Abstract
The question of unitarity of representations in the analytic continuation of discrete series from a Borel-de Siebenthal chamber is considered for those linear equal-rank classical simple Lie groups G that have not been treated fully before. Groups treated earlier by other authors include those for which G has real rank one or has a symmetric space with an invariant complex structure . Thus the groups in question are locally isomorphic to SO(2m, n)0 with m ≥ 2 and n ≥ 3, or to Sp(m, n) with m ≥ 2 and n ≥ 2.
The representations under study are obtained from cohomological induction. One starts from a finite-dimensional irreducible representation of a compact subgroup L of G associated to a Borel-de Siebenthai chamber, forms an upside-down generalized Verma module, applies a derived Bernstein functor, and passes to a specific irreducible quotient. Enright, Parthasarthy, Wallach, and Wolf had previously identified all cases where the representation of L is I-dimensional and the generalized Verma-like module is irreducible; for these cases they proved that unitarity is automatic. B. Gross and Wallach had proved unitarity for additional cases for a restricted class of groups when the representation of L is I-dimensional.
The present work gives results for all groups and allows higher-dimensional representations of L. In the case of I-dimensional representations of L, the results address unitarily and nonunitarity and are conveniently summarized in a table that indicates how close the results are to being the best possible. In the case of higher-dimensional representations of L, the method addresses only unitarity and in effect proceeds by reducing matters to what happens for a 1-dimensional representation of L and a lower-dimensional group G.
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References
Baldoni Silva, M.W., The unitary dual of Sp(n, 1), n ≥ 2, Duke Math. J. 48 (1981), 549–583.
Baldoni Silva, M.W., and D. Barbasch, The unitary spectrum for real rank one groups, Invent. Math. 72 (1983), 27–55.
Baldoni-Silva, M.W, and A.W. Knapp, Unitary representations induced from maximal parabolic subgroups, J. Funct. Anal. 69 (1986), 21–120.
Binegar, B., and R. Zierau, Unitarization of a singular representation of SO(p, q), Commun. Math. Phys. 138 (1991), 245–258.
Borel, A., and J. de Siebenthal, Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helvetici 23 (1949), 200–221
Brylinski, R., and B. Kostant, Minimal representat ions, geometric quantization, and unitarity, Proc. Nat. Acad. Sci. USA 91 (1994), 6026–6029.
Enright, T., R. Howe, and N. Wallach, A classification of unitary highest weight modules, Representation Theory of Reductive Groups, (P.C. Trombi, ed.), Birkhäuser, Boston, 1983, pp. 97–143.
Enright, T. J., R. Parthasarathy, N.R. Wallach, J.A. Wolf, Unitary derived functor modules with small spectrum, Acta Math. 154 (1985), 105–136.
Friedman, P.D., The Langlands parameters of subquotients of certain derived functor modules, J. Funct. Anal. 157 (1998), 210–241.
Friedman, P.D., Langlands parameters of derived functor modules and Vogan diagrams, Math. Scand. 92 (2003), 31–67.
Goodman, R., and N.R. Wallach, Representations and Invariants of the Classical Groups, Encyclopedia of Mathematics and Its Applications, Vol. 68, Cambridge University Press, Cambridge, 1998.
Gross, B.H., and N.R. Wallach, A distinguished family of unitary representations for the exceptional groups of real rank = 4, Lie Theory and Geometry: in Honor of Bertram Kostant, (J.-L. Brylinski, R. Brylinski, V. Guillemin, and V. Kac, eds.), Birkhäuser, Boston, 1994, pp. 289–304.
Gross, B.H., and N.R. Wallach, On quaternionic discrete series representations, and their continuations, J. Reine Angew. Math 481 (1996), 73–123.
Harish-Chandra, Representations of semisimple Lie groups IV, Amer. J. Math. 77 (1955), 743–777; V, 78 (1956), 1–41; VI, 78 (1956), 564–628.
Harish-Chandra, Discrete series for semisimple Lie groups I, Acta Math. 113 (1965), 241–318; Two theorems on semi-simple Lie groups, Ann. of Math. 83 (1965), 74–128; Discrete series for semisimple Lie groups II, Acta Math. 116 (1966), 1–111.
Hirai, T., On irreducible representations of the Lorentz group of n-th order, Proc. Japan. Aacd. 38 (1962), 258–262.
Jakobsen, H.P., Hermitian symmetric spaces and their unitary highest weight modules, J. Funct. Anal. 52 (1983), 385–412.
Kazhdan, D., and G. Savin, The smallest representations of simply laced groups, Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of His Sixtieth Birthday, Part I, (S. Gelbart, R. Howe, and P. Sarnak, eds.), Israel Mathematical Conference Proceedings, Vol. 2, Weizmann Science Press of Israel, Jerusalem, 1990, pp. 209–223.
Knapp, A.W., Lie Groups Beyond an Introduction, Birkhäuser, Boston, 1996; second edition, 2002.
Knapp, A.W., Exceptional unitary representations of semisimple Lie groups, Representation Theory 1 (1997), 1–24.
Knapp, A.W., Intertwining operators and small unitary representations, The Mathematical Legacy of Harish-Chandra, (R.S. Doran and V.S. Varadarajan, eds.), Proceedings of Symposia in Pure Mathematics, Vol. 68, American Mathematical Society, 2000, pp. 403–431.
Knapp, A.W, and D.A. Vogan, Cohomological Induction and Unitary Representations, Princeton University Press, Princeton, NJ, 1995.
Kobayashi, T., Singular unitary representations and discrete series for indefinite Stiefel manifolds U(p, q; F)/U(p - m, q; F), Memoirs Amer. Math. Soc. 95 (1992), Number 462.
Kostant, B., The principle of triality and a distinguished unitary representation of SO(4,4), Differential Geometrical Methods in Theoretical Physics (Como, 1987), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Vol. 250, Kluwer, Dordrecht, 1988, pp. 65–108.
Kostant, B., The vanishing of scalar curvature and the minimal representation of SO(4,4), Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989), Birkhäuser, Boston, 1990, pp. 85–124.
Kraljević, H., Representations of the universal covering group of the group SU(n, 1), Glasnik Mat. 8 (1973), 23–72.
Li, Jian-Shu, Singular unitary representations of classical groups, Invent. Math. 97 (1989), 237–255.
Li, Jian-Shu, On the classification of irreducible low rank unitary representations of classical groups , Compositio Math. 71 (1989) 29–48.
Ottoson, U., A classification of the unitary irreducible representations of SU(N, 1), Commun. Math. Physics 10 (1968), 114–131.
Vogan, D. A., Unitarizability of certain series of representations, Ann. of Math. 120 (1984), 141–187.
Wallach, N., The analytic continuation of the discrete series II, Trans. Amer. Math. Soc. 251 (1979), 19–37.
Wallach, N. R., On the unitarizability of derived functor modules, Invent. Math. 78 (1984), 131–141.
Wallach, N., Real Reductive Groups I, Academic Press, San Diego, 1988.
Zhu, Chen-Bo, and Jing-Song Huang, On certain small representations of indefinite orthogonal groups, Representation Theory 1 (1997), 190–206.
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Knapp, A.W. (2004). Analytic continuation of nonholomorphic discrete series for classical groups. In: Delorme, P., Vergne, M. (eds) Noncommutative Harmonic Analysis. Progress in Mathematics, vol 220. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8204-0_10
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