Dirac cohomology of one-$W$-type representations
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- by Dan Ciubotaru and Allen Moy
- Proc. Amer. Math. Soc. 143 (2015), 1001-1013
- DOI: https://doi.org/10.1090/S0002-9939-2014-12303-3
- Published electronically: November 12, 2014
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Abstract:
The smooth hermitian representations of a split reductive $p$-adic group whose restriction to a maximal hyperspecial compact subgroup contain a single $K$-type with Iwahori fixed vectors have been studied in a paper by Barbasch and Moy (1999) in the more general setting of modules for graded affine Hecke algebras with parameters. We show that every such one $K$-type module has nonzero Dirac cohomology (in the sense of a paper by Barbasch, Ciubotaru and Trapa), and use Dirac operator techniques to determine the semisimple part of the Langlands parameter for these modules, thus completing their classification.References
- Dan Barbasch, Dan Ciubotaru, and Peter E. Trapa, Dirac cohomology for graded affine Hecke algebras, Acta Math. 209 (2012), no. 2, 197–227. MR 3001605, DOI 10.1007/s11511-012-0085-3
- Dan Barbasch and Allen Moy, Reduction to real infinitesimal character in affine Hecke algebras, J. Amer. Math. Soc. 6 (1993), no. 3, 611–635. MR 1186959, DOI 10.1090/S0894-0347-1993-1186959-0
- Dan Barbasch and Allen Moy, Unitary spherical spectrum for $p$-adic classical groups, Acta Appl. Math. 44 (1996), no. 1-2, 3–37. Representations of Lie groups, Lie algebras and their quantum analogues. MR 1407038, DOI 10.1007/BF00116514
- Dan Barbasch and Allen Moy, Classification of one $\textrm {K}$-type representations, Trans. Amer. Math. Soc. 351 (1999), no. 10, 4245–4261. MR 1473430, DOI 10.1090/S0002-9947-99-02171-6
- D. Barbasch, D. Ciubotaru, Hermitian forms for affine Hecke algebras, preprint, arXiv:1312.3316.
- —, Unitary Hecke algebra modules with nonzero Dirac cohomology, Symmetry in Representation Theory and Its Applications: In honor of Nolan Wallach, Progr. Math. Birkhäuser (2014), 1–21.
- Armand Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976), 233–259. MR 444849, DOI 10.1007/BF01390139
- Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication. MR 1266626
- Dan Ciubotaru, Spin representations of Weyl groups and the Springer correspondence, J. Reine Angew. Math. 671 (2012), 199–222. MR 2983200, DOI 10.1515/crelle.2011.160
- D. Ciubotaru, E. Opdam, P. Trapa, Algebraic and analytic Dirac induction for graded affine Hecke algebras, J. Inst. Math. Jussieu 13 (2014), no. 3, 447–486.
- Anthony W. Knapp, Representation theory of semisimple groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001. An overview based on examples; Reprint of the 1986 original. MR 1880691
- George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599–635. MR 991016, DOI 10.1090/S0894-0347-1989-0991016-9
- George Lusztig, Cuspidal local systems and graded Hecke algebras. I, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 145–202. MR 972345
- A. O. Morris, Projective representations of reflection groups. II, Proc. London Math. Soc. (3) 40 (1980), no. 3, 553–576. MR 572019, DOI 10.1112/plms/s3-40.3.553
- Soichi Okada, Applications of minor summation formulas to rectangular-shaped representations of classical groups, J. Algebra 205 (1998), no. 2, 337–367. MR 1632816, DOI 10.1006/jabr.1997.7408
- E. W. Read, On projective representations of the finite reflection groups of type $B_{l}$ and $D_{l}$, J. London Math. Soc. (2) 10 (1975), 129–142. MR 367047, DOI 10.1112/jlms/s2-10.2.129
- John R. Stembridge, Shifted tableaux and the projective representations of symmetric groups, Adv. Math. 74 (1989), no. 1, 87–134. MR 991411, DOI 10.1016/0001-8708(89)90005-4
- Marko Tadić, Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case), Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 335–382. MR 870688
Bibliographic Information
- Dan Ciubotaru
- Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
- Address at time of publication: Mathematical Institute, University of Oxford, Oxford, OX26GG, UK
- MR Author ID: 754534
- Email: ciubo@math.utah.edu, dan.ciubotaru@maths.ox.ac.uk
- Allen Moy
- Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong
- MR Author ID: 127665
- Email: amoy@ust.hk
- Received by editor(s): August 23, 2012
- Received by editor(s) in revised form: July 6, 2013
- Published electronically: November 12, 2014
- Additional Notes: This paper was partly written while the first author visited Hong Kong University of Science and Technology. The first author thanks Xuhua He and the Department of Mathematics for their invitation and hospitality
The authors were supported in part by NSF-DMS 0968065 and Hong Kong Research Grants Council grant CERG #602408. - Communicated by: Kailash C. Misra
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 1001-1013
- MSC (2010): Primary 20C08, 22E50
- DOI: https://doi.org/10.1090/S0002-9939-2014-12303-3
- MathSciNet review: 3293718