A classification of unitary highest weight modules.

*(English)*Zbl 0535.22012
Representation theory of reductive groups, Proc. Conf., Park City/Utah 1982, Prog. Math. 40, 97-143 (1983).

Let \(G\) be a connected, simply connected semisimple Lie group with centre \(Z\) and let \(K\) be a closed maximal subgroup of \(G\) with \(K/Z\) compact. The paper under review contains a solution of the classification problem of unitary highest weight modules of \(G\).

Harish-Chandra showed [Am. J. Math. 77, 743–777 (1955; Zbl 0066.35603), ibid. 78, 1–41 (1956; Zbl 0070.11602)] that nontrivial such modules exists precisely when \((G,K)\) is a Hermitian symmetric pair. Many other authors contributed to the theory. In later years for example H. Rossi and M. Vergne [Acta Math. 136, 1–59 (1976; Zbl 0356.32020)], M. Kashiwara and M. Vergne [Invent. Math. 44, 1–47 (1978; Zbl 0375.22009)], N. Wallach [Trans. Am. Math. Soc. 251, 1–17 (1979; Zbl 0419.22017), ibid. 251, 19–37 (1979; Zbl 0419.22018], R. Parthasarathy [Proc. Indian Acad. Sci. 89, 1–24 (1980; Zbl 0434.22011)], G. I. Ol’shanskiĭ [Funct. Anal. Appl. 14, 190–200 (1980; Zbl 0439.22019)], H. P. Jakobsen [Invent. Math. 62, 67–78 (1980; Zbl 0466.22016)], T. Enright and R. Parthasarathy [Lect. Notes Math. 880, 74–90 (1981; Zbl 0492.22012)]. Then around 1981 came two independent solutions to the classification problem: H. P. Jakobsen [Math. Ann. 256, 439–447 (1981; Zbl 0478.22007), and J. Funct. Anal. 52, 385–412 (1983; Zbl 0517.22014)] and T. Enright, R. Howe and N. Wallach in the paper under review. Besides relying on some of the papers mentioned above, the two solutions are different in that Jakobsen relies heavily on I. N. Bernstein, I. M. Gelfand and S. I. Gelfand [Lie Groups Represent., Proc. Summer Sch. Bolyai János Math. Soc., Budapest 1971, 21–64 (1975; Zbl 0338.58019)], together with some interesting combinatorics. The authors’ solution relies on J. C. Jantzen [Math. Ann. 226, 53–65 (1977; Zbl 0372.17003)] and on Howe’s theory of dual pairs together with much case-by-case analysis.

It seems to the reviewer that in fact both proofs when traced back through the necessary prerequisites are rather long and complicated, compared to how easy it is to describe the problem. – Anyway, this seems to be a general feature in connection with representation theory of semisimple Lie groups.

[For the entire collection see Zbl 0516.00013.]

Harish-Chandra showed [Am. J. Math. 77, 743–777 (1955; Zbl 0066.35603), ibid. 78, 1–41 (1956; Zbl 0070.11602)] that nontrivial such modules exists precisely when \((G,K)\) is a Hermitian symmetric pair. Many other authors contributed to the theory. In later years for example H. Rossi and M. Vergne [Acta Math. 136, 1–59 (1976; Zbl 0356.32020)], M. Kashiwara and M. Vergne [Invent. Math. 44, 1–47 (1978; Zbl 0375.22009)], N. Wallach [Trans. Am. Math. Soc. 251, 1–17 (1979; Zbl 0419.22017), ibid. 251, 19–37 (1979; Zbl 0419.22018], R. Parthasarathy [Proc. Indian Acad. Sci. 89, 1–24 (1980; Zbl 0434.22011)], G. I. Ol’shanskiĭ [Funct. Anal. Appl. 14, 190–200 (1980; Zbl 0439.22019)], H. P. Jakobsen [Invent. Math. 62, 67–78 (1980; Zbl 0466.22016)], T. Enright and R. Parthasarathy [Lect. Notes Math. 880, 74–90 (1981; Zbl 0492.22012)]. Then around 1981 came two independent solutions to the classification problem: H. P. Jakobsen [Math. Ann. 256, 439–447 (1981; Zbl 0478.22007), and J. Funct. Anal. 52, 385–412 (1983; Zbl 0517.22014)] and T. Enright, R. Howe and N. Wallach in the paper under review. Besides relying on some of the papers mentioned above, the two solutions are different in that Jakobsen relies heavily on I. N. Bernstein, I. M. Gelfand and S. I. Gelfand [Lie Groups Represent., Proc. Summer Sch. Bolyai János Math. Soc., Budapest 1971, 21–64 (1975; Zbl 0338.58019)], together with some interesting combinatorics. The authors’ solution relies on J. C. Jantzen [Math. Ann. 226, 53–65 (1977; Zbl 0372.17003)] and on Howe’s theory of dual pairs together with much case-by-case analysis.

It seems to the reviewer that in fact both proofs when traced back through the necessary prerequisites are rather long and complicated, compared to how easy it is to describe the problem. – Anyway, this seems to be a general feature in connection with representation theory of semisimple Lie groups.

[For the entire collection see Zbl 0516.00013.]

Reviewer: Mogens Flensted-Jensen (København)

##### MSC:

22E46 | Semisimple Lie groups and their representations |

43A80 | Analysis on other specific Lie groups |