Abstract
For \(p\) odd, the Lie group \(G^\sharp =\mathrm{SO}_0(p+1,p+1)\) has a family of complementary series representations realized on the space of real \((p+1)\times (p+1)\) skew symmetric matrices, similar to the Stein’s complementary series for \(\mathrm{SL}(2n, {\mathbb C})\). We consider their restriction on the subgroup \(G_0=\mathrm{SO}_0(p+1,p)\) and prove that they are still irreducible and is equivalent to (a unitarization of) the principal series representation of \(G=\mathrm{SO}(p+1, p)\), and also irreducible under a maximal parabolic subgroup of \(G\).
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References
Baldoni Silva, M., Knapp, A.W.: Unitary representations induced from maximal parabolic subgroups. J. Funct. Anal. 69, 21–120 (1986)
Barchini,L., Sepanski, M., Zierau, R.: Positivity of zeta distributions and small unitary representations. In: The ubiquitous heat kernel, Contemporary Mathematics, vol. 398, pp. 1–46. American Mathematical Society (2006)
Baruch, E.M.: A proof of Kirillov’s conjecture. Ann. Math. 158, 207–252 (2003)
Benson, C., Jenkins, J., Ratcliff, G.: On Gelfand pairs associated with solvable Lie groups. Trans. Am. Math. Soc. 321(1), 85–116 (1990)
Bopp, N., Rubenthaler, H.: Local zeta functions attached to the minimal spherical series for a class of symmetric spaces, Mem. Amer. Math. Soc. vol. 174, No. 821. American Mathematical Society (2005)
Corwin, L., Greenleaf, F.P.: Representations of nilpotent Lie groups and their applications. Cambridge University Press, Cambridge (1990)
Dixmier, J.: Les \(C^\ast \)-algébres et leurs Représentations, 2nd edn. Gauthier-Villars, Paris (1964)
Fischer, V.: Etude de deux classes de groupe nilpotents de pas deux. PhD thesis Orsay Paris-Sud XI. http://arxiv.org/abs/0810.4173
Folland, G.B.: Harmonic analysis in phase space. In: Annals of Mathematics Studies, vol 122. Princeton University Press, Princeton, NJ (1989)
Gelbart, S.: A theory of Stiefel harmonics. Trans. Am. Math. Soc. 192, 29–50 (1974)
Helgason, S.: Groups and geometric analysis. Academic Press, New York (1984)
Howe, R., Lee, S.T.: Degenerate principal series representations of \(\text{ GL }_{n} ({\bf C})\) and \(\text{ GL }_{n} ({\bf R})\). J. Funct. Anal. 166(2), 244–309 (1999)
Kobayashi, T.: Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs. Representation Theory and Automorphic Forms, Progress in Mathematics, pp. 45–109. Birkhuser, Boston (2008)
Kobayashi, T.: Branching problems of Zuckerman derived functor modules. Representation theory and mathematical physics, Contemporary Mathematics, vol 557, American Mathematical Society, Providence, RI, pp 23–40 (2011)
Johnson, K.D.: Degenerate principal series and compact groups. Math. Ann. 287, 703–718 (1990)
Knapp, A.: Representation theory of semisimple groups. Princeton University Press, Princeton (1986)
Lee, S.T., Loke, H.Y.: Degenerate principal series representations of \(U(p, q)\) and \(Spin(p, q)\). Compositio Math. 132(3), 311–348 (2002)
Molčanov, V.F.: Restriction of a representation of the complementary series of the pseudo-orthogonal group to a pseudo-orthogonal group of lower dimension. Dokl. Akad. Nauk SSSR 237(4), 782–785 (1977). [English translation: Soviet Math. Dokl. 18(6), 1493–1497 (1977)]
Neretin, YuA, Olshanski, G.I.: Boundary values of holomorphic functions, singular unitary representations of the groups \(\text{ O }(p, q)\) and their limits as \(q\rightarrow \infty \). J. Math. Sci. (New York) 87(6), 3983–4035 (1997)
Ørsted, B., Zhang, G.: Generalized principal series representations and tube domains. Duke Math. J 78, 335–357 (1995)
Sahi, S.: Jordan algebras and degenerate principal series. J. Reine Angew. Math. 462, 1–18 (1995)
Sahi, S., Stein, E.M.: Analysis in matrix space and Speh’s representations. Invent. Math. 101, 379–393 (1990)
Speh, B., Venkataramana, T.: Discrete components of some complementary series representations Indian. J. Pure Appl. Math. 41(1), 145–151 (2010)
Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematics Series. Princeton University Press, Princeton (1993)
Strichartz, R.S.: \(L^p\) harmonic analysis and Radon transforms on the Heisenberg group. J. Funct. Anal 96, 350–406 (1991)
Thangavelu, S.: An introduction to the uncertainty principle: Hardy’s theorem on Lie groups. Progress in Mathematics. Birkhäuser, Boston (2004)
Vinberg, E., Kimelfeld, B.: Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups. Funct. Anal. Appl. 12, 168–174 (1978)
Wallach, N.: Real reductive groups II. Pure and Applied Mathematics. Academic Press, New York (1992)
Zhang, G.: Jordan algebras and generalized principal series representations. Math. Ann. 302, 773–786 (1995)
Zhang, G.: Degenerate principal series representations and their holomorphic extensions. Adv. Math. 223(5), 1495–1520 (2010)
Acknowledgments
We would like to thank Professors Pierre Cartier, Jacques Faraut and Jean Ludwig for some fruitful discussions. We are grateful to Professor Robert Stanton for drawing us attention to the work [1]. The authors would also like to thank the referee for the valuable comments which helped to improve the manuscript.
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Communicated by K. Gröchenig.
Genkai Zhang acknowledges the partially support by Swedish Research Council (VR). Veronique Fischer acknowledges the support of the London Mathematical Society (LMS).
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Fischer, V., Zhang, G. Degenerate principal series representations of \(\mathrm{SO}(p+1,p)\) . Monatsh Math 176, 87–105 (2015). https://doi.org/10.1007/s00605-014-0671-x
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DOI: https://doi.org/10.1007/s00605-014-0671-x