# American Institute of Mathematical Sciences

2010, 17: 138-154. doi: 10.3934/era.2010.17.138

## The equivariant index theorem for transversally elliptic operators and the basic index theorem for Riemannian foliations

 1 Institut für Mathematik, Humboldt Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany 2 Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, United States 3 Department of Mathematics, Texas Christian University, Fort Worth, Texas 76129, United States

Received  August 2010 Revised  October 2010 Published  December 2010

In this expository paper, we explain a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a $G$-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications is an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.
Citation: Jochen Brüning, Franz W. Kamber, Ken Richardson. The equivariant index theorem for transversally elliptic operators and the basic index theorem for Riemannian foliations. Electronic Research Announcements, 2010, 17: 138-154. doi: 10.3934/era.2010.17.138
##### References:
 [1] P. Albin and R. Melrose, Equivariant cohomology and resolution,, preprint \arXiv{0907.3211v2} [math.DG]., (). Google Scholar [2] M. F. Atiyah, "Elliptic Operators and Compact Groups,", Lecture Notes in Mathematics \textbf{401}, 401 (1974). Google Scholar [3] M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry. I,, Math. Proc. Camb. Phil. Soc., 77 (1975), 43. doi: 10.1017/S0305004100049410. Google Scholar [4] M. F. Atiyah and G. B. Segal, The index of elliptic operators. II,, Ann. of Math. (2), 87 (1968), 531. doi: 10.2307/1970716. Google Scholar [5] N. Berline, E. Getzler and M. Vergne, "Heat Kernels and Dirac Operators,", Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 298 (1992). Google Scholar [6] N. Berline and M. Vergne, The Chern character of a transversally elliptic symbol and the equivariant index,, Invent. Math., 124 (1996), 11. doi: 10.1007/s002220050045. Google Scholar [7] N. Berline and M. Vergne, L'indice équivariant des opérateurs transversalement elliptiques,, (French) [The equivariant index of transversally elliptic operators], 124 (1996), 51. doi: 10.1007/s002220050046. Google Scholar [8] B. Booss-Bavnbek and K. P. Wojciechowski, "Elliptic Boundary Problems for Dirac Operators,", Mathematics: Theory & Applications. Birkhäuser Boston, (1993). Google Scholar [9] G. Bredon, "Introduction to Compact Transformation Groups,", Pure and Applied Mathematics, (1972). Google Scholar [10] J. Brüning and E. Heintze, Representations of compact Lie groups and elliptic operators,, Inv. Math., 50 (): 169. doi: 10.1007/BF01390288. Google Scholar [11] J. Brüning and E. Heintze, The asymptotic expansion of Minakshisundaram-Pleijel in the equivariant case,, Duke Math. Jour., 51 (1984), 959. Google Scholar [12] J. Brüning, F. W. Kamber and K. Richardson, The eta invariant and equivariant index of transversally elliptic operators,, preprint \arXiv{1005.3845v1} [math.DG]., (). Google Scholar [13] J. Brüning, F. W. Kamber and K. Richardson, Index theory for basic Dirac operators on Riemannian foliations,, preprint \arXiv{1008.1757v1} [math.DG]., (). Google Scholar [14] H. Donnelly, Eta invariants for $G$-spaces,, Indiana Univ. Math. J., 27 (1978), 889. doi: 10.1512/iumj.1978.27.27060. Google Scholar [15] A. El Kacimi-Alaoui, Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications,, (French) [Transversely elliptic operators on a Riemannian foliation, 73 (1990), 57. Google Scholar [16] A. El Kacimi, G. Hector and V. Sergiescu, La cohomologie basique d'un feuilletage riemannien est de dimension finie,, (French) [The basic cohomology of a Riemannian foliation is finite-dimensional], 188 (1985), 593. doi: 10.1007/BF01161658. Google Scholar [17] J. F. Glazebrook and F. W. Kamber, Transversal Dirac families in Riemannian foliations,, Comm. Math. Phys., 140 (1991), 217. doi: 10.1007/BF02099498. Google Scholar [18] A. Gorokhovsky and J. Lott, The index of a transverse Dirac-type operator: The case of abelian Molino sheaf,, preprint \arXiv{1005.0161v2} [math.DG]., (). Google Scholar [19] G. Habib and K. Richardson, A brief note on the spectrum of the basic Dirac operator,, Bull. London Math. Soc., 41 (2009), 683. doi: 10.1112/blms/bdp042. Google Scholar [20] F. W. Kamber and K. Richardson, $G$-equivariant vector bundles on manifolds of one orbit type,, preprint., (). Google Scholar [21] F. W. Kamber and Ph. Tondeur, "Foliated Bundles and Characteristic Classes,", Lecture Notes in Math. \textbf{493}, 493 (1975). Google Scholar [22] F. W. Kamber and Ph. Tondeur, Foliations and metrics,, Differential geometry (College Park, 32 (1981), 103. Google Scholar [23] F. W. Kamber and Ph. Tondeur, Duality for Riemannian foliations,, Singularities, (1981), 609. Google Scholar [24] F. W. Kamber and Ph. Tondeur, De Rham-Hodge theory for Riemannian foliations,, Math. Ann., 277 (1987), 415. doi: 10.1007/BF01458323. Google Scholar [25] K. Kawakubo, "The Theory of Transformation Groups,", Translated from the 1987 Japanese edition. The Clarendon Press, (1987). Google Scholar [26] T. Kawasaki, The index of elliptic operators over $V$ -manifolds,, Nagoya Math. J., 84 (1981), 135. Google Scholar [27] H. B. Lawson and M.-L. Michelsohn, "Spin Geometry,", Princeton Mathematical Series, (1989). Google Scholar [28] P. Molino, "Riemannian Foliations,", Translated from the French by Grant Cairns. With appendices by Cairns, (1988). Google Scholar [29] I. Prokhorenkov and K. Richardson, Natural equivariant Dirac operators,, to appear in Geom. Dedicata, (). Google Scholar [30] B. Reinhart, "Differential Geometry of Foliations -- The Fundamental Integrability Problem,", Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], (1983). Google Scholar [31] Ph. Tondeur, "Geometry of Foliations,", Monographs in Mathematics, 90 (1997). Google Scholar

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##### References:
 [1] P. Albin and R. Melrose, Equivariant cohomology and resolution,, preprint \arXiv{0907.3211v2} [math.DG]., (). Google Scholar [2] M. F. Atiyah, "Elliptic Operators and Compact Groups,", Lecture Notes in Mathematics \textbf{401}, 401 (1974). Google Scholar [3] M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry. I,, Math. Proc. Camb. Phil. Soc., 77 (1975), 43. doi: 10.1017/S0305004100049410. Google Scholar [4] M. F. Atiyah and G. B. Segal, The index of elliptic operators. II,, Ann. of Math. (2), 87 (1968), 531. doi: 10.2307/1970716. Google Scholar [5] N. Berline, E. Getzler and M. Vergne, "Heat Kernels and Dirac Operators,", Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 298 (1992). Google Scholar [6] N. Berline and M. Vergne, The Chern character of a transversally elliptic symbol and the equivariant index,, Invent. Math., 124 (1996), 11. doi: 10.1007/s002220050045. Google Scholar [7] N. Berline and M. Vergne, L'indice équivariant des opérateurs transversalement elliptiques,, (French) [The equivariant index of transversally elliptic operators], 124 (1996), 51. doi: 10.1007/s002220050046. Google Scholar [8] B. Booss-Bavnbek and K. P. Wojciechowski, "Elliptic Boundary Problems for Dirac Operators,", Mathematics: Theory & Applications. Birkhäuser Boston, (1993). Google Scholar [9] G. Bredon, "Introduction to Compact Transformation Groups,", Pure and Applied Mathematics, (1972). Google Scholar [10] J. Brüning and E. Heintze, Representations of compact Lie groups and elliptic operators,, Inv. Math., 50 (): 169. doi: 10.1007/BF01390288. Google Scholar [11] J. Brüning and E. Heintze, The asymptotic expansion of Minakshisundaram-Pleijel in the equivariant case,, Duke Math. Jour., 51 (1984), 959. Google Scholar [12] J. Brüning, F. W. Kamber and K. Richardson, The eta invariant and equivariant index of transversally elliptic operators,, preprint \arXiv{1005.3845v1} [math.DG]., (). Google Scholar [13] J. Brüning, F. W. Kamber and K. Richardson, Index theory for basic Dirac operators on Riemannian foliations,, preprint \arXiv{1008.1757v1} [math.DG]., (). Google Scholar [14] H. Donnelly, Eta invariants for $G$-spaces,, Indiana Univ. Math. J., 27 (1978), 889. doi: 10.1512/iumj.1978.27.27060. Google Scholar [15] A. El Kacimi-Alaoui, Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications,, (French) [Transversely elliptic operators on a Riemannian foliation, 73 (1990), 57. Google Scholar [16] A. El Kacimi, G. Hector and V. Sergiescu, La cohomologie basique d'un feuilletage riemannien est de dimension finie,, (French) [The basic cohomology of a Riemannian foliation is finite-dimensional], 188 (1985), 593. doi: 10.1007/BF01161658. Google Scholar [17] J. F. Glazebrook and F. W. Kamber, Transversal Dirac families in Riemannian foliations,, Comm. Math. Phys., 140 (1991), 217. doi: 10.1007/BF02099498. Google Scholar [18] A. Gorokhovsky and J. Lott, The index of a transverse Dirac-type operator: The case of abelian Molino sheaf,, preprint \arXiv{1005.0161v2} [math.DG]., (). Google Scholar [19] G. Habib and K. Richardson, A brief note on the spectrum of the basic Dirac operator,, Bull. London Math. Soc., 41 (2009), 683. doi: 10.1112/blms/bdp042. Google Scholar [20] F. W. Kamber and K. Richardson, $G$-equivariant vector bundles on manifolds of one orbit type,, preprint., (). Google Scholar [21] F. W. Kamber and Ph. Tondeur, "Foliated Bundles and Characteristic Classes,", Lecture Notes in Math. \textbf{493}, 493 (1975). Google Scholar [22] F. W. Kamber and Ph. Tondeur, Foliations and metrics,, Differential geometry (College Park, 32 (1981), 103. Google Scholar [23] F. W. Kamber and Ph. Tondeur, Duality for Riemannian foliations,, Singularities, (1981), 609. Google Scholar [24] F. W. Kamber and Ph. Tondeur, De Rham-Hodge theory for Riemannian foliations,, Math. Ann., 277 (1987), 415. doi: 10.1007/BF01458323. Google Scholar [25] K. Kawakubo, "The Theory of Transformation Groups,", Translated from the 1987 Japanese edition. The Clarendon Press, (1987). Google Scholar [26] T. Kawasaki, The index of elliptic operators over $V$ -manifolds,, Nagoya Math. J., 84 (1981), 135. Google Scholar [27] H. B. Lawson and M.-L. Michelsohn, "Spin Geometry,", Princeton Mathematical Series, (1989). Google Scholar [28] P. Molino, "Riemannian Foliations,", Translated from the French by Grant Cairns. With appendices by Cairns, (1988). Google Scholar [29] I. Prokhorenkov and K. Richardson, Natural equivariant Dirac operators,, to appear in Geom. Dedicata, (). Google Scholar [30] B. Reinhart, "Differential Geometry of Foliations -- The Fundamental Integrability Problem,", Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], (1983). Google Scholar [31] Ph. Tondeur, "Geometry of Foliations,", Monographs in Mathematics, 90 (1997). Google Scholar
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