## Dirac cohomology, unitary representations and a proof of a conjecture of Vogan

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- by Jing-Song Huang and Pavle Pandžić
- J. Amer. Math. Soc.
**15**(2002), 185-202 - DOI: https://doi.org/10.1090/S0894-0347-01-00383-6
- Published electronically: September 6, 2001
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## Abstract:

Let $G$ be a connected semisimple Lie group with finite center. Let $K$ be the maximal compact subgroup of $G$ corresponding to a fixed Cartan involution $\theta$. We prove a conjecture of Vogan which says that if the Dirac cohomology of an irreducible unitary $(\mathfrak {g},K)$-module $X$ contains a $K$-type with highest weight $\gamma$, then $X$ has infinitesimal character $\gamma +\rho _{c}$. Here $\rho _{c}$ is the half sum of the compact positive roots. As an application of the main result we classify irreducible unitary $(\mathfrak {g},K)$-modules $X$ with non-zero Dirac cohomology, provided $X$ has a strongly regular infinitesimal character. We also mention a generalization to the setting of Kostant’s cubic Dirac operator.## References

- S. Bergmann and J. Marcinkiewicz,
*Sur les fonctions analytiques de deux variables complexes*, Fund. Math.**33**(1939), 75–94 (French). MR**57**, DOI 10.4064/fm-33-1-75-94 - C. J. Everett Jr.,
*Annihilator ideals and representation iteration for abstract rings*, Duke Math. J.**5**(1939), 623–627. MR**13** - William Casselman and M. Scott Osborne,
*The ${\mathfrak {n}}$-cohomology of representations with an infinitesimal character*, Compositio Math.**31**(1975), no. 2, 219–227. MR**396704** - Ryoshi Hotta,
*On a realization of the discrete series for semisimple Lie groups*, J. Math. Soc. Japan**23**(1971), 384–407. MR**306405**, DOI 10.2969/jmsj/02320384 - Ryoshi Hotta and R. Parthasarathy,
*A geometric meaning of the multiplicity of integrable discrete classes in $L^{2}(\Gamma \backslash G)$*, Osaka Math. J.**10**(1973), 211–234. MR**338265**
[K]K B. Kostant, - 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
[L]L J.-S. Li, - R. Parthasarathy,
*Dirac operator and the discrete series*, Ann. of Math. (2)**96**(1972), 1–30. MR**318398**, DOI 10.2307/1970892 - 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 - Wilfried Schmid,
*On the characters of the discrete series. The Hermitian symmetric case*, Invent. Math.**30**(1975), no. 1, 47–144. MR**396854**, DOI 10.1007/BF01389847 - David A. Vogan Jr.,
*Representations of real reductive Lie groups*, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. MR**632407** - 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
[V3]V3 D. A. Vogan, Jr., - David A. Vogan Jr. and Gregg J. Zuckerman,
*Unitary representations with nonzero cohomology*, Compositio Math.**53**(1984), no. 1, 51–90. MR**762307** - Nolan R. Wallach,
*Real reductive groups. I*, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR**929683**

*A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups*, Duke Math. Jour.

**100**(1999), 447-501. [K2]K2 B. Kostant,

*Dirac cohomology for the cubic Dirac operator*, in preparation.

*On the first eigenvalue of Laplacian on locally symmetric manifolds*, First International Congress of Chinese Mathematicians (Beijing, 1998), AMS/IP Stud. Adv. Math., 20, Amer. Math. Soc., Providence, RI, 2001, 271-278.

*Dirac operator and unitary representations*, 3 talks at MIT Lie groups seminar, Fall of 1997. [V4]V4 D. A. Vogan, Jr.,

*On the smallest eigenvalue of the Laplacian on a locally symmetric space*, Lecture at the Midwest Conference on Representation Theory and Automorphic Forms, Chicago, June, 2000.

## Bibliographic Information

**Jing-Song Huang**- Affiliation: Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
- MR Author ID: 304754
- Email: mahuang@ust.hk
**Pavle Pandžić**- Affiliation: Department of Mathematics, University of Zagreb, PP 335, 10002 Zagreb, Croatia
- ORCID: 0000-0002-7405-4381
- Email: pandzic@math.hr
- Received by editor(s): August 28, 2000
- Received by editor(s) in revised form: February 27, 2001
- Published electronically: September 6, 2001
- Additional Notes: The first author’s research was partially supported by RGC-CERG grants of Hong Kong SAR. A part of this work was done during his visit to the University of Zagreb

A part of this work was done during the second author’s visit to The Hong Kong University of Science and Technology - © Copyright 2001 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**15**(2002), 185-202 - MSC (2000): Primary 22E46, 22E47
- DOI: https://doi.org/10.1090/S0894-0347-01-00383-6
- MathSciNet review: 1862801