# The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part II

### Abstract

We present a new solution to the index problem for hypoelliptic operators in the Heisenberg calculus on contact manifolds, by constructing the appropriate topological $K$-theory cocycle for such operators. Its Chern character gives a cohomology class to which the Atiyah-Singer index formula can be applied. Such a $K$-cocycle has already been constructed by Boutet de Monvel for Toeplitz operators, and, more recently, by Melrose and Epstein for the class of Hermite operators. Our construction applies to general hypoelliptic pseudodifferential operators in the Heisenberg calculus. As in the Hermite Index Formula of Melrose and Epstein, our construction gives a vector bundle automorphism of the symmetric tensors of the contact hyperplane bundle. This automorphism is constructed directly from the invertible Heisenberg symbol of the operator, and is easily computed in the case of differential operators.

• [BG] R. Beals and P. Greiner, Calculus on Heisenberg manifolds, Princeton, NJ: Princeton Univ. Press, 1988.
@book {BG, MRKEY = {953082},
AUTHOR = {Beals, Richard and Greiner, Peter},
TITLE = {Calculus on {H}eisenberg manifolds},
SERIES = {Annals of Math. Studies},
NUMBER = {119},
PUBLISHER = {Princeton Univ. Press},
YEAR = {1988},
PAGES = {x+194},
ISBN = {0-691-08500-5; 0-691-08501-3},
MRCLASS = {35S05 (32F20 35-02 47G05 58G15)},
MRNUMBER = {89m:35223},
MRREVIEWER = {Kenneth G. Miller},
ZBLNUMBER = {0654.58033},
}
• [Bo] L. Boutet de Monvel, "On the index of Toeplitz operators of several complex variables," Invent. Math., vol. 50, iss. 3, pp. 249-272, 1978/79.
@article {Bo, MRKEY = {520928},
AUTHOR = {Boutet de Monvel, Louis},
TITLE = {On the index of {T}oeplitz operators of several complex variables},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {50},
YEAR = {1978/79},
NUMBER = {3},
PAGES = {249--272},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {58G10 (32F20 58B15)},
MRNUMBER = {80j:58063},
MRREVIEWER = {V. S. Rabinovich},
DOI = {10.1007/BF01410080},
ZBLNUMBER = {0398.47018},
}
• [CGGP] M. Christ, D. Geller, P. Głowacki, and L. Polin, "Pseudodifferential operators on groups with dilations," Duke Math. J., vol. 68, iss. 1, pp. 31-65, 1992.
@article {CGGP, MRKEY = {1185817},
AUTHOR = {Christ, Michael and Geller, Daryl and G{\l}owacki, Pawe{\l} and Polin, Larry},
TITLE = {Pseudodifferential operators on groups with dilations},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {68},
YEAR = {1992},
NUMBER = {1},
PAGES = {31--65},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {35S05 (22E30 47G30 58G15)},
MRNUMBER = {94b:35316},
MRREVIEWER = {Xue Bo Luo},
DOI = {10.1215/S0012-7094-92-06802-5},
ZBLNUMBER = {0764.35120},
}
• [EM] C. Epstein and R. Melrose, "Contact degree and the index of Fourier integral operators," Math. Res. Lett., vol. 5, iss. 3, pp. 363-381, 1998.
@article {EM, MRKEY = {1637844},
AUTHOR = {Epstein, Charles and Melrose, Richard},
TITLE = {Contact degree and the index of {F}ourier integral operators},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {5},
YEAR = {1998},
NUMBER = {3},
PAGES = {363--381},
ISSN = {1073-2780},
MRCLASS = {58J20 (35S30 58J40)},
MRNUMBER = {2000a:58062},
MRREVIEWER = {Rafe Mazzeo},
ZBLNUMBER = {0929.58012},
}
• [EM2] C. Epstein and R. Melrose, The Heisenberg algebra, index theory and homology, 2003.
@misc{EM2,
author={Epstein, Charles and Melrose, Richard},
TITLE={The Heisenberg algebra, index theory and homology},
NOTE={preprint},
YEAR={2003},
}
• [Ep] C. Epstein, "Lectures on indices and relative indices on contact and CR-manifolds," in Woods Hole Mathematics, World Sci. Publ., Hackensack, NJ, 2004, pp. 27-93.
@incollection {Ep, MRKEY = {2123367},
AUTHOR = {Epstein, Charles},
TITLE = {Lectures on indices and relative indices on contact and {CR}-manifolds},
BOOKTITLE = {Woods {H}ole Mathematics},
SERIES = {Ser. Knots Everything},
NUMBER = {34},
PAGES = {27--93},
PUBLISHER = {World Sci. Publ., Hackensack, NJ},
YEAR = {2004},
MRCLASS = {58J20 (32V20 47B35 47G30)},
MRNUMBER = {2006a:58028},
MRREVIEWER = {Evgeni{\u\i} V. Troitski{\u\i}},
}
• [Er1] E. van Erp, "The Atiyah-Singer ormula for subelliptic operators on a contact manifold. Part I," Ann. of Math., vol. 171, pp. 1647-1681, 2010.
@article{Er1,
author={van Erp, E.},
TITLE={The Atiyah-Singer ormula for subelliptic operators on a contact manifold. Part I},
JOURNAL={Ann. of Math.},
VOLUME={171},
YEAR={2010},
PAGES={1647--1681},
}
• [FS1] G. B. Folland and E. M. Stein, "Estimates for the $\bar \partial _{b}$ complex and analysis on the Heisenberg group," Comm. Pure Appl. Math., vol. 27, pp. 429-522, 1974.
@article {FS1, MRKEY = {0367477},
AUTHOR = {Folland, G. B. and Stein, E. M.},
TITLE = {Estimates for the {$\bar \partial \sb{b}$} complex and analysis on the {H}eisenberg group},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {27},
YEAR = {1974},
PAGES = {429--522},
ISSN = {0010-3640},
MRCLASS = {35N15 (22E30 32K15 47G05)},
MRNUMBER = {51 \#3719},
MRREVIEWER = {S. G. Gindikin},
DOI = {10.1002/cpa.3160270403},
ZBLNUMBER = {0293.35012},
}
• [Ta1] M. E. Taylor, "Noncommutative Microlocal Analysis. I." , 1984, vol. 52, p. iv.
@incollection{Ta1, MRKEY = {764508},
AUTHOR = {Taylor, Michael E.},
TITLE = {{\it Noncommutative Microlocal Analysis}. {\rm I}},
SERIES = {Mem. Amer. Math. Soc.},
FJOURNAL = {Memoirs of the American Mathematical Society},
VOLUME = {52},
YEAR = {1984},
PAGES = {iv+182},
ISSN = {0065-9266},
CODEN = {MAMCAU},
MRCLASS = {58G15 (35A27 35H05 35S99)},
MRNUMBER = {86f:58156},
MRREVIEWER = {Gunther A. Uhlmann},
ZBLNUMBER={0554.35025},
}

## Authors

Erik van Erp

Dartmouth College
Department of Mathematics
Kemeny Hall, 27 N. Main St.
Hanover, NH 03755-3551
United States