##
**The \((\mathfrak g,K)\)-module structures of principal series of \(\text{SU}(2,2)\).**
*(English)*
Zbl 1231.22014

Principal series representations are building blocks in representation theory of real reductive groups. Moreover, Harish-Chandra proved that the study of irreducible unitary representations of such groups can be deduced from the study of irreducible \((g,K)\)-modules. In the paper under review, the author describes explicitly the \((g,K)\)-modules of principal series representations, induced from maximal parabolic subgroups, of the group \(G=SU(2,2)\). Here \(g\) is the complexification of the Lie algebra of \(SU(2,2)\) and \(K=S(U(2)\times U(2))\) is a maximal compact subgroup of \(G\). The strategy is to write in terms of elementary functions the marked basis for each \(K\)-isotopic component of the principal series representations, compute the Clebsch-Gordan coefficients of finite dimensional representations of \(K=S(U(2)\times U(2))\) and deduce the explicit form of some matrix of intertwining constants. The main reason for the author to have such explicit formulas is motivated by the arithmetic of automorphic forms, e.g. the Whittaker models that he had studied before. It should be noted that the Hermitian group \(SU(2,2)\) plays an important role in physics as it is isomorphic to the conformal group \(Spin(4,2)\) via twistors. Therefore the formulas computed by the author will certainly be of some interest to physicists.

Reviewer: Salah Mehdi (Metz)

### MSC:

22E46 | Semisimple Lie groups and their representations |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

PDF
BibTeX
XML
Cite

\textit{G. Bayarmagnai}, J. Math. Soc. Japan 61, No. 3, 661--686 (2009; Zbl 1231.22014)

### References:

[1] | G. Bayarmagnai, Explicit evaluation of certain Jacquet integrals on \(SU(2,2)\), preprint, 2008. · Zbl 1184.22004 |

[2] | D. A. Vogan, Representations of real reductive Lie groups, Progr. Math., 15 , Birkhäuser, Boston, Basel, Stuttgart, 1981. · Zbl 0469.22012 |

[3] | T. Hayata, Differential equations for principal series Whittaker functions on \(SU(2,2)\), Indag. Math. (N.S.), 8 (1997), 493-528. · Zbl 0895.22007 |

[4] | R. Howe, \(K\)-type structure in the principal series of \(GL(3)\). I, : Analysis on homogeneous spaces and representation theory of Lie groups, Okayama-Kyoto (1997), Adv. Stud. Pure Math., 26 , Math. Soc. Japan, Tokyo, 2000, pp. 77-98. · Zbl 0967.22008 |

[5] | T. Ishii, On principal series Whittaker functions on \(Sp(2,\mbi{R})\), J. Func. Anal., 225 (2005), 1-32. · Zbl 1078.11031 |

[6] | A. U. Klimyk and B. Gruber, Structure and matrix elements of the degenerate series representations of \(U(p+q)\) and \(U(p,\,q)\) in a \(U(p)\times U(q)\) basis, J. Math. Phys., 23 (1982), 1399-1408. · Zbl 0548.22011 |

[7] | A. U. Klimyk and B. Gruber, Infinitesimal operators and structure of the most degenerate representationas of the groups \(Sp(p+q)\) and \(Sp(p,\,q)\) in a \(U(p)\times U(q)\) basis, J. Math. Phys., 25 (1984), 743-750. · Zbl 0548.22009 |

[8] | S.-T. Lee and H. Y. Loke, Degenerate principal series representations of \(U(p,q)\) and \(\textit{0}(p,q)\), Compositio Math., 132 (2002), 311-348. · Zbl 0997.22011 |

[9] | T. Miyazaki, The \((\mathfrak{g},K)\)-module structures of principal series representations \(Sp(3,\mbi{R})\), 2007, |

[10] | T. Miyazaki and T. Oda, Principal series Whittaker functions on \(Sp(2,\mbi{R})\), Explicit formulae of differential equations, Proceeding of the 1993 Workshop, Automorphic forms and related topics, Pyungsan Inst. Math. Sci., Seoul, pp. 59-92. · Zbl 0958.11037 |

[11] | V. F. Molchanov, Representations of pseudo-orthogonal group associated with a cone, Math. USSR Sbornik, 10 (1970), 333-347. |

[12] | T. Oda, Standard \((\mathfrak{g},K)\)-modules of \(Sp(2,\mbi{R})\), Preprint Series, The University of Tokyo, UTMS 2007-3. |

[13] | E. Thieleker, On the quasi-simple irreducible representations of the Lorentz groups, Trans. Amer, Math. Soc., 179 (1973), 465-505. · Zbl 0258.22016 |

[14] | N. R. Wallach, Real reductive groups, I, Pure and Applied Mathematics, 132 , Academic Press Inc., Boston, MA, 1988. · Zbl 0666.22002 |

[15] | H. Yamashita, Embedding of discrete series into induced representations of semisimple Lie groups, I, General theory and the case of \(SU(2,2)\), Japan J. Math., 16 (1990), 31-95. · Zbl 0759.22019 |

[16] | H. Yamashita, Embedding of discrete series into induced representations of semi-simple Lie groups, II, Generalized Whittaker models for \(SU(2,2)\), J. Math. Kyoto Univ., 31 (1991), 543-571. · Zbl 0745.22012 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.