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\(L^ 2\)-index and the Selberg trace formula. (English) Zbl 0537.58039
A method is developed for computing the \(L^ 2\)-index of a locally symmetric elliptic differential operator \(D_{\Gamma}\), acting on a locally symmetric manifold \(M_{\Gamma}=\Gamma \backslash G/K\) with G semisimple of real-rank 1 and \(\Gamma\) of finite co-volume, based on applying the Selberg trace formula to the difference of the two heat kernels associated to \(D_{\Gamma}\). The applications include an extension of the Osborne-Warner multiplicity formule to certain non- integrable discrete series and showing the existence, in some cases, of non-invariant \(L^ 2\)-cohomology classes in the middle dimension.
Reviewer: V.Deundjak

MSC:
58J20 Index theory and related fixed-point theorems on manifolds
22E46 Semisimple Lie groups and their representations
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[1] Arthur, J, Some tempered distributions on semisimple groups of real rank one, Ann. of math., 100, 553-584, (1974) · Zbl 0258.22014
[2] Atiyah, M.F; Bott, R; Patodi, V.K, On the heat equation and the index theorem, Inventiones math., 19, 279-330, (1973) · Zbl 0257.58008
[3] Atiyah, M.F; Patodi, V.K; Singer, I.M, Spectral asymmetry and Riemannian geometry. I, (), 43-69 · Zbl 0297.58008
[4] Atiyah, M.F; Schmid, W, A geometric construction of the discrete series for sem simple Lie groups, Inventiones math., 42, 1-62, (1977) · Zbl 0373.22001
[5] Atiyah, M.F; Singer, I.M; Atiyah, M.F; Singer, I.M, The index of elliptic operators. III, Ann. of math., Ann. of math., 87, 546-604, (1968) · Zbl 0164.24301
[6] Barbasch, D, Fourier inversion for unipotent invariant integrals, Trans. amer. math. soc, 249, 51-83, (1979) · Zbl 0378.22010
[7] Barbasch, D; Baldoni-Silva, W, The unitary dual for real-rank one semisimple Lie groups, Inventiones math., 72, 27-55, (1983) · Zbl 0561.22009
[8] Borel, A, Stable and L2-cohomology of arithmetic groups, Bull. amer. math. soc. (N. S.), 3, 1025-1027, (1980) · Zbl 0472.22002
[9] Bott, R, The index theorem for homogeneous differential operators, (), 167-185
[10] Connes, A; Moscovici, H, The L2-index theorem for homogeneous spaces of Lie groups, Ann. of math., 115, 291-330, (1982) · Zbl 0515.58031
[11] DeGeorge, D, On a theorem of osborne and warner. multiplicities in the cuspidal spectrum, J. funct. anal., 48, 81-94, (1982) · Zbl 0503.22008
[12] DeGeorge, D; Wallach, N, Limit formulas for multiplicities in L2(γβG), Ann. of math., 107, 133-150, (1978) · Zbl 0397.22007
[13] Gangolli, R; Warner, G, Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one, Nagoya math. J., 78, 1-44, (1980)
[14] Garland, H; Raghunathan, M.S, Fundamental domains for lattices in \(R\)-rank I semisimple Lie groups, Ann. of math., 92, 279-326, (1970) · Zbl 0206.03603
[15] Harish-Chandra; Harish-Chandra, Discrete series for semisimple Lie groups. II, Acta math., Acta math., 116, 1-111, (1966) · Zbl 0199.20102
[16] Hirzebruch, F, Automorphe formen und der satz von Riemann-Roch, (), 129-144
[17] Knapp, A.W; Stein, E.M, Intertwining operators for semisimple groups, Ann. of math., 93, 849, (1971) · Zbl 0257.22015
[18] Langlands, R.P, On the functional equations satisfied by Eisenstein series, () · Zbl 0204.09603
[19] Miatello, R.J, The minakshisundaram-pleijel coefficients for the vector valued heat kernel on compact locally symmetric spaces of negative curvature, Trans. amer. math. soc., 260, 1-33, (1980) · Zbl 0444.58015
[20] Miatello, R.J; Miatello, R.J, Alternating sum formulas for multiplicities in L2(γβG). II, Trans. amer. math. soc., Math. Z., 182, 35-44, (1983) · Zbl 0489.22016
[21] Moscovici, H, L2-index of elliptic operators on locally symmetric spaces of finite volume, (), 129-138
[22] Müller, W, Spectral theory of non-compact Riemannian manifolds with cusps and a related trace formula, (1980), I.H.E.S, Preprint
[23] Nelson, E, Analytic vectors, Ann. of math., 70, 572-615, (1959) · Zbl 0091.10704
[24] Osborne, M.S; Warner, G, Multiplicities of the integrable discrete series: the case of a non-uniform lattice in an \(R\)-rank one semi-simple group, J. funct. anal., 30, 287-310, (1978) · Zbl 0397.22002
[25] Parthasarathy, R, Dirac operators and the discrete series, Ann. of math., 96, 1-30, (1972) · Zbl 0249.22003
[26] Poulsen, N.S, On C∞-vectors and intertwining bilinear forms for representations of Lie groups, J. funct. anal., 9, 87-120, (1972) · Zbl 0237.22013
[27] Ragozin, D; Warner, G, On a method for computing multiplicities in L2(γβG), ()
[28] Sally, P; Warner, G, The Fourier transform on semisimple Lie groups of real-rank one, Acta math., 131, 1-26, (1973) · Zbl 0305.43007
[29] Schmid, W, Some properties of square-integrable representations of semisimple Lie groups, Ann. of math., 102, 535-564, (1975) · Zbl 0347.22011
[30] Selberg, A, Harmonic analysis and discontinuous groups in weakly symmetric spaces with applications to Dirichlet series, J. Indian math. soc., 20, 47-87, (1956) · Zbl 0072.08201
[31] Varadarajan, V.S, The theory of characters and the discrete series for semisimple Lie groups, (), 45-100
[32] {\scD. Vogan and G. Zuckerman}, Unitary representations with cohomology, preprint. · Zbl 0692.22008
[33] Warner, G, Selberg’s trace formula for nonuniform lattices: the \(R\)-rank one case, Advan. in math. suppl. stud., Vol. 6, 1-142, (1979)
[34] {\scF. Williams}, Discrete series multiplicities in L2(ΓβG), Amer. J. Math.
[35] Zucker, S, L2-cohomology of warped products and arithmetic groups, Inventiones math., 70, 169-218, (1982) · Zbl 0508.20020
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