Jean-Michel Bismut is professor of mathematics at the Université Paris-Sud, Orsay.

Hypoelliptic Laplacian and Orbital Integrals

By Jean-Michel Bismut

PRINCETON UNIVERSITY PRESS

Copyright © 2011 Princeton University Press
All right reserved.
ISBN: 978-0-691-15130-4

Contents

Introduction………………………………………………………………………………….11. Clifford and Heisenberg algebras……………………………………………………………..122. The hypoelliptic Laplacian on X = G/K…………………………………………………………223. The displacement function and the return map…………………………………………………..484. Elliptic and hypoelliptic orbital integrals……………………………………………………765. Evaluation of supertraces for a model operator…………………………………………………926. A formula for semisimple orbital integrals…………………………………………………….1137. An application to local index theory………………………………………………………….1208. The case where [t(γ), p₀] = 0…………………………………………………1389. A proof of the main identity…………………………………………………………………14210. The action functional and the harmonic oscillator……………………………………………..16111. The analysis of the hypoelliptic Laplacian……………………………………………………18712. Rough estimates on the scalar heat kernel…………………………………………………….21213. Refined estimates on the scalar heat kernel for bounded b………………………………………24814. The heat kernel [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for bounded b…………………26215. The heat kernel [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for b large…………………..290Bibliography………………………………………………………………………………….317Subject Index…………………………………………………………………………………323Index of Notation……………………………………………………………………………..325

Chapter One

Clifford and Heisenberg algebras

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 space V equipped with a symmetric bilinear form B.

In section 1.2, we specialize the construction of the Clifford algebra to the case of V [direct sum] V*.

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

In section 1.4, we specialize the construction of the Heisenberg algebra to the case of V [direct sum] V.

In section 1.5, we consider the combination of the Clifford and Heisenberg algebras for V [direct sum] V*, and we construct the complex (Λ (V*) [cross product] S’ (V*), [bar.d]), which is the subcomplex of polynomial forms in the de Rham complex.

Finally, in section 1.6, when V is equipped with a scalar product, this complex is related to a Witten complex over V.

1.1 The Clifford algebra of a real vector space

Let V be a finite dimensional real vector space of dimension n, let B be a real-valued symmetric bilinear form on V.

Let c (V) be the Clifford algebra associated to (V, B). Namely, c(V) is the algebra generated over R by 1, a [member of] V and the commutation relations for a, b [member of] V,

ab + ba = -2B (a, b). (1.1.1)

We denote by [??](V) the Clifford algebra associated to -B. Then c(V), [??](V) are filtered by length, and their corresponding Gr’ is just Λ(V). Also they are Z₂-graded algebras. If B is zero, then c(V), [??](V) coincide with [LAMBA](V).

In the sequel, we assume that B is nondegenerate, so that V and V* can be identified by the form B. Let φ: V [right arrow] V* denote the corresponding isomorphism, so that if a, b [member of] B,

(φa, b) = B (a,b). (1.1.2)

Let B* be the corresponding bilinear form on V*. Then B* induces a nondegenerate symmetric bilinear form on Λ (V*), which we still denote B*.

If a [member of] V, let c (a), [??](a) [member of] End (Λ (V*)) be given by

c (a) = φa [conjunction] – i_a, [??](a) = φa [conjunction] + i_a. (1.1.3)

Then c (a) and [??](a) are odd operators, which are respectively antisymmetric and symmetric with respect to B*.

If a, b [member of] V, then

[c (a) ,c (b)] = -2B (a,b), [[??](a), [??](b)] = 2B (a, b), [c (a), [??](b)] = 0. (1.1.4)

By (1.1.4), we find that Λ (V*) is a c(V) and a [?](V) Clifford module. Also if a [member of] V, [??](a) is the negative of c(a) associated with –B. This is a discrepancy that is unavoidable, given the fact that the definition of the operator [??](a) is well established by tradition. If A lies in c(V) or [??](V), we denote by c(A), [??](A) the corresponding endomorphisms of Λ(V*) associated with the representation of the Clifford algebras c(V), [??](V) defined in (1.1.3).

The maps A [member of] c(V) [right arrow] c(A) 1 [member of] Λ (V*), Λ [member of] c(V) [right arrow] (-1)^deg(A’) [??](Λ) 1 [member of] Λ (V*) identify c (V), [??](V) as vector spaces to Λ (V*). If a G Λ (V*), we will denote by c (a), [??](a) the corresponding elements in c (V), [??](V). There will be no risk of confusion with the above definition of c(A), [??](A).

Let e₁, …, e_n be a basis of V, and let e^*₁,…, e^*_n be the dual basis of V with respect to B, so that B (e_i, e^*_j) = δ_ij. Note that if a [member of] V,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.5)

Let e¹, …, eⁿ be the basis of V* which is dual to the basis e₁, …, e_n. Of course for 1 ≤ i ≤ n, eⁱ = φe^*_i. Take a G Ap (V*), so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.6)

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.7)

Note that when replacing B by -B, e^*_i is changed to -e^*_i and c (e_i) into -[??](e_i), which explains why there is no sign in the second formula in (1.1.7).

Moreover, the map f [right arrow] if induces the canonical isomorphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.8)

It follows from (1.1.8) that the complexified Clifford algebra c (V) [cross [product]_R C does not depend on the choice of B, which can as well be taken to be a scalar product. However, the Clifford representation which to a [member of] V associates the corresponding element in c (V) depends on B.

Let A (V) be the Lie algebra of the endomorphisms of V that are antisymmetric with respect to B. Then A(V) embeds as a Lie algebra in c(V). If A [member of] A(V), the image c(A) of A in c(V) is given by

c (A) = 1/4 B (Ae^*_i, e^*_j) c (e_i) c (e_j). (1.1.9)

Moreover, if a [member of] V,

[c (A), c (a)] = c (Aa). (1.1.10)

When replacing B by -B, we denote by [??](A) the corresponding element in [??](V). By (1.1.10), we obtain

[??](A) = -1/4B (Ae^*_i, e^*_j) [??](e_i) c (e_j). (1.1.11)

Instead of (1.1.10), we now have

[[??](A), [??](a)] = [??](Aa). (1.1.12)

When acting on Λ (V*) as indicated in (1.1.3), c(A), [??](A) are antisymmetric operators with respect to B*.

Recall that if A [member of] End (V), A acts on Λ(V*), and this action is given by

A|_{Λ x (V*)} = – (Ae_i, e^j> eⁱi_ej. (1.1.13)

Then one verifies easily that if A [member of] A (V),

A|_{Λ x (V*)} = c(A) + [??](A). (1.1.14)

Let N^{Λ x(V*)} be the number operator acting on Λ(V*), i.e., N^Λ(V*) acts on Λ^p (V*) by multiplication by p. Then one verifies easily that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.15)

Let (V’,B’) be another pair like (V, B). Then (V [direct sum] V’, B [direct sum] B’) is still another such a pair, and moreover

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.16)

In (1.1.16), [??] denotes the Z₂-graded tensor product.

Let V, V’ be real Euclidean vector spaces, let c(V), [??](V) , c(V’), [??](V’) be the corresponding Euclidean Clifford algebras. We denote by V^‘_ the vector space V‘ equipped with the negative of its Euclidean product. By (1.1.16), we find that if W = V [direct sum] V^‘_, then

c(W) = c(V) [??] (V’), [??](W)= [??](V) [??]c (V’). (1.1.17)

1.2 The Clifford algebra of V [direct sum] V*

Let V be a real vector space, and let V* be its dual. Put W = V [direct sum] V*. Then W is canonically equipped with minus half of the symmetric bilinear pairing between V and V*. If a [member of] V, b [member of] V, put

c(a) = i_a, c (b) = bΛ. (1.2.1)

Then (1.2.1) turns Λ (V*) into a c(W) Clifford module. Moreover, inspection of (1.1.3) and (1.2.1) shows that once a nondegenerate symmetric bilinear form B is fixed on V,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.2)

Let I [member of] End (V [direct sum] V*) be the map that is -1 on V and 1 on V*. Then I [member of] A (V [direct sum] V*). Let e₁, …, e_n be a basis of V, and let e¹, …, eⁿ be the corresponding dual basis of V*. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.3)

Recall that N^Λ(V*) is the number operator of Λ(V*). By (1.2.3), we have the identity of operators acting on Λ(V*),

c (I) = NΛ(V*} – n/2. (1.2.4)

Assume now that B is a scalar product on V. Let S^V be the Hermitian vector space of V spinors. Then S^V is a c(V) Clifford module. Moreover, if a [member of] V, c (a) acts as a skew-adjoint automorphism of S^V. Let V_ denote the vector space V equipped with the bilinear form -B. If V is replaced by V_, S^V is of course a c (V_) Clifford module. However, if a [member of] V, the associated Clifford action c_ (a) is given by

c_ (a) = ic (a), (1.2.5)

so that c_ (a) acts like a self-adjoint automorphism of S^V. If A [member of] End (V) is antisymmetric, c(A) and c_(A) coincide.

1.3 The Heisenberg algebra

Let V be a finite dimensional real vector space of dimension n. Let S'(V*) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the symmetric algebra of V*, i.e., the polynomial algebra of V.

Let ω be an antisymmetric bilinear form on V. The Heisenberg algebra b (V) is the algebra generated by 1, a [member of] V and the commutation relations for e, f [member of] V,

ef – fe = 2ω (e,f). (1.3.1)

Similarly we denote by [??](V) the Heisenberg algebra associated with the form -[oemga]. Again b(V), [??](V) are filtered by length, and the corresponding Gr’ is just S’ (V). If u vanishes, b(V), [??](V) coincide with S’ (V). In the sequel, we assume that ω is nondegenerate, i.e., ω is a symplectic form on V, so that n is even.

If e [member of] V, let e* [member of] V* correspond to e by the symplectic form, so that if f [member of] V,

(e*,f) = ω(e,f). (1.3.2)

If e [member of] V, f [member of] V*, then e and f act on S’ (V*) as annihilation and creation operators. More precisely e acts on S'(V*) by the derivation ∇_e, and f by multiplication. If e [member of] V, set

b (e)= e* – ∇_e, [??](e) = e* + ∇_e. (1.3.3)

The operators b(e), [??](e) act on S'(V*).

In the sequel, the algebras b(V), [??](V) will be considered as trivially Z₂-graded, so that supercommutators in this algebra are just ordinary commutators. Similarly the vector space S'(V*) will be considered as even, or as trivially Z₂-graded. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.4)

Equation (1.3.4) indicates that S'(V*) is a b(V) and a [??](V) Heisenberg module. Moreover, [??](e) is the negative of the b(e) associated with -ω. The maps A [member of] b (V) [right arrow] b (A) 1 [member of] S'(V*), A’ [member of] [??](V) [right arrow] (-1)^deg(A’) [??](A’)1 [member of] S'(V*) are isomorphisms of vector spaces. If a [member of] S'(V*), we denote by b(a), [??](a) the corresponding elements in b(V), [??](V). Then we have the obvious analogue of (1.1.7).

(Continues…)

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