Abstract
Let F be a local non-Archimedean field and let ψ: F → ℂ× be a non-trivial additive character of F. To this data one associates a meromorphic function γψ: π → γψ(π) on the set of irreducible representations of the group GL(n, F) in the following way. Consider the invariant distribution Φψ:= ψ(tr(g))|det(g)|n|dg| on GL(n,F), where |dg| denotes a Haar distribution on GL(n,F). Although the support of Φψ is not compact, it is well known that for a generic irreducible representation π of GL(n,F) the action of Φψ in π is well defined and thus it defines a number γψ(π). These gamma-functions and the associated L-functions were studied by J. Tate for n = 1 and by R. Go dement and H. Jacquet for arbitrary n.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, in “Analysis and Topology on Singular Spaces, I (Luminy, 1981)”, Astérisque 100 (1982), 5–171.
A. Berenstein, D. Kazhdan, Geometric and unipotent crystals, GAFA2000, in this issue.
J. Bernstein, P. Deligne, Le centre de Bernstein, Travaux en Cours, Representations of Reductive Groups over a Local Field, 1–32, Hermann, Paris, 1984.
J. Bernstein, A. Zelevinsky, Representations of the group GL(n,F), where F is a local non-Archimedean field (in Russian), Uspehi Mat. Nauk 31 (1976), 5–70.
J. Bernstein, A. Zelevinsky, Induced representations of reductive p- adic groups I, Ann. Sci. École Norm. Sup. (4)10 (1977), 441–472.
A. Borel, H. Jacquet, Automorphic forms and automorphic representations, with a supplement, “On the notion of an automorphic representation” by R.P. Langlands, Proc. Sympos. Pure Math. XXXIII, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Amer. Math. Soc, Providence, R.I. (1979), 189–207.
A. Braverman, D. Kazhdan, On the Schwartz space of the basic affine space, Selecta Math. (N.S.) 5:1 (1999), 1–28.
I. Gelfand, M. Graev, I. Piatetskii-Shapiro, Representation Theory and Automorphic Functions, Philadelphia, PA-London-Toronto, 1969.
I.M. Gelfand, D. Kazhdan, Representations of the group GL(n,K) where K is a local field, in “Lie Groups and Their Representations” (Proc. Summer School, Bolyai Jänos Math. Soc, Budapest, 1971), Halsted, New York, 1975, 95–118.
S. Gelfand, D. Kazhdan, Conjectural algebraic formulas for representations of GL(n), in “Sir Michael Atiyah: a Great Mathematician of the Twentieth Century,” Asian J. Math. 3:1 (1999), 17–48.
R. Godement, H. Jacquet, Zeta-functions of simple algebras, Springer Lecture Notes in Mathematics 260 (1972).
M. Harris, The local Langlands conjecture for GL(n) over a p-adic field, n < p, Invent. Math. 134:1 (1998), 177–210.
M. Harris, R. Taylor, On the geometry of cohomology of some simple Shimura varieties, preprint.
H. Jacquet, I. Piatetskii-Shapiro, J. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105:2 (1983), 367–464.
D. Kazhdan, On lifting, Lie group representations, II (College Park, Md., 1982/1983), Springer Lecture Notes in Math. 1041 (1984), 209–249
D. Kazhdan, Forms of the principle series for GL(n) in “Functional Analysis on the Eve of the 21st Century, Vol. 1 (New Brunswick, NJ, 1993), Birkhäuser Progr. Math. 131, (1995), 153–171.
D. Kazhdan, An algebraic integration, in “Mathematics: Frontiers and Perspectives” (V. Arnold, M. Atiyah, P. Lax, B. Mazur, eds.), AMS, 2000.
R.P. Langlands, Problems in the theory of automorphic forms, in “Lectures in Modern Analysis and Applications, III,” Springer Lecture Notes in Math. 170 (1970), 18–61.
G. Laumon, Faisceaux charactérs (d’aprés Lusztig) (in French), Séminaire Bourbaki, Vol. 1988/89. Astérisque 177–178 (1989), 231–260.
G. Laumon, M. Rapoport, U. Stuhler, D-elliptic sheaves and the Langlands correspondence, Invent. Math. 113:2 (1993), 217–338.
G. Lusztig, Character sheaves I, Adv. in Math. 56:3 (1985), 193–237.
I. Piatetskii-Shapiro, Multiplicity one theorems, in “Automorphic Forms, Representations and L-functions” (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math. XXXIII, Amer. Math. Soc, Providence, R.I. (1979), 209–212.
E.B. Vinberg, On reductive algebraic semi-groups, in “Lie Groups and Lie Algebras: E.B. Dynkin’s Seminar,” Amer. Math. Soc. Transl. Ser. 2, 169 (1995), 145–182.
J. Tate, Fourier analysis in number fields, and Hecke’s zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) (1967), 305–347.
N. Wallach, Real Reductive Groups I, Pure and Applied Mathematics, 132, Academic Press, Inc., Boston, MA, 1988.
A. Weil, Adéles and Algebraic Groups, Birkhäuser Progress in Mathematics 23, 1982.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Birkhäuser, Springer Basel AG
About this chapter
Cite this chapter
Braverman, A., Kazhdan, D. (2010). γ-Functions of Representations and Lifting. In: Alon, N., Bourgain, J., Connes, A., Gromov, M., Milman, V. (eds) Visions in Mathematics. Modern Birkhäuser Classics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0422-2_9
Download citation
DOI: https://doi.org/10.1007/978-3-0346-0422-2_9
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0421-5
Online ISBN: 978-3-0346-0422-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)