×

zbMATH — the first resource for mathematics

A local trace formula. (English) Zbl 0741.22013
This paper develops an idea that was put forward about 10 years ago by D. A. Kazhdan. According to this for a locally compact group \(G\) the diagonal embedding of \(G\) into \(G\times G\) is analogous to that of a discrete group \(\Gamma\) into a locally compact group \(H\). In particular if \(F\) is a local field and \(G\) consists of the \(F\)-points of a linear reductive group defined over \(F\) then the decomposition of \(L^ 2(G\setminus G\times G)\) into irreducible representations is given by Harish-Chandra’s Plancherel theorem. The spectrum consists of a discrete part and a continuous part. One can therefore ask for a description of the character of the representation on the discrete part, analogous to the formula given by the Selberg trace formula. If \(G\) is compact then the formula follows from the Peter-Weyl theorem.
The author solves precisely this problem. His final formula is an identity between a sum of distributions parametrized by elliptic (and so semisimple) conjugacy classes in \(G\) and in the Levi components of its parabolic subgroups, and a sum of distributions parametrized by the discrete spectrum of \(G\) and also that of the Levi components of the parabolic subgroups. The author points out that the distributions are not invariant except in some special cases but asserts that it is not difficult to find an invariant form. The proof of the formula makes use of Harish-Chandra’s theory and techniques developed by the author in his work on the general trace formula.

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] J. Arthur, The characters of discrete series as orbital integrals,Invent. Math.,32 (1976), 205–261. · Zbl 0359.22008
[2] J. Arthur, A trace formula for reductive groups I: terms associated to classes in G(Q),Duke Math. J.,45 (1978), 911–952. · Zbl 0499.10032
[3] J. Arthur, A trace formula for reductive groups II: applications of a truncation operator,Compos. Math.,40 (1980), 87–121. · Zbl 0499.10033
[4] J. Arthur, The trace formula in invariant form,Ann. of Math.,114 (1981), 1–74. · Zbl 0495.22006
[5] J. Arthur, On the inner product of truncated Eisenstein series,Duke Math. J.,49 (1982), 35–70. · Zbl 0518.22012
[6] J. Arthur, On a family of distributions obtained from Eisenstein series II: Explicit formulas,Amer. J. Math.,104 (1982), 1289–1336. · Zbl 0562.22004
[7] J. Arthur, A Paley-Wiener theorem for real reductive groups,Acta Math.,150 (1983), 1–89. · Zbl 0514.22006
[8] J. Arthur, The local behaviour of weighted orbital integrals,Duke Math. J.,56 (1988), 223–293. · Zbl 0649.10020
[9] J. Arthur, The characters of supercuspidal representations as weighted orbital integrals,Proc. Indian Acad. Sci.,97 (1987), 3–19. · Zbl 0652.22009
[10] J. Arthur, The invariant trace formula I. Local theory,J. Amer. Math. Soc.,1 (1988), 323–383. · Zbl 0682.10021
[11] J. Arthur, Intertwining operators and residues I. Weighted characters,J. Funct. Anal.,84 (1989), 19–84. · Zbl 0679.22011
[12] J. Arthur, The trace formula and Hecke operators, inNumber Theory, Trace Formulas and Discrete Groups, Academic Press, 1989, 11–27.
[13] J. Arthur, Towards a local trace formula, inAlgebraic Analysis, Geometry and Number Theory, The Johns Hopkins University Press, 1989, 1–24.
[14] J. Arthur, Some problems in local harmonic analysis, to appear inHarmonic Analysis on Reductive Groups, Birkhäuser.
[15] Harish-Chandra, A formula for semisimple Lie groups,Amer. J. Math.,79 (1957), 733–760. · Zbl 0080.10201
[16] Harish-Chandra, Spherical functions on a semisimple Lie group. I,Amer. J. Math.,80 (1958), 241–310. · Zbl 0093.12801
[17] Harish-Chandra, Two theorems on semisimple Lie groups,Ann. of Math.,83 (1966), 74–128. · Zbl 0199.46403
[18] Harish-Chandra, Harmonic Analysis on Reductivep-adic Groups,Springer Lecture Notes 162, 1970. · Zbl 0202.41101
[19] Harish-Chandra, Harmonic analysis on reductivep-adic groups, in Harmonic Analysis on Homogeneous Spaces,Proc. Sympos. Pure Math.,26, A.M.S., 1973, 167–192. · Zbl 0289.22018
[20] Harish-Chandra, Harmonic analysis on real reductive groups I. The theory of the constant term,J. Funct. Anal.,19 (1975), 104–204. · Zbl 0315.43002
[21] Harish-Chandra, Harmonic analysis on real reductive groups II. Wave packets in the Schwartz space,Invent. Math.,36 (1976), 1–55. · Zbl 0341.43010
[22] Harish-Chandra, Harmonic analysis on real reductive groups III. The Maass-Selberg relations and the Plancherel formula,Ann. of Math.,104 (1976), 117–201. · Zbl 0331.22007
[23] Harish-Chandra, The Plancherel formula for reductivep-adic groups, inCollected Papers, Vol. IV, Springer-Verlag, 353–367.
[24] S. Helgason,Differential Geometry and Symmetric Spaces, Academic Press, 1962. · Zbl 0111.18101
[25] D. Keys, L-indistinguishability and R-groups for quasi-split groups: Unitary groups of even dimension,Ann. Scient. Éc. Norm. Sup., 4e Sér., 20 (1987), 31–64. · Zbl 0634.22014
[26] R. P. Langlands, Eisenstein series, the trace formula and the modern theory of automorphic forms, inNumber Theory, Trace Formulas and Discrete Groups, Academic Press, 1989, 125–155.
[27] I. G. Macdonald,Spherical Functions on a Group of p-Adic Type, Publications of the Ramanujan Institute, Madras, 1971. · Zbl 0302.43018
[28] W. Müller, The trace class conjecture in the theory of automorphic forms,Ann. of Math.,130 (1989), 473–529. · Zbl 0701.11019
[29] A. Silberger,Introduction to Harmonic Analysis on Reductive p-Adic Groups, Mathematical Notes, Princeton University Press, 1979. · Zbl 0458.22006
[30] J. Tits, Reductive groups over local fields, inAutomorphic Forms, Representations and L-functions, Proc. Sympos. Pure Math.,33, Part I, A.M.S., 1979, 29–69.
[31] J. L. Waldspurger, Intégrales orbitales sphériques pour GL(N) sur un corpsp-adique,Astérisque, 171–172 (1989), 279–337.
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.