Representations of low copolarity

Gomes, André; Gorodski, Claudio

doi:10.1090/proc/16114

Representations of low copolarity
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by André Magalhães de Sá Gomes and Claudio Gorodski PDF

Proc. Amer. Math. Soc. 150 (2022), 5439-5447 Request permission

Abstract:

We classify irreducible representations of compact connected Lie groups whose orbit space is isometric to the orbit space of a representation of a compact Lie group of dimension $7$, $8$ or $9$. They turn out to be closely related to symmetric spaces, with one exception only.

References

E. M. Andreev, È. B. Vinberg, and A. G. Èlašvili, Orbits of highest dimension of semisimple linear Lie groups, Funkcional. Anal. i Priložen. 1 (1967), no. 4, 3–7 (Russian). MR 0267040
Jürgen Berndt, Sergio Console, and Carlos Olmos, Submanifolds and holonomy, Chapman & Hall/CRC Research Notes in Mathematics, vol. 434, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1990032, DOI 10.1201/9780203499153
Jiri Dadok, Polar coordinates induced by actions of compact Lie groups, Trans. Amer. Math. Soc. 288 (1985), no. 1, 125–137. MR 773051, DOI 10.1090/S0002-9947-1985-0773051-1
Jiri Dadok and Victor Kac, Polar representations, J. Algebra 92 (1985), no. 2, 504–524. MR 778464, DOI 10.1016/0021-8693(85)90136-X
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
A. M. de Sá Gomes, Representations of low copolarity and biquotients, Ph.D. thesis, University of São Paulo, 2021.
Claudio Gorodski and Francisco J. Gozzi, Representations with $\mathsf {Sp}(1)^k$-reductions and quaternion-Kähler symmetric spaces, Math. Z. 290 (2018), no. 1-2, 561–575. MR 3848446, DOI 10.1007/s00209-017-2031-8
C. Gorodski, A. Kollross, and B. Wilking, Action on positively curved manifolds and boundary in the orbit space, Preprint, arXiv:2112.00513v2, 2021.
Claudio Gorodski and Alexander Lytchak, On orbit spaces of representations of compact Lie groups, J. Reine Angew. Math. 691 (2014), 61–100. MR 3213548, DOI 10.1515/crelle-2012-0084
Claudio Gorodski and Alexander Lytchak, Representations whose minimal reduction has a toric identity component, Proc. Amer. Math. Soc. 143 (2015), no. 1, 379–386. MR 3272762, DOI 10.1090/S0002-9939-2014-12259-3
Claudio Gorodski, Carlos Olmos, and Ruy Tojeiro, Copolarity of isometric actions, Trans. Amer. Math. Soc. 356 (2004), no. 4, 1585–1608. MR 2034320, DOI 10.1090/S0002-9947-03-03427-5
F. Panelli, Representations admitting a toric reduction, Ph.D. thesis, University of Florence, 2017.
Gerald W. Schwarz, Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 37–135. MR 573821, DOI 10.1007/BF02684776
Eldar Straume, On the invariant theory and geometry of compact linear groups of cohomogeneity $\leq 3$, Differential Geom. Appl. 4 (1994), no. 1, 1–23. MR 1264906, DOI 10.1016/0926-2245(94)00007-7
Joseph A. Wolf, Spaces of constant curvature, 6th ed., AMS Chelsea Publishing, Providence, RI, 2011. MR 2742530, DOI 10.1090/chel/372