Rad HAZU, Matematičke znanosti, Vol. 19 (2015), 27-53.

REMARK ON REPRESENTATION THEORY OF GENERAL LINEAR GROUPS OVER A NON-ARCHIMEDEAN LOCAL DIVISION ALGEBRA

Marko Tadić

Abstract.   In this paper we give a simple (local) proof of two principal results about irreducible tempered representations of general linear groups over a non-archimedean local division algebra. We give a proof of the parameterization of the irreducible square integrable representations of these groups by segments of cuspidal representations, and a proof of the irreducibility of the tempered parabolic induction. Our proofs are based on Jacquet modules (and the Geometric Lemma, incorporated in the structure of a Hopf algebra). We use only some very basic general facts of the representation theory of reductive p-adic groups (the theory that we use was completed more then three decades ago, mainly in 1970-es). Of the specific results for general linear groups over A, basically we use only a very old result of G. I. Ol’šanskii, which says that there exist complementary series starting from Ind(ρ ⊗ ρ) whenever ρ is a unitary irreducible cuspidal representation. In appendix of [11], there is also a simple local proof of these results, based on a slightly different approach.

2010 Mathematics Subject Classification.   22E50.

Key words and phrases.   Non-archimedean local fields, division algebras, general linear groups, Speh representations, parabolically induced representations, reducibility, unitarizability.

References:

1. A. I. Badulescu, Jacquet-Langlands et unitarisabilité, J. Inst. Math. Jussieu 6 (2007), 349-379.
   MathSciNet     CrossRef

2. A. I. Badulescu, Un résultat d'irréductibilité en caractéristique non nulle, Tohoku Math. J. (2) 56 (2004), 583-592.
   MathSciNet

3. A. I. Badulescu and D. A. Renard, Sur une conjecture de Tadić, Glas. Mat. Ser III 39 (2004), 49-54.
   MathSciNet     CrossRef

4. A. I. Badulescu, On p-adic Speh representations, Bull. Soc. Math. France 142 (2014), 255-267.
   MathSciNet

5. A. I. Badulescu, G. Henniart, B. Lemaire and V. Sécherre, Sur le dual unitaire de GL_r(D), Amer. J. Math. 132 (2010), 1365-1396.
   MathSciNet     CrossRef

6. J. Bernstein, P-invariant distributions on GL(N) and the classification of unitary representations of GL(N) (non-archimedean case), in: Lie Group Representations II, Lecture Notes in Math. 1041, Springer-Verlag, Berlin, 1984, pp. 50-102.
   MathSciNet     CrossRef

7. W. Casselman, Introduction to the theory of admissible representations of p-adic reductive groups, preprint
   (http://www.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf).

8. P. Deligne, D. Kazhdan and M.-F. Vignéras, Repréacute;sentations des algébres centrales sim ples p-adiques, in: Représentations des Groupes Réductifs sur un Corps Local by J.-N. Bernstein, P. Deligne, D. Kazhdan and M.-F. Vignéras, Hermann, Paris, 1984.

9. V. Heiermann, Décomposition spectrale et représentations spéciales d'un groupe réductif p-adique, J. Inst. Math. Jussieu 3 (2004), 327-395.
   MathSciNet     CrossRef

10. C. Jantzen, On square-integrable representations of classical p-adic groups, Canad. J. Math. 52 (2000), 539-581.
    MathSciNet     CrossRef

11. E. Lapid and A. Mínguez, On parabolic induction on inner forms of the general linear group over a non-archimedean local field, preprint
    (http://arxiv-web3.library.cornell.edu/pdf/1411.6310v1.pdf).

12. A. Mínguez and V. Sécherre, Représentations banales de GL(m,D), Compos. Math. 149 (2013), 679-704.
    MathSciNet     CrossRef

13. A. Mínguez and V. Sécherre, Représentations lisses modulo l de GL(m,D), Duke Math. J. 163 (2014), 795-887.
    MathSciNet     CrossRef

14. G. I. Ol'šanskii, Intertwining operators and complementary series in the class of representations induced from parabolic subgroups of the general linear group over a locally compact division algebra (in Russian), Mat. Sb. (N.S.) 93(135) (1974), 218-253 (English translation: Math. USSR Sbornik 22 (1974), 217-254).
    MathSciNet

15. F. Rodier, Représentations de GL(n,k) où k est un corps p-adique, Séminaire Bourbaki no 587 (1982), Astérisque 92-93 (1982), 201-218.
    MathSciNet

16. V. Sécherre, Proof of the Tadić conjecture (U0) on the unitary dual of GL_m(D), J. Reine Angew. Math. 626 (2009), 187-203.
    MathSciNet     CrossRef

17. F. Shahidi, Local coefficients and normalization of intertwining operators for GL(n), Compositio Math. 48 (1983), 271-295.
    MathSciNet

18. F. Shahidi, Poles of Intertwining Operators via Endoscopy; the Connection with Prehomogeneous Vector Spaces. With an Appendix, `Basic Endoscopic Data', by Diana Shelstad, Compositio Math. 120 (2000), 291-325.
    MathSciNet     CrossRef

19. A. Silberger, Special representations of reductive p-adic groups are not integrable, Ann. of Math. 111 (1980), 571-587.
    MathSciNet     CrossRef

20. M. Tadić, Classification of unitary representations in irreducible representations of general linear group (non-archimedean case), Ann. Sci. École Norm. Sup. 19 (1986), 335-382.
    MathSciNet

21. M. Tadić, Geometry of dual spaces of reductive groups (non-archimedean case), J. Analyse Math. 51 (1988), 139-181.
    MathSciNet     CrossRef

22. M. Tadić, Induced representations of GL(n,A) for p-adic division algebras A, J. Reine Angew. Math. 405 (1990), 48-77.
    MathSciNet     CrossRef

23. M. Tadić, An external approach to unitary representations, Bull. Amer. Math. Soc. (N.S.) 28 (1993), 215-252.
    MathSciNet     CrossRef

24. A. V. Zelevinsky, Induced representations of reductive p-adic groups II. On irreducible representations of GL(n), Ann. Sci. École Norm. Sup. 13 (1980), 165-210.
    MathSciNet