Classification of discrete series by minimal 𝐾-type

Parthasarathy, Rajagopalan

doi:10.1090/ert/467

Classification of discrete series by minimal $K$-type
HTML articles powered by AMS MathViewer

by Rajagopalan Parthasarathy

Represent. Theory 19 (2015), 167-185

DOI: https://doi.org/10.1090/ert/467

Published electronically: October 7, 2015

PDF | Request permission

Abstract:

Following the proof by Hecht and Schmid of Blattner’s conjecture for $K$ multiplicities of representations belonging to the discrete series it turned out that some results which were earlier known with some hypothesis on the Harish-Chandra parameter of the discrete series representation could be extended removing those hypotheses. For example this was so for the geometric realization problem. Occasionally a few other results followed by first proving them for Harish-Chandra parameters which are sufficiently regular and then using Zuckerman translation functors, wall crossing methods, etc. Recently, Hongyu He raised the question (private communication) of whether the characterization of a discrete series representation by its lowest $K$-type, which was proved by this author and R. Hotta with some hypothesis on the Harish-Chandra parameter of the discrete series representations, can be extended to all discrete series representations excluding none, using a combination of these powerful techniques. In this article we will answer this question using Dirac operator methods and a result of Susana Salamanca-Riba.

References

R. Hotta and R. Parthasarathy, Multiplicity formulae for discrete series, Invent. Math. 26 (1974), 133–178. MR 348041, DOI 10.1007/BF01435692
Jing-Song Huang and Pavle Pandžić, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), no. 1, 185–202. MR 1862801, DOI 10.1090/S0894-0347-01-00383-6
Jing-Song Huang and Pavle Pandžić, Dirac operators in representation theory, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006. MR 2244116
Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. MR 142696, DOI 10.2307/1970237
Harish-Chandra, On the theory of the Eisenstein integral, Conference on Harmonic Analysis (Univ. Maryland, College Park, Md., 1971), Lecture Notes in Math., Vol. 266, Springer, Berlin, 1972, pp. 123–149. MR 0399355
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
K. R. Parthasarathy, R. Ranga Rao, and V. S. Varadarajan, Representations of complex semi-simple Lie groups and Lie algebras, Ann. of Math. (2) 85 (1967), 383–429. MR 225936, DOI 10.2307/1970351
R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1–30. MR 318398, DOI 10.2307/1970892
R. Parthasarathy, Criteria for the unitarizability of some highest weight modules, Proc. Indian Acad. Sci. Sect. A Math. Sci. 89 (1980), no. 1, 1–24. MR 573381
R. Parthasarathy, A generalization of the Enright-Varadarajan modules, Compositio Math. 36 (1978), no. 1, 53–73. MR 515037
Susana A. Salamanca-Riba, On the unitary dual of real reductive Lie groups and the $A_g(\lambda )$ modules: the strongly regular case, Duke Math. J. 96 (1999), no. 3, 521–546. MR 1671213, DOI 10.1215/S0012-7094-99-09616-3
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
Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR 929683
V. S. Varadarajan, George Mackey and his work on representation theory and foundations of physics, Group representations, ergodic theory, and mathematical physics: a tribute to George W. Mackey, Contemp. Math., vol. 449, Amer. Math. Soc., Providence, RI, 2008, pp. 417–446. MR 2391814, DOI 10.1090/conm/449/08722
V. S. Varadarajan, Some mathematical reminiscences, Methods Appl. Anal. 9 (2002), no. 3, v–xviii. Special issue dedicated to Daniel W. Stroock and Srinivasa S. R. Varadhan on the occasion of their 60th birthday. MR 2023128, DOI 10.4310/MAA.2002.v9.n3.a2
Apoorva Khare, Representations of complex semi-simple Lie groups and Lie algebras, Connected at infinity. II, Texts Read. Math., vol. 67, Hindustan Book Agency, New Delhi, 2013, pp. 85–129. MR 3135123
A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups. II, Invent. Math. 60 (1980), no. 1, 9–84. MR 582703, DOI 10.1007/BF01389898
David A. Vogan Jr. and Gregg J. Zuckerman, Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), no. 1, 51–90. MR 762307
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
S. Kumaresan, On the canonical $k$-types in the irreducible unitary $g$-modules with nonzero relative cohomology, Invent. Math. 59 (1980), no. 1, 1–11. MR 575078, DOI 10.1007/BF01390311
Wilfried Schmid, On a conjecture of Langlands, Ann. of Math. (2) 93 (1971), 1–42. MR 286942, DOI 10.2307/1970751
Wilfried Schmid, Discrete series, Representation theory and automorphic forms (Edinburgh, 1996) Proc. Sympos. Pure Math., vol. 61, Amer. Math. Soc., Providence, RI, 1997, pp. 83–113. MR 1476494, DOI 10.1090/pspum/061/1476494
Henryk Hecht and Wilfried Schmid, A proof of Blattner’s conjecture, Invent. Math. 31 (1975), no. 2, 129–154. MR 396855, DOI 10.1007/BF01404112