to: Unitary representations of super Lie groups and applications to the classification and multiplet structure of super particles, Comm. Math. Phys. 307(2) (2011), 565566, doi:10.1007/s00220-011-1332-8.
15. Carmeli, C., Caston, L. and Fioresi, R., Mathematical Foundations of Supersymmetry, EMS Series of Lectures in Mathematics (European Mathematical Society (EMS), Zürich, 2011).
16. Chevalley, C. and Eilenberg, S., Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85124.
17. Clerc, J.-L., Eymard, P., Faraut, J., Raïs, M. and Takahashi, R., Analyse harmonique, in Les cours du C.I.M.P.A. (CIMPA/ICPAM, Nice, 1982).
18. Deligne, P. and Morgan, J. W., Notes on Supersymmetry, Quantum Fields and Strings: A Course for Mathematiciansm, Volume 1, pp. 4198 (American Mathematical Society, Providence, RI, 1999).
19. Demazure, M. and Gabriel, P., Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs (Masson & Cie, Éditeur, Paris, 1970).
20. Duistermaat, J. J., On global action-angle coordinates, Comm. Pure Appl. Math. 33(6) (1980), 687706, doi:10.1002/cpa.3160330602.
21. Fioresi, R. and Lledó, M. A., On algebraic supergroups, coadjoint orbits, and their deformations, Comm. Math. Phys. 245(1) (2004), 177200.
22. Frydryszak, A. M., Q-representations and unitary representations of the super-Heisenberg group and harmonic superanalysis, J. Phys. Conf. Ser. 563(1) (2014), 012010.
23. Gabriel, P., Construction de préschémas quotient, Schémas en Groupes (Sém. Géométrie Algébrique, Inst. Hautes Études Sci., 1963/64) pp. 251–286 (Inst. Hautes Études Sci., Paris, 1963) (French).
24. Grothendieck, A., Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2) 9 (1957), 119221.
25. Grothendieck, A. and Dieudonné, J. A., Éléments de géométrie algébrique. I, Grundlehren der Mathematischen Wissenschaften, Volume 166 (Springer, Berlin, 1971).
26. Guillemin, V. and Sternberg, S., Symplectic Techniques in Physics (Cambridge University Press, Cambridge, 1984).
27. Hilgert, J. and Neeb, K.-H., Structure and Geometry of Lie Groups, Springer Monographs in Mathematics (Springer, New York, 2012).
28. Hohnhold, H., Kreck, M., Stolz, S. and Teichner, P., Differential forms and 0-dimensional supersymmetric field theories, Quantum Topol. 2(1) (2011), 141, doi:10.4171/QT/12.
29. Kac, V. G., Lie superalgebras, Adv. Math. 26(1) (1977), 896.
30. Kirillov, A. A., Unitary representations of nilpotent Lie groups, Uspekhi Mat. Nauk 17(4(106)) (1962), 57110 (Russian); English transl. Russian Math. Surv., 17, 53–104 doi:10.1070/RM1962v017n04ABEH004118.
31. Kostant, B., Graded manifolds, graded Lie theory, and prequantization, in Differential Geometrical Methods in Mathematical Physics (Proc. Sympos., Univ. Bonn, Bonn), Lecture Notes in Mathematics, Volume 570, pp. 177306 (Springer, Berlin, 1977).
32. Kostant, B., Harmonic analysis on graded (or super) Lie groups, in Group Theoretical Methods in Physics (Sixth Internat. Colloq., Tübingen, 1977), Lecture Notes in Physics, Volume 79, pp. 4750 (Springer, Berlin–New York, 1978).
33. Leites, D. A., Introduction to the theory of supermanifolds, Uspekhi Mat. Nauk 1 (1980), 357 (Russian); English Transl., Russian Math. Surveys, 35(1) (1980), 1–64.
34. Mac Lane, S., Categories for the Working Mathematician, 2nd edn, Graduate Texts in Mathematics, Volume 5 (Springer, New York, 1998).
35. Manin, Y. I., Gauge Field Theory and Complex Geometry, 2nd edn, Grundlehren der Mathematischen Wissenschaften, Volume 289 (Springer, Berlin, 1997).
36. Mumford, D., Fogarty, J. and Kirwan, F., Geometric Invariant Theory, 3rd edn, Ergebnisse der Mathematik, Volume 34 (Springer, Berlin, 1994), doi:10.1007/978-3-642-57916-5.
37. Neeb, K.-H. and Salmasian, H., Lie supergroups unitary representations, and invariant cones, in Supersymmetry in Mathematics and Physics, Lecture Notes in Mathematics, Volume 2027, pp. 195239 (Springer, Heidelberg, 2011), doi:10.1007/978-3-642-21744-9_10.
38. Salmasian, H., Unitary representations of nilpotent super Lie groups, Comm. Math. Phys. 297(1) (2010), 189227, doi:10.1007/s00220-010-1035-6.
39. Tuynman, G. M., Super symplectic geometry and prequantization, J. Geom. Phys. 60(12) (2010), 19191939, doi:10.1016/j.geomphys.2010.06.009.
40. Tuynman, G. M., Super Heisenberg orbits: a case study, in XXIX Workshop on Geometric Methods in Physics, Volume 1307, pp. 181184 (American Institute of Physics, Melville, NY, 2010).
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