The Burger-Sarnak method and operations on the unitary dual of $\mathrm {GL}(n)$

Author:
Akshay Venkatesh

Journal:
Represent. Theory **9** (2005), 268-286

MSC (2000):
Primary 22E50; Secondary 11F70

DOI:
https://doi.org/10.1090/S1088-4165-05-00226-8

Published electronically:
March 31, 2005

MathSciNet review:
2133760

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the effect of restriction to Levi subgroups, induction from Levi subgroups, and tensor product, on unitary representations of $\mathrm {GL}(n)$ over a local field $k$. These results give partial answers to questions raised by Clozel.

- I. N. Bernšteĭn,
*All reductive ${\mathfrak p}$-adic groups are of type I*, Funkcional. Anal. i Priložen.**8**(1974), no. 2, 3–6 (Russian). MR**0348045** - M. Burger, J.-S. Li, and P. Sarnak,
*Ramanujan duals and automorphic spectrum*, Bull. Amer. Math. Soc. (N.S.)**26**(1992), no. 2, 253–257. MR**1118700**, DOI https://doi.org/10.1090/S0273-0979-1992-00267-7
clozelparkcity L. Clozel. Spectral theory of automorphic forms. - L. Clozel,
*Combinatorial consequences of Arthur’s conjectures and the Burger-Sarnak method*, Int. Math. Res. Not.**11**(2004), 511–523. MR**2038775**, DOI https://doi.org/10.1155/S1073792804132649 - Laurent Clozel and Emmanuel Ullmo,
*Équidistribution des points de Hecke*, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 193–254 (French). MR**2058609** - M. Cowling, U. Haagerup, and R. Howe,
*Almost $L^2$ matrix coefficients*, J. Reine Angew. Math.**387**(1988), 97–110. MR**946351** - Jacques Dixmier,
*Enveloping algebras*, Graduate Studies in Mathematics, vol. 11, American Mathematical Society, Providence, RI, 1996. Revised reprint of the 1977 translation. MR**1393197** - J. M. G. Fell,
*Weak containment and induced representations of groups*, Canadian J. Math.**14**(1962), 237–268. MR**150241**, DOI https://doi.org/10.4153/CJM-1962-016-6 - 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** - Robert P. Langlands,
*On the functional equations satisfied by Eisenstein series*, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. MR**0579181** - Wenzhi Luo, Zeév Rudnick, and Peter Sarnak,
*On the generalized Ramanujan conjecture for ${\rm GL}(n)$*, Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996) Proc. Sympos. Pure Math., vol. 66, Amer. Math. Soc., Providence, RI, 1999, pp. 301–310. MR**1703764** - George W. Mackey,
*The theory of unitary group representations*, University of Chicago Press, Chicago, Ill.-London, 1976. Based on notes by James M. G. Fell and David B. Lowdenslager of lectures given at the University of Chicago, Chicago, Ill., 1955; Chicago Lectures in Mathematics. MR**0396826** - C. Mœglin and J.-L. Waldspurger,
*Le spectre résiduel de ${\rm GL}(n)$*, Ann. Sci. École Norm. Sup. (4)**22**(1989), no. 4, 605–674 (French). MR**1026752** - C. Mœglin and J.-L. Waldspurger,
*Spectral decomposition and Eisenstein series*, Cambridge Tracts in Mathematics, vol. 113, Cambridge University Press, Cambridge, 1995. Une paraphrase de l’Écriture [A paraphrase of Scripture]. MR**1361168**
ms W. Mueller and B. Speh. On the absolute convergence of the spectral side of the Arthur trace formula for $\mathrm {GL}(n)$. arXiv:math.RT/0211030.
- Hee Oh,
*Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants*, Duke Math. J.**113**(2002), no. 1, 133–192. MR**1905394**, DOI https://doi.org/10.1215/S0012-7094-02-11314-3 - Yehuda Shalom,
*Explicit Kazhdan constants for representations of semisimple and arithmetic groups*, Ann. Inst. Fourier (Grenoble)**50**(2000), no. 3, 833–863 (English, with English and French summaries). MR**1779896** - 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** - Marko Tadić,
*Topology of unitary dual of non-Archimedean ${\rm GL}(n)$*, Duke Math. J.**55**(1987), no. 2, 385–422. MR**894588**, DOI https://doi.org/10.1215/S0012-7094-87-05522-0
vogan2 David Vogan. Isolated unitary representations. - David A. Vogan Jr.,
*The unitary dual of ${\rm GL}(n)$ over an Archimedean field*, Invent. Math.**83**(1986), no. 3, 449–505. MR**827363**, DOI https://doi.org/10.1007/BF01394418 - S. P. Wang,
*The dual space of semi-simple Lie groups*, Amer. J. Math.**91**(1969), 921–937. MR**259023**, DOI https://doi.org/10.2307/2373310

*Preprint*.

*To appear*.

Retrieve articles in *Representation Theory of the American Mathematical Society*
with MSC (2000):
22E50,
11F70

Retrieve articles in all journals with MSC (2000): 22E50, 11F70

Additional Information

**Akshay Venkatesh**

Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307

Address at time of publication:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012

MR Author ID:
693009

Email:
akshayv@math.mit.edu

Received by editor(s):
December 19, 2003

Received by editor(s) in revised form:
January 30, 2005

Published electronically:
March 31, 2005

Additional Notes:
The author was supported in part by NSF Grant DMS-0245606

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.