We consider a class of nonlocal operators associated with a compact Lie group G acting on a smooth manifold. A notion of symbol of such operators is introduced and an index formula for elliptic elements is obtained. The symbol in this situation is an element of a noncommutative algebra (crossed product by G) and to obtain an index formula, we define the Chern character for this algebra in the framework of noncommutative geometry.
@article{bwmeta1.element.doi-10_2478_s11533-011-0028-9,
author = {Anton Savin},
title = {On the index of nonlocal elliptic operators for compact Lie groups},
journal = {Open Mathematics},
volume = {9},
year = {2011},
pages = {833-850},
zbl = {1248.58013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0028-9}
}
Anton Savin. On the index of nonlocal elliptic operators for compact Lie groups. Open Mathematics, Tome 9 (2011) pp. 833-850. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0028-9/
[1] Antonevich A.B., Elliptic pseudodifferential operators with a finite group of shifts, Izv. Akad. Nauk SSSR Ser. Mat., 1973, 37(3), 663–675 (in Russian)
[2] Atiyah M.F., Elliptic Operators and Compact Groups, Lecture Notes in Math., 401, Springer, Berlin-New York, 1974
[3] Atiyah M.F., Segal G.B., The index of elliptic operators II, Ann. of Math., 1968, 87(3), 531–545 http://dx.doi.org/10.2307/1970716 | Zbl 0164.24201
[4] Baum P., Connes A., Chern character for discrete groups, In: A Fête of Topology, Academic Press, Boston, 1988, 163–232
[5] Berline N., Getzler E., Vergne M., Heat Kernels and Dirac Operators, Grundlehren Math. Wiss., 298, Springer, Berlin, 1992 | Zbl 0744.58001
[6] Berline N., Vergne M., The Chern character of a transversally elliptic symbol and the equivariant index, Invent. Math., 1996, 124(1–3), 11–49 http://dx.doi.org/10.1007/s002220050045 | Zbl 0847.46037
[7] Block J., Getzler E., Equivariant cyclic homology and equivariant differential forms, Ann. Sci. École Normale Sup., 1994, 27(4), 493–527 | Zbl 0849.55008
[8] Bredon G.E., Introduction to Compact Transformation Groups, Pure Appl. Math., 46, Academic Press, New York-London, 1972 | Zbl 0246.57017
[9] Brylinski J.-L., Nistor V., Cyclic cohomology of étale groupoids, K-Theory, 1994, 8(4), 341–365 http://dx.doi.org/10.1007/BF00961407 | Zbl 0812.19003
[10] Connes A., Cyclic cohomology and the transverse fundamental class of a foliation, In: Geometric Methods in Operator Algebras, Kyoto, 1983, Pitman Res. Notes Math. Ser., 123, Longman, Harlow, 1986, 52–144
[11] Connes A., Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math., 1985, 62, 257–360 http://dx.doi.org/10.1007/BF02698807
[12] Dave S., An equivariant noncommutative residue, preprint available at http://arxiv.org/abs/math/0610371 | Zbl 1286.58018
[13] Dave S., Equivariant homology for pseudodifferential operators, preprint available at http://arxiv.org/abs/1005.2282
[14] Karoubi M., Homologie Cyclique et K-théorie, Astérisque, 149, Soc. Math. France Inst. Henri Poincaré, Paris, 1987
[15] Kawasaki T., The index of elliptic operators over V-manifolds, Nagoya Math. J., 1981, 84, 135–157 | Zbl 0437.58020
[16] Kordyukov Yu.A., Index theory and non-commutative geometry on foliated manifolds, Russian Math. Surveys, 2009, 64(2), 273–391 http://dx.doi.org/10.1070/RM2009v064n02ABEH004616 | Zbl 1180.58006
[17] Koszul J.L., Sur certains groupes de transformations de Lie, In: Géométrie Différentielle, Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1953, Centre National de la Recherche Scientifique, Paris, 1953, 137–141
[18] Nazaikinskii V.E., Savin A.Yu., Sternin B.Yu., Elliptic Theory and Noncommutative Geometry, Oper. Theory Adv. Appl., 183, Birkhäuser, Basel, 2008 | Zbl 1158.58013
[19] Nistor V., Higher index theorems and the boundary map in cyclic cohomology, Doc. Math., 1997, 2, 263–295 | Zbl 0893.19002
[20] Pedersen G.K., C*-Algebras and Their Automorphism Groups, London Math. Soc. Monogr., 14, Academic Press, London-New York, 1979
[21] Quillen D., Superconnections and the Chern character, Topology, 1985, 24(1), 89–95 http://dx.doi.org/10.1016/0040-9383(85)90047-3
[22] Sternin B.Yu., On a class of nonlocal elliptic operators for compact Lie groups. Uniformization and finiteness theorem, Cent. Eur. J. Math., 2011, 9(4) | Zbl 1241.58012
[23] Tsygan B.L., The homology of matrix Lie algebras over rings and the Hochschild homology, Russian Math. Surveys, 1983, 38(2), 198–199 http://dx.doi.org/10.1070/RM1983v038n02ABEH003481 | Zbl 0526.17006
[24] Vergne M., Equivariant index formulas for orbifolds, Duke Math. J., 1996, 82(3), 637–652 http://dx.doi.org/10.1215/S0012-7094-96-08226-5 | Zbl 0874.57029
[25] Verona A., A de Rham type theorem for orbit spaces, Proc. Amer. Math. Soc., 1988, 104(1), 300–302 http://dx.doi.org/10.1090/S0002-9939-1988-0958087-8 | Zbl 0667.57021