##
**The Atiyah-Singer index formula for subelliptic operators on contact manifolds. II.**
*(English)*
Zbl 1206.19005

The present paper is a sequel to part I [E. van Erp, Ann. Math. (2) 171, No. 3, 1647–1681 (2010; Zbl 1206.19004)]. There, an index theorem of the form

\[ \mathrm{Index}(P)=\int_{TM}\mathrm{Ch}[\sigma_H(P)] \wedge \mathrm{Td}(M) \]

for a subelliptic operator \(P\) on a closed odd-dimensional contact manifold \(M\) was proven. However, the element \([\sigma_H(P)] \in K^0(T^*M)\) was defined in terms of an abstract isomorphism and not given explicitly. Here the author constructs a symbol class \([\sigma_H(P)] \in K^1(M)\) in a different way: The operator \(P\) defines for each \(m \in M\) a model operator \(P_m\) given by a distribution on a Heisenberg group \(G_m\). (These Heisenberg groups form the tangent groupoid \(T_HM\).) Via the Bargmann-Fok representation \(\pi_m\) the bundle of Heisenberg groups is represented on a bundle of Hilbert spaces \(V^{BF}\). The automorphism \(a(P)_m:=\pi_m(P_m)\pi_m(P_m^{\mathrm{op}})^{-1}\) on \(V_m^{BF}\) is invertible and \(a(P)_m-1\) is compact. One gets a well-defined element \([a(P),V^{BF}] \in K_1(C(M))\cong K^1(M)\).

In order to obtain an element directly in topological \(K\)-theory, the author constructs a finite-dimensional approximation of \(a(P)\): If \(H\) is the hyperplane bundle of the contact manifold endowed with a compatible almost complex structure, define for \(N \in {\mathbb N}\) the bundle

\[ V^N=\bigoplus_{j=0}^N \mathrm{Sym}^j H^{1,0}\subset V^{BF} \]

and let \(e_N:V^{BF} \to V^N\) be the projection. For \(N\) large one has \([e_N a(P)e_N,V^N]=[a(P),V^{BF}]\). The main result of the present paper says that for \(N\) large

\[ \mathrm{Index}(P)=\int_{TM}\mathrm{Ch}[e_N a(P)e_N,V^N] \wedge \mathrm{Td}(M). \]

The proof is based on the groupoid methods developed in the previous paper. It is first done for Hermite operators, from which the general case is derived. On the way, an Atiyah-Singer type exact sequence for the Heisenberg calculus is proven. The index theorem generalizes Boutet de Mouvel’s index theorem for Toeplitz operators on contact manifolds and (partially unpublished) results of Epstein and Melrose, among them the index theorem for Hermite operators.

\[ \mathrm{Index}(P)=\int_{TM}\mathrm{Ch}[\sigma_H(P)] \wedge \mathrm{Td}(M) \]

for a subelliptic operator \(P\) on a closed odd-dimensional contact manifold \(M\) was proven. However, the element \([\sigma_H(P)] \in K^0(T^*M)\) was defined in terms of an abstract isomorphism and not given explicitly. Here the author constructs a symbol class \([\sigma_H(P)] \in K^1(M)\) in a different way: The operator \(P\) defines for each \(m \in M\) a model operator \(P_m\) given by a distribution on a Heisenberg group \(G_m\). (These Heisenberg groups form the tangent groupoid \(T_HM\).) Via the Bargmann-Fok representation \(\pi_m\) the bundle of Heisenberg groups is represented on a bundle of Hilbert spaces \(V^{BF}\). The automorphism \(a(P)_m:=\pi_m(P_m)\pi_m(P_m^{\mathrm{op}})^{-1}\) on \(V_m^{BF}\) is invertible and \(a(P)_m-1\) is compact. One gets a well-defined element \([a(P),V^{BF}] \in K_1(C(M))\cong K^1(M)\).

In order to obtain an element directly in topological \(K\)-theory, the author constructs a finite-dimensional approximation of \(a(P)\): If \(H\) is the hyperplane bundle of the contact manifold endowed with a compatible almost complex structure, define for \(N \in {\mathbb N}\) the bundle

\[ V^N=\bigoplus_{j=0}^N \mathrm{Sym}^j H^{1,0}\subset V^{BF} \]

and let \(e_N:V^{BF} \to V^N\) be the projection. For \(N\) large one has \([e_N a(P)e_N,V^N]=[a(P),V^{BF}]\). The main result of the present paper says that for \(N\) large

\[ \mathrm{Index}(P)=\int_{TM}\mathrm{Ch}[e_N a(P)e_N,V^N] \wedge \mathrm{Td}(M). \]

The proof is based on the groupoid methods developed in the previous paper. It is first done for Hermite operators, from which the general case is derived. On the way, an Atiyah-Singer type exact sequence for the Heisenberg calculus is proven. The index theorem generalizes Boutet de Mouvel’s index theorem for Toeplitz operators on contact manifolds and (partially unpublished) results of Epstein and Melrose, among them the index theorem for Hermite operators.

Reviewer: Charlotte Wahl (Hannover)

### MSC:

19K56 | Index theory |

53D10 | Contact manifolds (general theory) |

58J20 | Index theory and related fixed-point theorems on manifolds |

### Keywords:

index theory; subelliptic operators; contact manifolds; Heisenberg calculus; Hermite operators; Heisenberg groups; tangent groupoid### Citations:

Zbl 1206.19004### References:

[1] | R. Beals and P. Greiner, Calculus on Heisenberg manifolds, Princeton, NJ: Princeton Univ. Press, 1988. · Zbl 0654.58033 · doi:10.1515/9781400882397 |

[2] | 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. · Zbl 0398.47018 · doi:10.1007/BF01410080 |

[3] | 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. · Zbl 0764.35120 · doi:10.1215/S0012-7094-92-06802-5 |

[4] | 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. · Zbl 0929.58012 · doi:10.4310/MRL.1998.v5.n3.a9 |

[5] | C. Epstein and R. Melrose, The Heisenberg algebra, index theory and homology, 2003. · Zbl 0929.58012 |

[6] | 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. |

[7] | 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. · Zbl 1206.19004 · doi:10.4007/annals.2010.171.1647 |

[8] | 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. · Zbl 0293.35012 · doi:10.1002/cpa.3160270403 |

[9] | M. E. Taylor, ”Noncommutative Microlocal Analysis. I.” , 1984, vol. 52, p. iv. · Zbl 0554.35025 |

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