## Exceptional Unitary Representations Of Semisimple Lie Groups

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- by A. W. Knapp
- Represent. Theory
**1**(1997), 1-24 - DOI: https://doi.org/10.1090/S1088-4165-97-00001-0
- Published electronically: November 4, 1996
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## Abstract:

Let $G$ be a noncompact simple Lie group with finite center, let $K$ be a maximal compact subgroup, and suppose that $\text {rank }G=\text {rank }K$. If $G/K$ is not Hermitian symmetric, then a theorem of Borel and de Siebenthal gives the existence of a system of positive roots relative to a compact Cartan subalgebra so that there is just one noncompact simple root and it occurs exactly twice in the largest root. Let $\mathfrak {q}=\mathfrak {l}\oplus \mathfrak {u}$ be the $\theta$ stable parabolic obtained by building $\mathfrak {l}$ from the roots generated by the compact simple roots and by building $\mathfrak {u}$ from the other positive roots, and let $L\subseteq K$ be the normalizer of $\mathfrak {q}$ in $G$. Cohomological induction of an irreducible representation of $L$ produces a discrete series representation of $G$ under a dominance condition. This paper studies the results of this cohomological induction when the dominance condition fails. When the inducing representation is one-dimensional, a great deal is known about when the cohomologically induced representation is infinitesimally unitary. This paper addresses the question of finding Langlands parameters for the natural irreducible constituent of these representations, and also it finds some cases when the inducing representation is higher-dimensional and the cohomologically induced representation is infinitesimally unitary.## References

- Dan Barbasch,
*Representations with maximal primitive ideal*, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 317–331. MR**1103595** - B. Binegar and R. Zierau,
*Unitarization of a singular representation of $\textrm {SO}(p,q)$*, Comm. Math. Phys.**138**(1991), no. 2, 245–258. MR**1108044**, DOI 10.1007/BF02099491 - Morgan Ward and R. P. Dilworth,
*The lattice theory of ova*, Ann. of Math. (2)**40**(1939), 600–608. MR**11**, DOI 10.2307/1968944 - Ranee Brylinski and Bertram Kostant,
*Minimal representations, geometric quantization, and unitarity*, Proc. Nat. Acad. Sci. U.S.A.**91**(1994), no. 13, 6026–6029. MR**1278630**, DOI 10.1073/pnas.91.13.6026 - Thomas Enright, Roger Howe, and Nolan Wallach,
*A classification of unitary highest weight modules*, Representation theory of reductive groups (Park City, Utah, 1982) Progr. Math., vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 97–143. MR**733809**, DOI 10.1007/978-1-4684-6730-7_{7} - T. J. Enright, R. Parthasarathy, N. R. Wallach, and J. A. Wolf,
*Unitary derived functor modules with small spectrum*, Acta Math.**154**(1985), no. 1-2, 105–136. MR**772433**, DOI 10.1007/BF02392820 - Benedict H. Gross and Nolan R. Wallach,
*A distinguished family of unitary representations for the exceptional groups of real rank $=4$*, Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 289–304. MR**1327538**, DOI 10.1007/978-1-4612-0261-5_{1}0 - 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,
*Harmonic analysis on real reductive groups. I. The theory of the constant term*, J. Functional Analysis**19**(1975), 104–204. MR**0399356**, DOI 10.1016/0022-1236(75)90034-8 - Hans Plesner Jakobsen,
*Hermitian symmetric spaces and their unitary highest weight modules*, J. Funct. Anal.**52**(1983), no. 3, 385–412. MR**712588**, DOI 10.1016/0022-1236(83)90076-9 - D. Kazhdan and G. Savin,
*The smallest representation of simply laced groups*, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 2, Weizmann, Jerusalem, 1990, pp. 209–223. MR**1159103** - Anthony W. Knapp,
*Representation theory of semisimple groups*, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR**855239**, DOI 10.1515/9781400883974 - A. W. Knapp,
*Lie Groups Beyond an Introduction*, Birkhäuser, Boston, 1996. - Anthony W. Knapp and David A. Vogan Jr.,
*Cohomological induction and unitary representations*, Princeton Mathematical Series, vol. 45, Princeton University Press, Princeton, NJ, 1995. MR**1330919**, DOI 10.1515/9781400883936 - William M. McGovern,
*Dixmier algebras and the orbit method*, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 397–416. MR**1103597** - William M. McGovern,
*Rings of regular functions on nilpotent orbits. II. Model algebras and orbits*, Comm. Algebra**22**(1994), no. 3, 765–772. MR**1261003**, DOI 10.1080/00927879408824874 - David A. Vogan Jr.,
*The algebraic structure of the representation of semisimple Lie groups. I*, Ann. of Math. (2)**109**(1979), no. 1, 1–60. MR**519352**, DOI 10.2307/1971266 - David A. Vogan Jr.,
*Unitarizability of certain series of representations*, Ann. of Math. (2)**120**(1984), no. 1, 141–187. MR**750719**, DOI 10.2307/2007074 - Nolan R. Wallach,
*The analytic continuation of the discrete series. I, II*, Trans. Amer. Math. Soc.**251**(1979), 1–17, 19–37. MR**531967**, DOI 10.1090/S0002-9947-1979-0531967-2 - Nolan R. Wallach,
*On the unitarizability of derived functor modules*, Invent. Math.**78**(1984), no. 1, 131–141. MR**762359**, DOI 10.1007/BF01388720

## Bibliographic Information

**A. W. Knapp**- Affiliation: Department of Mathematics, State University of New York, Stony Brook, New York 11794
- MR Author ID: 103200
- Email: aknapp@ccmail.sunysb.edu
- Received by editor(s): June 19, 1996
- Received by editor(s) in revised form: August 5, 1996
- Published electronically: November 4, 1996
- Additional Notes: Presented to the Society August 7, 1995 at the AMS Summer Meeting in Burlington, Vermont.
- © Copyright 1997 American Mathematical Society
- Journal: Represent. Theory
**1**(1997), 1-24 - MSC (1991): Primary 22E46, 22E47
- DOI: https://doi.org/10.1090/S1088-4165-97-00001-0
- MathSciNet review: 1429371