×

zbMATH — the first resource for mathematics

On characters of irreducible unitary representations of general linear groups. (English) Zbl 0856.22026
The author reduces the determination of the characters of irreducible unitary representations of \(\text{GL}(n)\) over a non-Archimedean local field to the determination of the characters of the irreducible square integrable representations of \(\text{GL}(m)\) with \(m\leq n\). He also obtains a formula expressing the characters of irreducible unitary representations in terms of the characters of the segment representations of Zelevinsky. This formula generalizes the classical formula for the Steinberg character of \(\text{GL}(n)\). To obtain his results, he uses a clever trick based on the formal similarity of the unitary duals in the Archimedean and the non-Archimedean case and a result of Gregg Zuckerman expressing the character of the trivial representation as a linear combination of standard characters for \(\text{GL}(n)\) over complex numbers.

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A.-M. Aubert. Dualité dans le groupe de Grothendieck de la catégorie des représentations lises de longeur finie d’un groupe réductifp-adique, preprint 1994.
[2] 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, Proceedings, University of Maryland 1982–83, 50–102. Lecture Notes in Math.1041. Springer-Verlag, Berlin 1984.
[3] A. Borel andN. Wallach.Continuous cohomology, discrete subgroups, and representations of reductive groups. Princeton University Press, Princeton 1980. · Zbl 0443.22010
[4] C.J. Bushnell andP. Kutzko.The admissible dual of GL(N)via compact open subgroups. Princeton University Press, Princeton 1993. · Zbl 0787.22016
[5] W. Casselman. The Steinberg character as a true character. In:Symp. Pure Math. 26, 413–417. Amer. Math. Soc., Providence, Rhone Island 1973. · Zbl 0289.22017
[6] W. Casselman. Characters and Jacquet modules.Math.Ann. 230 (1977), 101–105. · Zbl 0337.22019 · doi:10.1007/BF01370657
[7] L. Corwin. A construction of the supercuspidal representations of GL n (F),F p-adic.Trans. Amer. Math. Soc. 337 (1993), 1–58. · Zbl 0789.22032 · doi:10.2307/2154308
[8] L. Corwin, A. Moy andP.J. Sally. Degrees and formal degrees for division algebras and GL n over ap-adic field.Pacific J. Math. 141, No. 1 (1990), 21–45. · Zbl 0689.22009
[9] G. van Dijk. Computation of certain induced characters ofp-adic groups.Math. Ann. 199 (1972), 229–240. · Zbl 0231.22018 · doi:10.1007/BF01429876
[10] I.M. Gelfand andM.A. Naimark.Unitare Derstellungen der Klassischen Gruppen (German translation of Russian publication from 1950). Akademie Verlag, Berlin 1957.
[11] H. Jacquet. Generic representations. In:Non-Commutative Harmonic Analysis, 91–101. Lecture Notes in Math.587. Springer Verlag, Berlin 1977.
[12] P. Kutzko. Characters formulas for supercuspdal representations of GLt, a prime.Amer. J. Math. 109 (1987), 201–221. · Zbl 0618.22006 · doi:10.2307/2374571
[13] G. Laumon, M. Rapoport andU., Stuhler.D-elliptic sheaves and the Langlands correspondence.Invent. Math. 113 (1993), 217–338. · Zbl 0809.11032 · doi:10.1007/BF01244308
[14] D. Miličić. OnC *-algebras with bounded, trace.Glasnik Mat. 8 (28) (1973), 7–21.
[15] C. Moeglin andJ.-L. Waldspurger. Sur l’involution de Zelevinsky.J. reine angew. Math. 372 (1986), 136–177. · Zbl 0594.22008
[16] K. Procter.The Zelevinsky duality conjecture for GL n . Thesis, King’s College, London 1994.
[17] F. Rodier. Représentations de GL(n, k) oùk est un corpsp-adique, Séminaire Bourbaki no 587 (1982).Astérisque 92–93 (1982), 201–218.
[18] S. Sahi. Jordan algebras and degenerate principal, series, preprint. · Zbl 0822.22006
[19] S. Sahi. Letter.
[20] P.J. Sally. Some remarks on discrete series characters for reductivep-adic groups. In:Representations of Lie Groups, Kyoto, Hiroshima, 1986, 337–348. Advanced Studies in Pure Mathematics14, 1988.
[21] E.M. Stein. Analysis in matrix spaces and some new representations of SL(N, \(\mathbb{C}\)).Ann. of Math. 86 (1967), 461–490. · Zbl 0188.45303 · doi:10.2307/1970611
[22] M. Tadić. Unitary dual ofp-adic GL(n), Proof, of Bernstein Conjectures.Bulletin Amer. Math. Soc. 13 No. 1 (1985), 39–42. · Zbl 0583.22008 · doi:10.1090/S0273-0979-1985-15355-8
[23] M. Tadić. Proof of a conjecture of Bernstein.Math. Ann. 272 (1985), 11–16. · Zbl 0563.22012 · doi:10.1007/BF01455923
[24] M. Tadić. Unitary representations of general linear group over real and complex field, preprint MPI/SFB 85-22 Bonn (1985).
[25] M. Tadić. Classification of unitary representations of general linear group (non-archimedean case).Ann. Sci. École Norm. Sup. 19 (1986), 335–382. · Zbl 0614.22005
[26] M. Tadić. Topology of unitary dual of non-archimedean GL(n).Duke Math. J. 55 (1987), 385–422. · Zbl 0668.22006 · doi:10.1215/S0012-7094-87-05522-0
[27] M. Tadić. Induced representations of GL(n, A) forp-adic division algebras.A. J. reine angew. Math. 405 (1990), 48–77.
[28] M. Tadić. An external approach to unitary representations.Bulletin Amer. Math. Soc. 28 No. 2 (1993), 215–252. · Zbl 0799.22010 · doi:10.1090/S0273-0979-1993-00372-0
[29] D.A. Vogan.Representations of real reductive groups. Birkhäuser, Boston 1987. · Zbl 0681.22013
[30] D.A. Vogan. The unitary dual of GL(n) over an archimedean field.Invent. Math. 82 (1986), 449–505. · Zbl 0598.22008 · doi:10.1007/BF01394418
[31] D.A. Vogan. Letter.
[32] A.V. Zelevinsky. Induced representations of reductivep-adic groups II, On irreducible representations of GL(n).Ann. Sci. École Norm Sup. 13 (1980), 165–210. · Zbl 0441.22014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.