Classification of discrete series by minimal type
Author:
Rajagopalan Parthasarathy
Journal:
Represent. Theory 19 (2015), 167185
MSC (2010):
Primary 22E46; Secondary 22D30
Published electronically:
October 7, 2015
MathSciNet review:
3405535
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Following the proof by Hecht and Schmid of Blattner's conjecture for multiplicities of representations belonging to the discrete series it turned out that some results which were earlier known with some hypothesis on the HarishChandra 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 HarishChandra 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 type, which was proved by this author and R. Hotta with some hypothesis on the HarishChandra 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 SalamancaRiba.
 [1]
R.
Hotta and R.
Parthasarathy, Multiplicity formulae for discrete series,
Invent. Math. 26 (1974), 133–178. MR 0348041
(50 #539)
 [2]
JingSong
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 (electronic). MR 1862801
(2002h:22022), 10.1090/S0894034701003836
 [3]
JingSong
Huang and Pavle
Pandžić, Dirac operators in representation
theory, Mathematics: Theory & Applications, Birkhäuser
Boston, Inc., Boston, MA, 2006. MR 2244116
(2007j:22025)
 [4]
Bertram
Kostant, Lie algebra cohomology and the generalized BorelWeil
theorem, Ann. of Math. (2) 74 (1961), 329–387.
MR
0142696 (26 #265)
 [5]
HarishChandra,
On the theory of the Eisenstein integral, Conference on Harmonic
Analysis (Univ. Maryland, College Park, Md., 1971), Springer, Berlin,
1972, pp. 123–149. Lecture Notes in Math., Vol. 266. MR 0399355
(53 #3200)
 [6]
HarishChandra,
Harmonic analysis on real reductive groups. I. The theory of the
constant term, J. Functional Analysis 19 (1975),
104–204. MR 0399356
(53 #3201)
 [7]
K.
R. Parthasarathy, R.
Ranga Rao, and V.
S. Varadarajan, Representations of complex semisimple Lie groups
and Lie algebras, Ann. of Math. (2) 85 (1967),
383–429. MR 0225936
(37 #1526)
 [8]
R.
Parthasarathy, Dirac operator and the discrete series, Ann. of
Math. (2) 96 (1972), 1–30. MR 0318398
(47 #6945)
 [9]
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
(82c:22020)
 [10]
R.
Parthasarathy, A generalization of the EnrightVaradarajan
modules, Compositio Math. 36 (1978), no. 1,
53–73. MR
515037 (80f:22016)
 [11]
Susana
A. SalamancaRiba, On the unitary dual of real reductive Lie groups
and the 𝐴_{𝑔}(𝜆) modules: the strongly regular
case, Duke Math. J. 96 (1999), no. 3,
521–546. MR 1671213
(2000a:22023), 10.1215/S0012709499096163
 [12]
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
(96c:22023)
 [13]
Nolan
R. Wallach, Real reductive groups. I, Pure and Applied
Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR 929683
(89i:22029)
 [14]
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
(2009i:22005), 10.1090/conm/449/08722
 [15]
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
(2005d:60001), 10.4310/MAA.2002.v9.n3.a2
 [16]
Apoorva
Khare, Representations of complex semisimple 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
 [17]
A.
W. Knapp and E.
M. Stein, Intertwining operators for semisimple groups. II,
Invent. Math. 60 (1980), no. 1, 9–84. MR 582703
(82a:22018), 10.1007/BF01389898
 [18]
David
A. Vogan Jr. and Gregg
J. Zuckerman, Unitary representations with nonzero cohomology,
Compositio Math. 53 (1984), no. 1, 51–90. MR 762307
(86k:22040)
 [19]
David
A. Vogan Jr., Unitarizability of certain series of
representations, Ann. of Math. (2) 120 (1984),
no. 1, 141–187. MR 750719
(86h:22028), 10.2307/2007074
 [20]
S.
Kumaresan, On the canonical 𝑘types in the irreducible
unitary 𝑔modules with nonzero relative cohomology, Invent.
Math. 59 (1980), no. 1, 1–11. MR 575078
(83c:17011), 10.1007/BF01390311
 [21]
Wilfried
Schmid, On a conjecture of Langlands, Ann. of Math. (2)
93 (1971), 1–42. MR 0286942
(44 #4149)
 [22]
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
(98k:22051), 10.1090/pspum/061/1476494
 [23]
Henryk
Hecht and Wilfried
Schmid, A proof of Blattner’s conjecture, Invent. Math.
31 (1975), no. 2, 129–154. MR 0396855
(53 #715)
 [1]
 R. Hotta and R. Parthasarathy, Multiplicity formulae for discrete series, Invent. Math. 26 (1974), 133178. MR 0348041 (50 #539)
 [2]
 JingSong Huang and Pavle Pandžić, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), no. 1, 185202 (electronic). MR 1862801 (2002h:22022), https://doi.org/10.1090/S0894034701003836
 [3]
 JingSong Huang and Pavle Pandžić, Dirac operators in representation theory, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006. MR 2244116 (2007j:22025)
 [4]
 Bertram Kostant, Lie algebra cohomology and the generalized BorelWeil theorem, Ann. of Math. (2) 74 (1961), 329387. MR 0142696 (26 #265)
 [5]
 HarishChandra, On the theory of the Eisenstein integral, Conference on Harmonic Analysis (Univ. Maryland, College Park, Md., 1971), Springer, Berlin, 1972, pp. 123149. Lecture Notes in Math., Vol. 266. MR 0399355 (53 #3200)
 [6]
 HarishChandra, Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Functional Analysis 19 (1975), 104204. MR 0399356 (53 #3201)
 [7]
 K. R. Parthasarathy, R. Ranga Rao, and V. S. Varadarajan, Representations of complex semisimple Lie groups and Lie algebras, Ann. of Math. (2) 85 (1967), 383429. MR 0225936 (37 #1526)
 [8]
 R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 130. MR 0318398 (47 #6945)
 [9]
 R. Parthasarathy, Criteria for the unitarizability of some highest weight modules, Proc. Indian Acad. Sci. Sect. A Math. Sci. 89 (1980), no. 1, 124. http://repository.ias.ac.in/82621/. MR 573381 (82c:22020)
 [10]
 R. Parthasarathy, A generalization of the EnrightVaradarajan modules, Compositio Math. 36 (1978), no. 1, 5373. http://repository.ias.ac.in/82622/. MR 515037 (80f:22016)
 [11]
 Susana A. SalamancaRiba, On the unitary dual of real reductive Lie groups and the modules: the strongly regular case, Duke Math. J. 96 (1999), no. 3, 521546. MR 1671213 (2000a:22023), https://doi.org/10.1215/S0012709499096163
 [12]
 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 (96c:22023)
 [13]
 Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR 929683 (89i:22029)
 [14]
 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. 417446. MR 2391814 (2009i:22005), https://doi.org/10.1090/conm/449/08722
 [15]
 V. S. Varadarajan, Some mathematical reminiscences, Methods Appl. Anal. 9 (2002), no. 3, vxviii. MR 2023128 (2005d:60001), https://doi.org/10.4310/MAA.2002.v9.n3.a2
 [16]
 Apoorva Khare, Representations of complex semisimple Lie groups and Lie algebras, Connected at infinity. II, Texts Read. Math., vol. 67, Hindustan Book Agency, New Delhi, 2013, pp. 85129. arXiv:1208.0416v2 [math.RT]. MR 3135123
 [17]
 A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups. II, Invent. Math. 60 (1980), no. 1, 984. MR 582703 (82a:22018), https://doi.org/10.1007/BF01389898
 [18]
 David A. Vogan Jr. and Gregg J. Zuckerman, Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), no. 1, 5190. MR 762307 (86k:22040)
 [19]
 David A. Vogan Jr., Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), no. 1, 141187. MR 750719 (86h:22028), https://doi.org/10.2307/2007074
 [20]
 S. Kumaresan, On the canonical types in the irreducible unitary modules with nonzero relative cohomology, Invent. Math. 59 (1980), no. 1, 111. MR 575078 (83c:17011), https://doi.org/10.1007/BF01390311
 [21]
 Wilfried Schmid, On a conjecture of Langlands, Ann. of Math. (2) 93 (1971), 142. MR 0286942 (44 #4149)
 [22]
 Wilfried Schmid, Discrete series, Representation theory and automorphic forms (Edinburgh, 1996) Proc. Sympos. Pure Math., vol. 61, Amer. Math. Soc., Providence, RI, 1997, pp. 83113. MR 1476494 (98k:22051), https://doi.org/10.1090/pspum/061/1476494
 [23]
 Henryk Hecht and Wilfried Schmid, A proof of Blattner's conjecture, Invent. Math. 31 (1975), no. 2, 129154. MR 0396855 (53 #715)
Similar Articles
Retrieve articles in Representation Theory of the American Mathematical Society
with MSC (2010):
22E46,
22D30
Retrieve articles in all journals
with MSC (2010):
22E46,
22D30
Additional Information
Rajagopalan Parthasarathy
Affiliation:
Raja Ramanna Fellow Bharathiar University Coimbatore
Email:
sarathy.math.tifr@gmail.com
DOI:
https://doi.org/10.1090/ert/467
Keywords:
Dirac operator inequality,
Dirac cohomology,
HarishChandra class group,
Mackey's criterion,
$A_{\mathfrak q,
\lambda}$
Received by editor(s):
July 29, 2014
Received by editor(s) in revised form:
December 4, 2014, and August 14, 2015
Published electronically:
October 7, 2015
Additional Notes:
This research was supported by Raja Ramanna Fellowship from DAE
The author thanks the referee for suggestions to improve the article by addressing the case of general groups of HarishChandra class. His comments on the initial proof of Theorem 1.1 in Section 3 have greatly helped in adding considerable clarity to the original submission.
Article copyright:
© Copyright 2015
American Mathematical Society
