zbMATH — the first resource for mathematics

\(L^ 2\)-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

58J20 Index theory and related fixed-point theorems on manifolds
22E46 Semisimple Lie groups and their representations
Full Text: DOI
[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
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.