.

.

Spectrum of the sub-Laplacian and Laplacian onnilmanifolds attached to Clifford modules andisospectral, non-diffeomorphic manifolds
Himeji Conference on partial differential equations(偏微分方程式姫路研究集会)

Kenro Furutani（古谷賢朗）
Tokyo University of Science(東京理科大学)

05-07/March/2015

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
1 /and44 isIsospectral problem

Isospectral Problem
Notations:1.
(M, g) : a closed{ Riemannian manifold,} ∆ : its Laplacian.
.2 Spec(M, g) = 0 = λ0 < λ1 ≤ λ2 ≤ · · · the set of eigenvaluesof the Laplacian with non-zero eigenfunctions f k ∈ C∞ (M).
3.
Heat kernel K = K(t, x, y) ∈ C∞ (R+ × M × M), i.e., thesolution to the heat equation()∂+ ∆ K = 0, lim K(t, x, ∗) = δ xt↓0∂t. Wave kernel W = W(t, x, y) ∈ D′ (R × M × M), i.e., thesolution to the wave equation

4

(

√ )
∂
+∆ W =± i ∆ W = 0, lim W(t, x, ∗) = δ x
t→0
∂t
∂t 2
∂

)

(

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
2 /and44 isIsospectral problem

By the method of the separation of variables these kernels are
expressed as∑e−tλ k f k (x) f k (y)K(t, x, y) =and

W(t, x, y) =

∑

±it

√

e

λk

f k (x) f k (y)with the eigenvalues λ k and corresponding orthonormal
eigen-functions f k of the Laplacian ∆. In principle, from these
expressions we can determine the heat kernel and wave kernel if
we can solve the eigenvalue problem completely:∆ f = λ f.
At least, if we know the heat trace∫

K(t, x, x)d g =∑e−tλ kM

we can determine the eigenvalues completely.

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
3 /and44 isIsospectral problem

So the wave trace

∑

e

it

√

λk

=

∑

cos t

√
λk

is determined.
With these facts in mind I explain the problem asked by Mark Kac in
the paper“Can one hear the shape of a drum?”
American Mathematical Monthly(1966).
This paper can be considered as the initiation to the spectral
geometry (although its origin goes back to Hermann Weyl).
In this paper he showed an asymptotic expansion of the heat trace
for polygonal domains D under Dirichlet condition:∑e−tλ k ∼

|D|
2πt

−

1− g
6

√
+ O( t ) as t ↓ 0.

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
4 /and44 isIsospectral problemSpec(M1 , g1 ) = Spec(M2 , g2 )Problem=⇒M1

isometric ? diffeo ?



M2



This problem can be illustrated in the picture:

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
5 /and44 isIsospectral problem

It must have been believed from the beginning that the answer isnegative (especially in the higher dimensions) and one of the main
problem is to find pairs of isospectral but non-isometric or evennon-diffeomorphic manifolds. In fact, soon after this paper was
published J. Milnor pointed out that there exist such pairs amongthe 16-dimensional flat tori in the paper
Eigenvalues of the Laplace operator on certain manifolds,Proc.
Nat. Acad. Sci. U.S.A. 51 (1964), 542.
Nowadays, many examples of pairs of such manifolds were found,for example,
A. Ikeda, On lens spaces which which are isospectral but notisometric, Ann. Sci. École Norm Sup. (4) 13:3 (1980), 303-315.
A. Ikeda, On spherical space forms which are isospectral but notisometric, J. Math. Soc. of Japan, 35 1983), 437-444.

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
6 /and44 isIsospectral problem

T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of
Math. (2) 121, no. 1 (1985), 169-186.
C. Gordon, D. Webb, S. Welpert, Isospectral plane domains andsurfaces via Riemanian orbifolds, Invent. Math. 110, no. 1 (1992),
1-22.
R. Gorneth, A new construction of isospectral Riemannian
nilmanifolds with examples, Michigan Math. J. 43 (1996), 159-188.
Even so, in Riemannian geometry it must be still an interesting
problem to find such pairs or family of manifolds in a systematicway.

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
7 /and44 isIsospectral problem

So one of our purpose is to find a new class of isospectral (withrespect to sub-Laplacian), but non-diffeomorphic manifolds among
compact nilmanifolds.
(1) First, we introduce a class of nilpotent Lie groups with a lattice,
and also Laplacian and sub-Laplacian on these groups.
(2) Secondly, we determined eigenvalues of the sub-Laplacian and
Laplacian on this class of nilmanifolds explicitly.
(3) Then, we tried to find out isospecral, but non-diffeomorphic pairs
in the class.
As a result we know that there are infinitely many isospectral, but
non-diffeomorphic nilmanifolds among them with respect to thesub-Laplacians and Laplacians.

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
8 /and44 isClifford algebra and pseudo H-type algebra

Clifford module and Pseudo H-type algebras
I introduce a class of nilmanifolds for which we can calculate thespectrum of sub-Laplacians and Laplacians explicitly.
Let R r,s be the Euclidean space R r+s with the non-degeneratescalar product< x, y > r,s :=

r
∑
i=1

xi yi −s∑x r+ j y r+ j .

j=1

By Cℓ r,s  T (R r+s )/I r,s : the Clifford algebra generated by R r,s .
It is a quotient algebra of the tensor algebra

)
)
(k
(2
T (R r,s ) = R ⊕ R r+s ⊕ ⊗ R r+s ⊕ · · · ⊕ ⊗ R r+s ⊕ · · ·devided by the two sided ideal I r,s generated by the elements of theform x ⊗ x+ < x, x > r,s , x ∈ R r,s .

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
9 /and44 isClifford algebra and pseudo H-type algebra

Table of Clifford algebras

8

R(16)

C(16)

H(16)

H(16) ⊕ H(16)

H(32)

C(64)

R(128)

R(128) ⊕ R(128)

R(256)

7

C(8)

H(8)

H(8) ⊕ H(8)

H(16)

C(32)

R(64)

R(64) ⊕ R(64)R(128)C(128)

6

H(4)

H(4) ⊕ H(4)

H(8)

C(16)

R(32)

R(32) ⊕ R(32)

R(64)

C(64)

H(64)

5

H(2) ⊕ H(2)

H(4)

C(8)

R(16)

R(16) ⊕ R(16)

R(32)

C(32)

H(32)

H(32) ⊕ H(32)

4

H(2)

C(4)

R(8)

R(8) ⊕ R(8)

R(16)

C(16)

H(16)

H(16) ⊕ H(16)

H(32)

3

C(2)

R(4)

R(4) ⊕ R(4)

R(8)

C(8)

H(8)

H(8) ⊕ H(8)

H(16)

C(32)

2

R(2)

R(2) ⊕ R(2)

R(4)

C(4)

H(4)

H(4) ⊕ H(4)

H(8)

C(16)

R(32)

1

R⊕R

R(2)

C(2)

H(2)

H(2) ⊕ H(2)

H(4)

C(8)

R(16)

R(16) ⊕ R(16)

s= 0

R

C

H

H⊕H

H(2)

C(4)

R(8)

R(8) ⊕ R(8)

R(16)

s/r

r=0

1

2

3

4

5

6

7

8

H(x) is x × x matrix algebra of quaternions and so on.

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
10 /and44 isClifford algebra and pseudo H-type algebra

Let V be a module of Cℓ r,s . It is called usually Clifford module. We
can see what are the Clifford modules from the table of the Clifford
algebras.
Let V be a Clifford module. We denote the module action

J : Cℓ r,s × V −→ V, that is
J : (z, X) 7→ J(z, X) =: J z (X)J z 2 = − < z, z > r,s , ∀ z ∈ R r,s .
Definition

We call a module V admissible, if there is a non-degeneratescalar product < •, • >V on V satisfying the following condition:< J z (X), Y >V + < X, J z (Y) >V = 0 for ∀ z ∈ R r,s , ∀ X, ∀ Y ∈ V.



Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
11 /and44 isClifford algebra and pseudo H-type algebra

Pseudo H-type algebra
Let { Cℓ r,s , V, J, < •, • >V } be an admissible module. We define
an antisymmetric bilinear map [•, •]V × V −→ R r,s , (X, Y) 7−→ [X, Y],
by the relation< J z (X), Y >V =< z, [X, Y] > r,s .
With this bilinear map we can define a Lie algebra structure on
V ⊕ R r,s  N r,s (V), which we call pseudo H-type algebra,()()
N r,s (V) × N r,s (V) ∋ (X, z), (Y, w) 7−→ 0, [X, Y] ∈ R r+s  center.
It is a two step nilpotent Lie algebra and so we denote thecorresponding simply connected Lie group by G r,s (V).

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
12 /and44 isClifford algebra and pseudo H-type algebra

We identify the group G r,s (V) with N r,s (V) through the exponential
map. The Campbell-Hausdroff formula gives us the group structure∗ on G r,s (V) by()
1(X, z) ∗ (Y, w) = X + Y, z + w + [X, Y] .
2

s = 0 case was studied by A. Kaplan in the paper
A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE
generated by composition of quadratic forms, Trans. Amer. Math.
Soc. 258 (1980), 147-153 and also,
A. Kaplan, On the geometry of groups of Heisenberg type, Bull.
Lond. Math. Soc. 15(1) (1983), 35-42.
In this case, every irreducible module can be an admissible moduleand the scalar product < •, • >V on the Clifford module V is taken
to be positive definite, that is an inner product.

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
13 /and44 isClifford algebra and pseudo H-type algebra

For s > 0 cases, it was proved in the paper
P. Ciatti, Scalar products on Clifford modules and pseudo-H-type
Lie algebras, Ann. Mat. Pura Appl. (4) 178, 1-31(2000) that

Theorem

Irreducible modules need not be an admissible module, but insuch a case a double of an irreducible module (or sum of two
non-equivalent irreducible modules) can be an admissible module. In any case of s > 0, the scalar product < •, • >V on themodule V has the same dimensional positive and negative subspaces. So the dimension dim V of the module space V is evenalways. In fact it can be devided by 4 (except three lowest “dim”
cases).


Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
14 /and44 isClifford algebra and pseudo H-type algebra

Existence of the lattice in G r,s (V)So we proved in
Existence of Lattices in General H-type Groups, in Journal of Lie
Theory, Vol. 24, No. 4, 979-1011(2014) that

Theorem(K. F. and Irina Markina)

Let Cℓ r,s be a Clifford algebra and assume V is an admissible{ }module. Then for each orthonormal basis Z k in R r, s there ex{ }ists an orthonormal basis X j in the Clifford module V such thatfor each pair of the basis elements (Xi , X j ) there exists “at most”
one basis vector Z k = Z k(i, j) ∈ R r,s such that

[Xi , X j ] = σi, j Z k(i, j)with σi, j being 1 or −1.
Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
15 /and44 isClifford algebra and pseudo H-type algebra{}

We call this type of basis Xi , Z k as the integral basis on N r,s (V).
Then by Malćev’s Theorem and the Campbell-Hausdorff formula weknow that the subgroupΓ r,s (V) ={ ∑1∑
mi Xi +
nk Z k mi , nk ∈ Z
2}is a lattice (= uniform discrete subgroup) of G r,s (V), which we callan integral lattice.
RemarkThe basis vector {Xi } corrsponding to the orthonormal vector{Z k } in R r,s is not unique, but can be mapped to each other by a
Lie algebra automorphism, fixing the basis vectors {Z k }.







Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
16 /and44 isClifford algebra and pseudo H-type algebraThis is an extension of the property proved for the positive definite
cases N r, 0 (V) by
G. Crandall, J. Dodziuk, Integral structures on H-type Lie algebras,
J. Lie Theory 12 (2002), no. 1, 69-79.
RemarkEven the algebra Cℓ r,s  Cℓ r′ ,s′ , the pseudo H-type algebras
N r,s (V) and N r′ ,s′ (V) need not be isomorphic, also even if thealgebras Cℓ r,s and Cℓ r′ ,s′ are not isomorphic, in some cases
N r,s (V) and N r′ ,s′ (V ′ ) are isomorphic.







Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
17 /and44 isClifford algebra and pseudo H-type algebra

We have a partial classification theorem among the pseudo H-type
algebras:
Theorem (Christian Autenred, K.F. and Irina Markina

(1) If r + s , r′ + s′ , then N r,s (V) and N r′ ,s′ (V ′ ) are never isomorphic.
(2) Let r + s = r′ + s′ . Assume one of the admissible modules
V of Cℓ r,s or V ′ of Cℓ r′ ,s′ are minimal and dim V = dim V ′ . Then
only for the cases (r′ , s′ ) ∈ {(r, s), (s, r)}, N r,s (V) and N r′ ,s′ (V ′ )can be isomorphic.
So there is possibility of examples of isospectral, non-diffeomorphic
manifolds among pairs Γ r,s (V)\G r,s (V) and Γ s,r (U)\G s,r (U).

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
18 /and44 isClifford algebra and pseudo H-type algebra

Laplacian and sub-Laplacian on G r,s (V)Let G r,s (V) be a pseudo H-type group and we fix an integral lattice

Γ r,s (V) ={∑1∑
mi Xi +
kk Z k2}

associated with the orthonormal basis {Z k } in the center R r,s .We
denote the left invariant vector field defined by each vector Xi (andalso Z k ) by X̃i ( Z̃ k ).
In the definition of admissible modules the scalar product in themodule space V need not be positive definite, so we have two typesof invariant differential operators on G r,s (V),
Laplacian (sub-Laplacian) and ultra-hyperbolic operators.

()

Both come from Id map in Hom N r,s (V), N r,s (V) .

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
19 /and44 isClifford algebra and pseudo H-type algebra

We consider here only the Laplacian and sub-Laplacian:∆Gr,s (V) = ∆ = −∑X̃ j2 −∑

Z̃ k2 ,and sub-Laplacian is
G (V)∆ subr,s= ∆ sub = −∑X̃i 2 .

These two are descended to the quotient space Γ r,s (V)\G r,s (V)which are our operators considering here. Then we have
Theorem A
Let V and U be admissible modules of Cℓ r,s and Cℓ s,r , respectively and assume dim V = dim U, then two nilmanifolds
Γ r,s (V)\G r,s (V) and Γ s,r (U)\G s,r (U) are isospectral with respect
to the sub-Laplacians and Laplacians always.


Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
20 /and44 isClifford algebra and pseudo H-type algebra

Now we consider two conditions IR1 and IR2 in relation with theirreducibility and admissibility of the Clifford module for a pair ofClifford algebras Cℓ r+1,s and Cℓ s,r+1 with r + 1 , s:IR1: An irreducible module V of Cℓ r+1,s is admissible with a scalar
product < •, • >V and also assume the dimension of an irreducible
module of Cℓ r,s = dim V .

Hence irreducible modules of the algebra Cℓ r,s is admissible.
IR2: The dimension of the minimal admissible module of Cℓ s,r+1≤ dim V .

Then under these conditions we may take an admissible
module {U, J̃} of Cℓ s,r+1 with dim U = dim V .

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
21 /and44 isClifford algebra and pseudo H-type algebra

Then we have
Theorem BThe pseudo H-type algebras N r+1,s (V) and N s,r+1 (U) are not isomorphic.
Hence, Theorem A + Theorem B imply
Main TheoremFor r + 1 , s, we assume that the Clifford algebras Cℓ r+1,s ,Cℓ r,s and Cℓ s,r+1 satisfy the two conditions IR1 and IR2. So let
V be an irreducible module of Cℓ r+1,s (which is admissible by theconditions) and we take an admissible module U of Cℓ s,r+1 with
dim U = dim V , then two nilmanifolds

Γ r+1,s (V)\G r+1,s (V) and Γ s,r+1 (U)\G s,r+1 (U)are isospectral, but non-diffeomorphic.
Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
22 /and44 isClifford algebra and pseudo H-type algebra

A note for the proof of Main Theorem
First we note a general fact in nilpotent Lie group theory:

Lemma



Let G and G′ be two simply connected nilpotent Lie groups withlattices Γ and Γ′ respectively. If the nilmanifolds Γ\G and Γ′ \G′
are diffeomorphic, then G and G′ are isomorphic.
This can be seen by the arguments:′′

The given diffeomorphism between Γ\G and Γ \G induces anisomorphism of the fundamental groups π1 (Γ\G)  Γ and′′π1 (Γ \G )  Γ′ , which can be extended to the whole groups.
Hence this lemma together with Theorem B implies thenon-diffeomorphy assertion in the Main Theorem .

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
23 /and44 isClifford algebra and pseudo H-type algebra

Table of dimensions of minimal admissible modules
Dimensions of minimal admissible modules of Cℓ r,s for r, s ≤ 8
8
7
6
5
4
3
2
1
0

16
16
16
16
8
8
4

32
32
♭
16×2
16
8
8

64 64×2
64
64
32
32
#
16 16
8
8×2
8
8

s/r

128
128
64
32

2
1

4×2
4
2

8
8
4

8
8
4×2

16
16
16
16
8

0

1

2

3

4

128
128
64×2
64
32
#

32
16×2
16
8
5

128 128×2
128 128
128 128
128 128
64
64×2
64
64

32
16
8

32
16
8×2

6

7

♭

256
256
256
256
128
128
64
32

16
8

black = irreducible, red = double of irreducible,∗×2 = two minimal dimensional admissible modules

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
24 /and44 isClifford algebra and pseudo H-type algebra

Low dimensional examples

Now we have examples of isospectral, but non-diffeomorphic
nilmanifolds from the table of the dimensions of minimal admissible
modules.
Corollary
The following pairs satisfy the conditions IR1 and IR2:

Cℓ4+1,2 and Cℓ2,4+1 ,Cℓ5+1,1 and Cℓ1,5+1 ,These are low dimensional cases and we have many isospectral
but non-diffeomorphic pairs of nilmanifolds of pseudo H-typegroups. In fact

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
25 /and44 isClifford algebra and pseudo H-type algebra

Higher dimensional examples

Theorem

The pairs of nilmanifolds

Γ3+ℓ,ℓ \G3+ℓ,ℓ and Γℓ,3+ℓ \Gℓ,3+ℓ
for ℓ = 2 + 4k with k = 0, 1, . . . ,andΓ5+4k+1,4k+1 \G5+4k+1,4k+1 and Γ4k+1,5+4k+1 \G4k+1,5+4k+1 ,k = 0, 1, . . .
are isospectral, but non-diffeomorphic.
Here we consider the minimal dimensional admissible modules ofCℓ3+ℓ,ℓ and Cℓ5+4k+1,4k+1 respectively and corresponding modules ofCℓℓ,3+ℓ and that of Cℓ4k+1,5+4k+1 with the same dimensions.

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
26 /and44 isClifford algebra and pseudo H-type algebra

Other examples
Let V  R4,4 be the irreducible module of Cℓ3,1  H(2), then it isadmissible and we take the minimal admissible module U  R4,4 ofCℓ1,3  R(4) which is the double of the irreducible module, then thepairs
Γ3,1 (V)\G3,1 (V) and Γ1,3 (U)\G1,3 (U)are isospectral, but non-diffeomorphic, although they do not satisfy
the conditions IR1. This case is a minimal dimensional example
among these nilmanifolds.
Also similar holds for the pairΓ3,2 (V)\G3,2 (V) and Γ2,3 (U)\G2,3 (U)with the irreducible module V  R4,4 of the algebra Cℓ3,2  C(4)and the sum of the non-equivalent irreducible modules of thealgebra Cℓ2,3  R(4) ⊕ R(4).

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
27 /and44 isProof of the Main Theorem

Heat kernel on two step nilpotent Lie groups
For the proof of Theorem A we recall an integral formula of the heat
kernel for a sub-Laplacian on simply connected two step nilpotent
Lie groups.

G : a simply connected two step nilpotent Lie group
N : Lie algebra exp : N  G. We assume that[]
N, N = center of NN
{Xi , Z k } : linear basis in N such that {Z k } dk=1 and {Xi }i=1
span thecenter [N, N] and a complement of the center, respectively.
Moreover, we assume that the basis vectors {Xi , Z k } are
orthonormal. Then the Lie algebra N is decomposed into an

orthogonal sum

[
[
]
N ]
N = {Xi }i=1⊕⊥ N, N  R N ⊕⊥ R d ,Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
28 /and44 isProof of the Main Theorem

Put[Xi , X j ] =d∑cik j Z k ,cik j are structure constants.

k=1

Denote by Ω(z) the skew-symmetric matrixΩ(z) =d∑k=1

where z =

(

zk
∑

)

cik j

∈ R(N) = N × N real matrices,z k Z k ∈ [N, N].


Remark

We identify the group G with R N × R d via the coordinatesG ∋ g → (x1 , . . . , x N , z1 , . . . , z d ) ∈ R N × R d .
≈Then the exponential map exp : N → G is the identity map.

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
29 /and44 isProof of the Main TheoremX̃i : the left invariant vector field on G
N1∑ 2G
sub-Laplacian : ∆sub = −X̃ .
2 i=1 iWe known that ∆G
is sub-elliptic and essentially selfadjoint insub
L2 (G) (with respect to the Haar measure).
Since the sub-Laplacian ∆G
is left-invariant it can be shown thatsubthe heat kernel K is of the form K(t, g, h) = kG (t, g−1 ∗ h) with asmooth function kG ∈ C∞ (R+ × G).

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
30 /and44 isProof of the Main Theorem

In the paper by R. Beals, B. Gaveau, P. Greiner,

The Green function of model step two hypoelliptic operators and theanalysis of certain tangential Cauchy Riemannian complexes, Adv.
in Math. 121 (1996), 288-345,
the explicit integral expression of the heat kernel is given:

Theorem (Beals-Gaveau-Greiner)K(t, g, h) = kG (t, g−1 ∗ h)∫f (τ, g−1 ∗ h)1=e− t
W(τ) dτ,(2πt) N/2+d RdKenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
31 /and44 isProof of the Main Theorem

where the functions f = f (τ, g) ∈ C∞ (R d × G) and W(τ) are given
as follows:
put g = (x, z) ∈ R N × R d ,√f (τ, g) = f (τ, x, z) = −1 < τ, z >
1⟨ √
( √
)
⟩
+ Ω( −1τ) coth Ω( −1τ) · x, x
2
√

1/2



Ω( −1τ) 
W(τ) = 
det



√

 ,
sinh Ω( −1τ)
where < z, z′ >=

d
∑
k=1

z k z′k and

⟨ √
( √
)
⟩
Ω( −1τ) coth Ω( −1τ) · x, x

are the Euclidean inner product on R d and on R N respectively.

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
32 /and44 isProof of the Main Theorem

Outline of the proof of Theorem A(i) The spectrum of the sub-Laplacian on the nilmanifold
Γ r,s (V)\G r,s (V) is given by the explicit determination of the heat
trace

Tr e
=

−t∆ sub

∫

∑

=

∑

(
)
K t, γ · (x, z), (x, z) dxdz

F r,s (V) γ∈Γ (V)r,se−tλ kλ k : eigenvalue

where K(t, (x, z), (y, w)) ∈ C∞ (R+ × G r,s (V) × G r,s (V)) is the kernel
function of the heat operator e−t∆sub and F r,s (V) is a fundamental
domain of the lattice Γ r,s (V).

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
33 /and44 isProof of the Main Theorem(ii) We do not employ Selberg trace( formula method,) instead we∞
decompose the function space C Γ r,s (V)\G r,s (V) into Fourier
coefficients. The nilmanifold Γ r,s (V)\G r,s (V) is the total space of aprincipal bundle

Γ r,s (V)\G r,s (V)



y

(
)
Γ r,s (V) ∩ V \V  Tdim V/()
with the structure group R r+s Γ r,s (V) ∩ R r,s  T r+s .
So we decompose() ∑ χ
C∞ Γ r,s (V)\G r,s (V) =F ,χ

where χ : T r+s → U(1) is a unitary character and}{()
F χ = f ∈ C∞ Γ r,s (V)\G r,s (V) f ((x, z) ∗ λ) = χ (λ) · f (x, z)

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
34 /and44 isProof of the Main Theorem(iii) Then the sub-Laplacian (also Laplacian) leaves invariant eachsubspace F χ . Hence if we denote by D(χ) the sub-Laplacian
restricted to the subspace F χ , D(χ) = ∆ sub χ , thenF∑

Tr e−tD(χ)= Tr e−t∆subχand we can determine each term Tr e−tD
describe below:(χ)

explicitly, which I shall

The unitary character χ is identified with integers
χ := 2(m1 , . . . , mr , n1 , . . . , ns ) = 2(µ, ν) ∈ 2Z r+s and the trace
Tr e−tD(χ)

is expressed as follows (we put N = dim V ):Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
35 /and44 isProof of the Main Theorem

Theorem (Wolfram Bauer, K. F. and Chisato Iwasaki)(1) If n = 0, then the trace of the operator e−tD is given by(0)(

Tr e−tD(0))=∑ ∥ℓ∥21e− 2t .
(2π t) N ℓ∈Z2N[]∗(2) Assume that n ∈ Γ r,s (V) ∩ A

with

r
∑
i=1

mi 2 =s∑n j 2 , thatj=1

is, ∥µ∥ = |ν∥ and let d0 > 0 be the greatest common divisor among non-zero components of the integers (µ, ν) =(m1 , . . . , mr , n1 , . . . , ns ). Then()(n)
Tr e−tD =1∑(π t) N/2 ℓ∈Z2Ne−∥µ∥2 ·∥ℓ∥2
d0 2 t

(

) N/2
2∥µ∥
.
(
)
sinh 8π t∥µ∥Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
36 /and44 isProof of the Main Theorem(3) For n = 2(µ + ν) with ∥µ∥ , ∥ν∥ the matrix Ω(n) is nonsingular.

(

Tr e−tD

(n)

(
=2

2N

·



Theorem A)∥µ∥2 − ∥ν∥2sinh{4π t(∥µ∥ + |ν∥)} sinh{4π t(∥µ∥ − |ν∥)}) N/2.
.

Let V and U be minimal admissible modules of Cℓ r,s and Cℓ s,rrespectively and assume dim V = dim U, then the nilmanifolds
Γ r,s (V)\G r,s (V) and Γ s,r (U)\G s,r (U) are isospectral with respect
to the sub-Laplacian (and the Laplacian, since ∆ = ∆ sub + Cχ on
F χ ).

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
37 /and44 isProof of the Main TheoremRemark• The pseudo H-type algebra N r,s (V min) with a minimal dimensional admissible module V min does not depend on the chosenminimal admissible module, even if the Clifford algebra has two
non-equivalent irreducible modules.

• Moreover, in the above determination we do not explicitly use
the assumption that the admissible module is minimal.

• This implies that if V is a sum of k minimal admissible modules, then the heat trace in each of the case of above three isthe k-th power of the corresponding heat trace for the manifold
Γ r,s (V min)\G r,s (V min).

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
38 /and44 isProof of the Main Theorem

Proof of Theorem B
Under the assumption IR1 and IR2 first I explain a Proposition:
Let Z 1 , . . . , Z r+1 , Z r+2 , . . . , Z r+s+1 be the orthonormal generators of
the Clifford algebra Cℓ r+1,s , Z i for i = 1, . . . , r + 1 positive and thelast s vectors negative.
Also let W1 , . . . , W r , W r+1 , . . . , W r+s be the orthonormal generatorsof Cℓ r,s , again the first r vectors being positive and so on.
We define a map I : R r,s −→ Cℓ r+1,s in such a way that

W1 7−→ Z 1 Z r+1 , . . . , W r 7−→ Z r Z r+1 ,W r+1 7−→ Z r+2 Z r+1 , . . . , W r+s 7−→ Z r+s+1 Z r+1 ,then it satisfies the property

I (Wi )2 = − < Wi , Wi > r,s .

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
39 /and44 isProof of the Main Theorem

Hence it can be extended to the algebra homomorphismCℓ r,s −→ Cℓ r+1,s ,which we also denote by I .
Let { V, J, < •, • >V } be an admissible module of Cℓ r+1,s . Then
Proposition

The action of Cℓ r,s on {V, < •, • >V } by the composition map
Ĵ := J ◦ I also gives us an admissible module.







Proof. This can be seen from< Ĵ Wi (X), Y >V =< J Z i J Z r+1 (X), Y >V=< X, J Z r+1 J Z i (Y) >V = − < X, J Z i J Z r+1 (Y) >V= − < X, Ĵ Wi (Y) >V .

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
40 /and44 isProof of the Main Theorem

Now let’s assume that there exists an algebra isomorphism

(

)
V
U
A 0
Φ=
: N r+1,s  ⊕⊥−→ ⊕⊥
 N s,r+1 ,
0 C
r+1,s
s,r+1
R
R
then the map)
VVAτ ◦ A 0−→ ⊕⊥ N r+1,s ,Φ ◦Φ=: N r+1,s  ⊕⊥0
−Id
R r+1,sR r+1,sτ(is an algebra automorphism, whereoperator defined byAτ denote the adjoint< A(X), Y >U =< X, Aτ (Y) >V
and satisfies the conditionsAτ ◦ Ĵ w ◦ A = J Cτ (w) , and Cτ ◦ C = −Id.

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
41 /and44 isProof of the Main Theorem

Then we have

( Aτ ◦ A) ◦ J z ◦ (Aτ ◦ A) = J −z , for z ∈ R r+1,s .

(1)

By some arguments we know it satisfies( Aτ ◦ A) ◦ Ĵ W = Ĵ W ◦ ( Aτ ◦ A), W ∈ R r,s .
By the irreducibility assumption of the module {Cℓ r,s , Ĵ, < •, • >V }we know that the intertwining operator Aτ ◦ A must be a constant= λ · Id (Schur’s lemma). Then by the relation (1) we have

λ2 J z = −J z , z ∈ R r+1,s .
Hence λ2 < 0 which is a contradiction. So the two algebras
N r+1,s (V) and N s,r+1 (U) are not isomorphic.

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
42 /and44 isProof of the Main Theorem

Existence of any number of isopsectral,non-diffeomorphic manifolds(•) Let denote irreducible modules of Cℓ3,0  H ⊕ H by V±  R4,0 .
In this case irreducible modules are admissible. Then two algebras
p

q

p′

q′

p

q

N3,0 (⊕ V+ ⊕ ⊕ V− ) and N3,0 (⊕ V+ ⊕ ⊕ V− ) are isomorphic for thecases of (p′ , q′ ) ∈ {( p, q), (q, p)}.
pq

Hence the nilmanifolds Γ3,0 (⊕ V+ ⊕ ⊕ V− )\G3,0 (⊕ V+ ⊕ ⊕ V− ) and

q
q′
( p
)
( p′
)
Γ3,0 ⊕ V+ ⊕ ⊕ V− \G3,0 ⊕ V+ ⊕ ⊕ V− for ( p′ , q′ ) < {( p, q), (q, p)}

are isospectral, but non-diffeomorphic.
(•) The same occurs for the case Cℓ7,0 and accordingly for thecases of Cℓ k,0 with k = 3 mod 4.

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
43 /and44 isProof of the Main Theorem

Kenro Furutani（古谷賢朗）(Tokyo University ofSpectrum
Science(東京理科大学
of the sub-Laplacian))and Laplacian on nilmanifolds attached05-07/March/2015
to Clifford modules
44 /and44 is