# Hypoelliptic Laplacian and Orbital Integrals (AM-177)

Jean-Michel Bismut
Pages: 320
https://www.jstor.org/stable/j.ctt7rrxw

1. Front Matter
(pp. i-vi)
(pp. vii-xii)
3. Introduction
(pp. 1-11)

The purpose of this book is to use the hypoelliptic Laplacian to evaluate semisimple orbital integrals, in a formalism that unifies index theory and the trace formula.

Let us explain how to think formally of such a unified treatment, while allowing ourselves a temporarily unbridled use of mathematical analogies. LetXbe a compact Riemannian manifold, and let ΔXbe the corresponding Laplace-Beltrami operator. Fort> 0, consider the trace of the heat kernel$\text{Tr} \left[ \exp \left( {t{{\Delta }^{X}}}/{2} \right) \right]$. If$L_{2}^{X}$is the Hilbert space of square-integrable functions onX,$\text{Tr} \left[ \exp \left( {t{{\Delta }^{X}}}/{2} \right) \right]$is the trace of the ‘group element’$\exp \left( {t{{\Delta }^{X}}}/{2} \right)$acting on$L_{2}^{X}$....

4. Chapter One Clifford and Heisenberg algebras
(pp. 12-21)

The purpose of this chapter is to recall various results on Clifford algebras and Heisenberg algebras. The results of this chapter will be used in our construction of the hypoelliptic Laplacian over a symmetric space.

This chapter is organized as follows. In section 1.1, we introduce the Clifford algebra of a vector spaceVequipped with a symmetric bilinear formB.

In section 1.2, we specialize the construction of the Clifford algebra to the case ofVV*.

In section 1.3, if (V,ω) is a symplectic vector space, we construct the associated Heisenberg algebra.

In section 1.4, we...

5. Chapter Two The hypoelliptic Laplacian on X = G/K
(pp. 22-47)

The purpose of this chapter is to construct the hypoelliptic Laplacian${\mathcal{L}}_{b}^{X},b > 0$acting on the total space of a vector bundle$TX \oplus N \simeq \mathfrak{g}$overX=G/K. The operator${\mathcal{L}}_{b}^{X}$will be obtained using general constructions involving Clifford algebras and Heisenberg algebras, and also the Dirac operator of Kostant [Ko97].

This chapter is organized as follows. In section 2.1, we introduce a pair (G,K), the symmetric spaceX=G/K, and the vector bundlesTX,NonX.

In section 2.2, we construct the canonical at connection on$TX \oplus N \simeq \mathfrak{g}.$

In section 2.3, we define Clifford algebras associated with$\mathfrak{g}.$

In...

6. Chapter Three The displacement function and the return map
(pp. 48-75)

The purpose of this chapter is to study the displacement functiondγonXthat is associated with a semisimple element γ ∈G. If φt,t∈ R denotes the geodesic flow on the total space χ of the tangent bundle ofX, the critical setX(γ) ⊂Xofdγcan be easily related to the fixed point setFγ⊂ χ of the symplectic transformation γ−1φ1of χ. We study the nondegeneracy of γ−1φ1− 1 alongFγ. More fundamentally, we give important quantitative estimates on how much φ1/2differs from φ−1/2γ away fromFγ. These...

7. Chapter Four Elliptic and hypoelliptic orbital integrals
(pp. 76-91)

The purpose of this chapter is to construct semisimple orbital integrals associated with the heat kernel for the hypoelliptic Laplacian${\cal{L}}_{b}^{X}$. By makingb→ 0, we show that the corresponding supertrace coincides with the orbital integral associated with the standard elliptic heat kernel. In the next chapters, the evaluation of these elliptic orbital integrals will be obtained by makingb→ +∞.

This chapter is organized as follows. In section 4.1, we introduce an algebraQof invariant kernelsq(x,x′) acting on the vector space of bounded continuous sections ofF.

In section 4.2, if γ...

8. Chapter Five Evaluation of supertraces for a model operator
(pp. 92-112)

In this chapter, given a semisimple element γ ∈G, we evaluate the supertrace of the heat kernel of a hypoelliptic operator acting over$\mathfrak{p} \times \mathfrak{g}$. In section 9.10, this operator will appear in the asymptotics asb→ +∞ of a rescaled version of the operator${\cal{L}}_{b}^{X}$.

In particular we obtain a function${{J}_{\gamma }} \left( Y_{0}^{\mathfrak{k}} \right), Y_{0}^{\mathfrak{k}} \in \mathfrak{k}\ \left( \gamma \right)$, which will play a fundamental role in our final formula for the orbital integrals.

This chapter is organized as follows. In section 5.1, if$Y_{0}^{\mathfrak{k}} \in \mathfrak{k} ( \gamma )$, we introduce the hypoelliptic operator${\cal{P}}_{a,Y_{0}^{\mathfrak{k}}}$, its heat kernel, and a corresponding supertrace$J_{\gamma } \left( {Y_0^\mathfrak{k} } \right)$.

In...

9. Chapter Six A formula for semisimple orbital integrals
(pp. 113-119)

This chapter is the central part of the book. First, we give an explicit formula for the orbital integrals associated with the heat kernel of${\cal{L}}_{A}^{X}$in terms of a Gaussian integral on$\mathfrak{k} \left( \text{ }\!\!\gamma\!\!\text{ } \right)$. In chapter 9, this formula will be obtained by the explicit computation of the asymptotics asb→ +∞ of the orbital integrals associated with${\cal{L}}_{A,b}^{X}$. From the formula for the heat kernel, we derive a corresponding formula for the semisimple orbital integrals associated with the wave operator of${\cal{L}}_{A}^{X}$.

This chapter is organized as follows. In section 6.1, we give the formula...

10. Chapter Seven An application to local index theory
(pp. 120-137)

The purpose of this chapter is to verify the compatibility of our formula in Theorem 6.1.1 for the orbital integrals of heat kernels to the index formula of Atiyah-Singer [AS68a, AS68b], to the fixed point formulas of Atiyah-Bott [ABo67, ABo68], and to the index formula for orbifolds of Kawasaki [Ka79]. Recall that the McKean-Singer formula [McKS67] expresses the index of a Dirac operator over a compact manifoldZas the supertrace of a heat kernel. IfZis the quotient ofXby a cocompact torsion free group, this supertrace can be evaluated explicitly by the formulas we gave in...

11. Chapter Eight The case where [𝖐(γ), 𝖕₀] = 0
(pp. 138-141)

The purpose of this chapter is to evaluate explicitly the Gaussian integral that appears in the right-hand side of our formula in (6.1.2) for the orbital integrals of the heat kernel, when γ is nonelliptic and$[ \mathfrak{k} (\gamma), {\mathfrak{p}}_{0}] = 0$. Our computations can be easily extended to the more general kernels considered in chapter 6. It is remarkable that the index formulas of chapter 7 play here a key role.

This chapter is organized as follows. In section 8.1, we consider the case whereG=K.

In section 8.2, we compute explicitly the Gaussian integral when γ is nonelliptic and$[ \mathfrak{k} (\gamma), {\mathfrak{p}}_{0}] = 0.$...

12. Chapter Nine A proof of the main identity
(pp. 142-160)

The purpose of this chapter is to establish Theorem 6.1.1. The proof consists in makingb→ +∞ in equation (4.6.1), that is, we evaluate the limit asb→ +∞ of$\text{T}{{\text{r}}_{\text{s}}}^{\left[ \gamma \right]}\left[ \exp \left( - {\cal L}_{A,b}^{X} \right) \right]$.

This chapter is organized as follows. In section 9.1, we state various estimates on the hypoelliptic heat kernels, which are valid forb≥ 1. The proofs of these estimates are deferred to chapter 15. They will be used for dominated convergence in the hypoelliptic orbital integrals asb→ +∞.

In section 9.2, we make a natural rescaling on the coordinates parametrizing$\hat{\cal{X}}$.

In...

13. Chapter Ten The action functional and the harmonic oscillator
(pp. 161-186)

The purpose of this chapter is to solve explicitly certain natural variational problems associated with a scalar hypoelliptic Laplacian, in the case where the underlying Riemannian manifold is an Euclidean vector space. The action is the one introduced in [B05, eq. (0.10)]. It depends on the parameterb> 0. The behavior of the minimum values as well as of the minimizing trajectories is studied whenb→ 0 and whenb→ +∞. Finally, certain heat kernels are computed in terms of the minimum value of the action.

The above variational problem has already been considered by Lebeau in [L05,...

14. Chapter Eleven The analysis of the hypoelliptic Laplacian
(pp. 187-211)

The purpose of this chapter is to construct a functional analytic machinery that is adapted to the analysis of the hypoelliptic Laplacian${\cal{L}}_{A,b}^{X}$.

Our constructions are inspired by our previous work with Lebeau [BL08, chapter 15]. There are two differences with [BL08]. The first difference is that the symmetric spaceXis noncompact, while the base manifold in [BL08] was assumed to be compact. A second difference is that in${\cal{L}}_{A,b}^{X}$, the quartic term$\frac{1}{2}{{\left| \left[ {{Y}^{N}},{{Y}^{TX}} \right] \right|}^{2}}$appears. Strictly speaking, it is not directly accessible to the methods of [BL08].

The analysis of the hypoelliptic Laplacian essentially consists in...

15. Chapter Twelve Rough estimates on the scalar heat kernel
(pp. 212-247)

The purpose of this chapter is to establish rough estimates on the heat kernel$r_{b,t}^{X}$for the scalar hypoelliptic operator${\cal{A}}_{b}^{X}$on$\cal{X}$defined in chapter 11. By rough estimates, we mean just uniform bounds on the heat kernel, and not a Gaussian-like decay like in (4.5.3). Such refined estimates will be obtained in chapter 13 for boundedb, and in chapter 15 forblarge. We will also obtain corresponding bounds for the heat kernels associated with operators$\mathfrak{A}_b^X,\text {A}_b^X$over$\hat{\cal{X}}$. The case of$\cal{X}$should be thought of as a warm-up for the case of$\hat{\cal{X}}$,...

16. Chapter Thirteen Refined estimates on the scalar heat kernel for bounded b
(pp. 248-261)

In this chapter, for boundedb> 0, we obtain uniform bounds for the kernels$r_{b,t}^{X}, \mathfrak{r}_{b,t}^{X}$, with the proper decay at infinity on$\cal{X}$or$\hat{\cal{X}}$. In chapter 14, these bounds will be used to obtain corresponding bounds for the kernel$q_{b,t}^{X}$, in order to prove Theorem 4.5.2. The arguments developed in section 12.3, which connect the hypoelliptic heat kernel with the wave equation, play an important role in proving the required estimates.

This chapter is organized as follows. In section 13.1, we establish estimates on the Hessian of the distance function onX.

In section 13.2, we...

17. Chapter Fourteen The heat kernel $q_{b,t}^{X}$ for bounded b
(pp. 262-289)

The purpose of this chapter is to establish the estimates of Theorem 4.5.2 for the hypoelliptic heat kernel$q_{b,t}^{X} \left( \left( x,Y \right), \left( {x}',{Y}' \right) \right)$on$\hat{\cal{X}}$. More precisely, we show that for boundedb> 0, the heat kernel verifies uniform Gaussian type estimates. Also we study the limit asb→ 0 of this heat kernel.

The method consists in using the techniques of chapters 12 and 13 for the scalar heat kernels over$\cal{X}$and$\hat{\cal{X}}$, and to control the kernel$q_{b,t}^{X}$by using a Feynman-Kac formula. Still, because the operator${\cal{L}}_{A,b}^{X}$contains matrix terms that themselves diverge asb→ 0,...

18. Chapter Fifteen The heat kernel $q_{b,t}^{X}$ for b large
(pp. 290-316)

The purpose of this chapter is to establish the estimates of Theorem 9.1.1 on the hypoelliptic heat kernel$\underline{\text{q}}_{b,t}^{X}$. More precisely, we show that asb→ +∞,$\underline{\text{q}}_{b,t}^{X}$exhibits the proper decay away from${\cal{\hat{F}}_{\gamma }}={{\hat{i}}_{a}}{\cal{N}}\left( {{k}^{-1}} \right)\subset \cal{\hat{X}}$.

To avoid being overburdened with technicalities at the very beginning, first, we prove similar estimates on scalar hypoelliptic heat kernels over$\cal{X}$. Such estimates are then extended to scalar hypoelliptic heat kernels over$\hat{\cal{X}}$. Ultimately, we extend the estimates to the kernel$\underline{\text{q}}_{b,t}^{X}$using the Feynman-Kac formula. The term$\frac{1}{2}{{\left| \left[ {{Y}^{N}},\ {{Y}^{TX}} \right] \right|}^{2}}$in the right-hand side of equation (2.13.5) for${\cal{L}}_{b}^{X}$plays...

19. Bibliography
(pp. 317-322)
20. Subject Index
(pp. 323-324)
21. Index of Notation
(pp. 325-330)