Journal of Lie Theory Volume 8 (1998) 95–110 ◯^{C}1998 Heldermann Verlag

Kazhdan Constants Associated with Laplacian on Connected Lie Groups.

M. E. B. Bekka, P-A. Cherix^{1}, P. Jolissaint

Communicated by A. Valette

Abstract. Let G be a finite dimensional connected Lie group. Fix a basis {X_{i}}_{i=1,···,n} of the Lie algebra 𝔤 and form the associated Laplace operator Δ = − ∑_{1≤i≤n} X_{i}^{2} in the enveloping algebra U(𝔤). Let π be a strongly continuous unitary representation of G; let \overline{dπ(Δ)} be the closure of the essentially self-adjoint operator dπ(Δ). We show that π almost has invariant vectors if and only if 0 belongs to the spectrum of \overline{dπ(Δ)}. From this, we deduce that G has Kazhdan’s property (T) if and only if there exists ϵ > 0 such that, for any unitary representation without non zero fixed vectors, one has ϵ < min\{Sp(\overline{dπ(Δ)})\}. This answers positively a question of Y. Colin de Verdière. It also allows us to define new Kazhdan constants, that we compare to the classical ones.

1. Introduction

In 1967, Kazhdan introduced property (T), a fixed point property of unitary representations for locally compact groups. More precisely,

Definition 1.1. Let G be a locally compact group.

1. Let π : G → U(H_{π}) be a strongly continuous unitary representation, ϵ > 0 and let K ⊂ G be a compact subset of G; a vector ξ ∈ H_{π}^{1}, the set of vectors of length 1 in H_{π}, is (ϵ, K)-invariant if

sup{∥π(g)ξ − ξ ∥ | g ∈ K} < ϵ.

2. π has almost invariant vectors if for every ϵ and K as above, there exists an (ϵ, K)-invariant vector.

3. G has property (T), if for all representations with almost invariant vectors, there exists a nonzero fixed vector.

1Supported by Grant 20-43215.95 of the Swiss National Fund for Scientific Research

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag

This property has strong consequences both from practical and theoretical points of view (see [10]). In graph theory, the property (T) permits the construction of family of expanders via the knowledge of the Kazhdan constant (see [11]). Let (π, H_{π}) be a representation of G and let K be a compact generating set of G. We define κ(G, K, π) as follows :

κ(G, K, π) = inf_{ξ∈H_{π}^{1}} max_{s∈K} ∥π(s)ξ − ξ ∥ .

The Kazhdan constant of the group G relatively to K is defined by:

κ(G, K) = inf{κ(G, K, π) | π ∈ G̃^{∗}}

where G̃^{∗} is the set of equivalence classes of unitary representations of G on separable Hilbert spaces, without nonzero fixed vectors.

Proposition 1.2. For G a locally compact group and K a compact generating set of G, the following assertions are equivalent :

1. G has property (T)

2. κ(G, K) > 0

A proof of this result is given in [4].

In general, it is difficult to see whether a group G has property (T). In this paper we give an equivalent definition of property (T) for connected Lie groups. This approach was suggested by Colin de Verdière [1].

2. Definitions and first properties

Let G be a connected Lie group with Lie algebra 𝔤 and enveloping algebra U(𝔤). We use on G a right invariant Haar measure dg.

Any unitary representation (π, H_{π}) of G induces a representation dπ of 𝔤 on the subspace C^{∞}(H_{π}) of C^{∞}-vectors of π, i.e. the subspace of vectors ξ ∈ H_{π} for which the function x ↦ π(x)ξ is a C^{∞} function. dπ extends to a representation of U(𝔤) on the same subspace C^{∞}(H_{π}).

Let ξ and η be two vectors in H_{π}, we denote by φ_{ξ,η} the function on G defined by φ_{ξ,η}(g) = ⟨π(g)ξ | η⟩. We call φ_{ξ,η} the coefficient of π associated to ξ and η.

Lemma 2.1. Using the same notations as before, if ξ is in C^{∞}(H_{π}), η is in H_{π} and X is in 𝔤, then φ_{ξ,η} satisfies :

1. Xφ_{ξ,η} = φ_{dπ(X)ξ,η} ,

2. Δφ_{ξ,η} = φ_{dπ(Δ)ξ,η} ,

3. Δ^{n}φ_{ξ,η} = φ_{dπ(Δ)^{n}ξ,η} for every n ≥ 1.

Proof. By definition, we get for every g in G :

(Xφ_{ξ,η})(g) = lim_{t→0} \frac{φ_{ξ,η}(g exp(tX)) − φ_{ξ,η}(g)}{t}
= lim_{t→0} \frac{1}{t} [⟨π(exp(tX))ξ | π(g^{−1})η⟩ − ⟨ξ|π(g^{−1})η⟩]
= ⟨dπ(X)ξ | π(g^{−1})η⟩ = φ_{dπ(X)ξ,η}(g)

This proves (1). The assertions (2) and (3) follow from the fact that dπ(X)ξ is a C^{∞} vector if ξ is.

Lemma 2.2. Let X be an element of 𝔤 and ψ be a C^{∞} function on G such that ψ and Xψ lie in L^{1}(G); then ∫_{G}(Xψ)(g)dg = 0.

Proof. Suppose first that ψ ∈ C_{0}^{∞}(G), the space of C^{∞} with compact support on G :

∫_{G}(Xψ)(g)dg = ∫_{G} ( lim_{t→0} \frac{ψ(g exp(tX)) − ψ(g)}{t}) dg
= lim_{t→0} ∫_{G} \frac{ψ(g exp(tX)) − ψ(g)}{t}dg = 0.

The two last equalities hold because Supp(ψ) is compact and because the measure dg is invariant by right multiplication. Now, if ψ ∈ C^{∞}(G) is such that ψ and Xψ lie in L^{1}(G), there exists a compact subset K of G such that ∫_{G−K} |ψ(g)|dg < ϵ/3 and ∫_{G−K} |(Xψ)(g)|dg < ϵ/3. So we have : | ∫_{G}(Xψ)(g)dg| < | ∫_{K}(Xψ)(g)dg|+ϵ/3.

For χ ∈ C_{0}^{∞}(G), χ ≡ 1 on K, 0 ≤ χ ≤ 1, |(Xχ)(g)| ≤ 1, we have

∫_{K}(Xψ)(g)dg = ∫_{K}(Xψ)(g)χ(g)dg
= ∫_{K}(X(ψχ))(g)dg (because Xχ = 0 on K)
= − ∫_{G−K}(X(ψχ))(g)dg (because ∫_{G} X(ψχ)(g)dg = 0)
= − ∫_{G−K}(Xψ)(g)χ(g)dg − ∫_{G−K}(Xχ)(g)ψ(g)dg

But | ∫_{G−K}(Xψ)(g)χ(g)dg| ≤ ∫_{G−K} |(Xψ)(g)|dg ≤ ϵ/3 and | ∫_{G−K}(Xχ)(g)ψ(g)dg| ≤ ∫_{G−K} |ψ(g)|dg ≤ ϵ/3.

Corollary 2.3. Let X be an element of 𝔤 and ψ be a C^{∞} function on G such that ψ and Xψ lie in L^{1}(G). If ξ belongs to C^{∞}(H_{π}) and η belongs to H_{π}, then :

∫_{G} ⟨ξ|π(g^{−1})η⟩ (Xψ)(g)dg = − ∫_{G} ⟨dπ(X)ξ|π(g^{−1})η⟩ ψ(g)dg.

Proof. The corollary follows from 2.2, 2.1 (1) and from the equality :

∫_{G} φ_{ξ,η}(g)(Xψ)(g)dg + ∫_{G}(Xφ_{ξ,η})(g)ψ(g)dg = ∫_{G}(X(φ_{ξ,η}ψ))(g)dg = 0.

Let X_{1}, …, X_{n} be a basis of 𝔤; set Δ = − ∑_{i=1}^{n} X_{i}^{2} ∈ U(𝔤) The operator dπ(Δ) is essentially selfadjoint [8], and we denote by \overline{dπ(Δ)} its closure and by D(\overline{dπ(Δ)}) the domain of \overline{dπ(Δ)}.

Corollary 2.4. If ψ is a C^{∞} function on G such that ψ, X_{i}ψ and X_{i}^{2}ψ belong to L^{1}(G) for every i = 1, …, n, then for every ξ in D(\overline{dπ(Δ)}) and for every η in H_{π} :

∫_{G} φ_{ξ,η}(g)(Δψ)(g)dg = ∫_{G} φ_{\overline{dπ(Δ)}ξ,η}(g)ψ(g)dg.

Proof. By lemmas 2.1 and 2.3, the equality is true for ξ ∈ C^{∞}(H_{π}). If ξ ∈ D(\overline{dπ(Δ)}), there exists a sequence (ξ_{k})_{k≥1} in C^{∞}(H_{π}) such that

∥ξ_{k} − ξ ∥ + ∥\overline{dπ(Δ)}ξ_{k} − \overline{dπ(Δ)}ξ ∥ ⟶_{k→∞} 0.

Then :  | ∫ φ_{ξ,η}(g)(Δψ)(g)dg − ∫ ⟨\overline{dπ(Δ)}ξ|π(g^{−1})η⟩ ψ(g)dg|
≤ | ∫ ⟨ξ − ξ_{k}|π(g^{−1})η⟩ (Δψ)(g)dg|

+ | ∫ ⟨ξ_{k}|π(g^{−1})η⟩ (Δψ)(g)dg

− ∫ ⟨\overline{dπ(Δ)}ξ_{k}|π(g^{−1})η⟩ ψ(g)dg|

+ | ∫ ⟨\overline{dπ(Δ)}ξ_{k} − \overline{dπ(Δ)}ξ|π(g^{−1})η⟩ ψ(g)dg|

≤ (∥ξ − ξ_{k} ∥∥Δψ ∥_{1} + ∥\overline{dπ(Δ)}(ξ_{k} − ξ)∥∥ψ ∥_{1}) ∥η ∥ ⟶_{k→∞} 0.

This finishes the proof.

Proposition 2.5. The following conditions are equivalent :

1. π has a non zero fixed vector.

2. 0 is an eigenvalue for dπ(Δ).

3. 0 is an eigenvalue for \overline{dπ(Δ)}.

Proof. The implications 1) ⇒ 2) and 2) ⇒ 3) are obvious.

We prove now 3) ⇒ 1) : Let ξ be a non zero vector in Ker(\overline{dπ(Δ)}). Let η ∈ H_{π} and ψ ∈ C_{0}^{∞}(G). Then, using Corollary 2.4, we have in the sense of weak derivatives :

(Δφ_{ξ,η}, ψ) = (φ_{ξ,η}, Δψ) = (φ_{\overline{dπ(Δ)}ξ,η}, ψ) = 0.

Therefore, Δφ_{ξ,η} = 0 as a distribution, and since Δ is hypo-elliptic (see [8]), φ_{ξ,η} is a C^{∞} function on G.

Since η is arbitrary, this implies that g ↦ π(g)ξ is weakly C^{∞} and so, by a lemma due to Poulsen (see [15]), g ↦ π(g)ξ is strongly C^{∞}. Therefore, \overline{dπ(Δ)}ξ = dπ(Δ)ξ = 0. Hence,

∑_{i=1}^{n} ∥dπ(X_{i})ξ ∥^{2}= ⟨dπ(Δ)ξ|ξ⟩ = 0.

Since {X_{i}}_{i=1,...,n} is a basis of 𝔤, this implies that dπ(X)ξ = 0 for every element of 𝔤. Hence, π(g)ξ = ξ for every g in V , a suitable neighbourhood of e in G. As G is connected, V generates G and so ξ is fixed under the action of G.

3. Spectral characterisation of almost invariant vectors

The goal of this section is to prove :

Theorem 3.1. For a unitary representation (π, H_{π}) of G, the following conditions are equivalent :

1. π almost has invariant vectors.

2. 0 is an approximate eigenvalue of dπ(Δ).

3. 0 is a spectral value of \overline{dπ(Δ)}.

Proof. The equivalence between 2) ⇔ 3) is clear, since \overline{dπ(Δ)} is the closure of dπ(Δ).

2) ⇒ 1)

If 0 is an approximate eigenvalue of dπ(Δ), there exists a sequence {ξ_{m}}_{m≥0} of unit vectors in C^{∞}(H_{π}) such that lim_{m→+∞} ∥dπ(Δ)ξ_{m} ∥= 0. So,

lim_{m→+∞} ∥dπ(X_{i})ξ_{m} ∥= 0 for every i = 1, …, n.

Let V defined by

V = { Π_{1≤i≤n} exp(t_{i}X_{i}) | − 1 ≤ t_{i} ≤ 1}.

V is a compact neighbourhood of e, see [5]. Moreover, for every X in 𝔤 and every t ≥ 0, we have :

π(exp(tX))ξ_{m} − ξ_{m} = ∫_{0}^{t} π(exp(sX))dπ(X)ξ_{m}ds .

Let ϵ > 0. Then, for 0 ≤ t ≤ 1, we have :

∥π(exp(tX_{i}))ξ_{m} − ξ_{m} ∥≤ t ∥dπ(X_{i})ξ_{m} ∥≤ ϵ/n

for i = 1, ⋯, n and m sufficiently large.

This implies that ∥π(g)ξ_{m} − ξ_{m} ∥≤ ϵ for every g in V as soon as m is large enough.

So we proved that, for every ϵ > 0, there exist (ϵ, V)-invariant vectors. As G is connected, V generates G and one easily deduces that π has (ϵ, K)-invariant vectors for any compact subset K of G.

The remainder of this section is devoted to the proof of the implication 1) ⇒ 3) which is much more involved. For this, we need to recall some facts about the heat kernel associated with Δ.

Let h denote the closure of Δ acting on L^{2}(G). As −h is a selfadjoint and negative definite operator, by the Hille-Yosida theorem (see [16]), −h is the infinitesimal generator of a strongly continous semi-group of contractions T (t).

It is known (see [13]) that, for every t > 0, T (t) is given by the convolution from the right with a function p_{t}, the heat kernel. The function p_{t} is a smooth function on G with positive values and integral 1. We briefly recall its construction.

Let δ denote the modular function associated with our right Haar measure, thus :

∫_{G} δ(x)f(xy)dy = ∫_{G} f(y)dy,

for every integrable function f over G. The right regular representation over L^{2}(G) is given by :

(ρ(x)ξ)(y) = ξ(yx).

The left regular representation is then given by :

(λ(x)ξ)(y) = δ(x^{−1})^{1/2}ξ(x^{−1}y).

If Δ = −∑_{i=1}^{n} X_{i}^{2} and if h is the closure of Δ on L^{2}(G), then h and λ(x) commute

for every x in G.

For every t ≥ 0, let T (t) = exp(−th).

We claim that for every t > 0, T (t) is a regularising operator, that is, for every ξ ∈ L^{2}(G), for every integer m ≥ 1, h^{m}T (t)ξ ∈ D(h). Indeed, let h = ∫_{0}^{∞} λdE(λ) be the spectral decomposition of h. As the function λ ↦ λ^{2m} exp(−2tλ) is bounded on ℝ_{+}, we have :

∫_{0}^{∞} λ^{2m} exp(−2tλ)dE_{ξ}(λ) =∥h^{m}T (t)ξ ∥^{2}< ∞, ∀ξ ∈ L^{2}(G).

By Sobolev’s lemma, T (t)ξ ∈ C^{∞}(G), ∀ξ ∈ L^{2}(G).

By Schwartz Kernel theorem, there exists a C^{∞} function p′_{t} : G × G → ℝ_{+} such that

(T (t)ξ)(x) = ∫_{G} p′_{t}(x, y)ξ(y)dy, ∀ξ ∈ L^{2}(G).

Hence ,

(λ(x)T (t)ξ)(x′) = δ(x^{−1})^{1/2}(T (t)ξ)(x^{−1}x′)
= δ(x^{−1})^{1/2} ∫_{G} p′_{t}(x^{−1}x′, y)ξ(y)dy

Moreover

(T (t)λ(x)ξ)(x′) = ∫_{G} p′_{t}(x′, y)(λ(x)ξ)(y)dy
= δ(x^{−1})^{1/2} ∫_{G} p′_{t}(x′, y)ξ(x^{−1}y)dy
= δ(x^{−1})^{1/2} ∫_{G} δ(x)p′_{t}(x′, xy)ξ(y)dy

As T (t) and λ(x) commute, we deduce that

(∗) p′_{t}(x^{−1}x′, y) = δ(x)p′_{t}(x′, xy) ∀x, x′, y ∈ G.

Set p_{t}(x) = p′_{t}(e, x). The above relation (∗) shows that, for all y ∈ G,

p_{t}(x^{−1}y)δ(x^{−1}) = p′_{t}(e, x^{−1}y)δ(x^{−1}) = p′_{t}(x, y).

Hence

(T (t)ξ)(x) = ∫ p′_{t}(x, y)ξ(y)dy = ∫ p_{t}(x^{−1}y)δ(x^{−1})ξ(y)dy
= ∫ p_{t}(y)ξ(xy)dy for ξ ∈ L^{2}(G).

Recall that if π is a representation of G, then π(f) is the operator defined for f ∈ L^{1}(G) by

π(f)η = ∫ f(y)π(y)ηdy for every η ∈ H_{π}.

Taking π = ρ and f = p_{t}, we see that :

(ρ(p_{t})ξ)(x) = ∫ p_{t}(y)(ρ(y)ξ)(x)dy = ∫ p_{t}(y)ξ(xy)dy = (T (t)ξ)(x).

As T (t) = T (t)^{∗}, we have p_{t} = p_{t}^{∗}, and p_{t} is a solution of the heat equation :

\frac{∂p_{t}}{∂t} = −Δp_{t}.

Lemma 3.2. Δp_{t} belongs to L^{1}(G) and

lim_{s→0^{+}} ∥ \frac{p_{t+s} − p_{t}}{s} + Δp_{t} ∥_{1}= 0, for every t > 0.

Proof. We denote by ρ_{1} the right regular representation of G on L^{1}(G). Following ([13], theorem 4, p.599), we set for t > 0 and for f ∈ L^{1}(G) :

P^{t}f = ∫_{G} p_{t}(y)ρ_{1}(y)fdy.

By lemma 7.1 and theorem 4 of [13], P^{t}f is an analytic vector in the following sense : there exists s > 0 such that

∑_{m=0}^{∞} \frac{1}{m!} ∑_{1≤i_{1},...,i_{m}≤n} ∥dρ_{1}(X_{i_{1}}) … dρ_{1}(X_{i_{m}})P^{t}f ∥_{1} s^{m} < ∞.

In particular, p_{t} = P^{t/2}p_{t/2} is analytic in the preceding sense, and hp_{t} ∈ L^{1}(G). The last assertion is a consequence of mean value theorem and of Lebesgue’s dominated convergence theorem.

Let now π be a unitary representation of G. Set

S(t) = π(p_{t}), if t > 0 and S(0) = 𝕀_{H_{π}}.

(S(t))_{t≥0} is a strongly continuous semi-group on H_{π} with infinitesimal generator A. Moreover S(t)^{∗} = S(t) for every t because p_{t} = p_{t}^{∗}.

By Corollary 10.6, p.41 of [14], A is selfadjoint. More precisely the following is true.

Lemma 3.3. With the same notations, one has A = −\overline{dπ(Δ)}, where \overline{dπ(Δ)} denotes the closure of dπ(Δ).

Proof. Let ξ be in D(\overline{dπ(Δ)}). We claim that, for every t > 0, S(t)ξ belongs to D(A) and

AS(t)ξ = −S(t)\overline{dπ(Δ)}ξ.

Indeed, let η in H_{π}. Then, for 0 < s ≤ 1 :

⟨\frac{S(s)S(t)ξ − S(t)ξ}{s} |η⟩ = \frac{1}{s} ⟨π(p_{t+s})ξ − π(p_{t})ξ|η⟩
= \frac{1}{s} ∫ (p_{t+s}(g) − p_{t}(g)) ⟨ξ|π(g^{−1})η⟩ dg
= ∫ \frac{p_{t+s}(g) − p_{t}(g)}{s}φ_{ξ,η}(g)dg.

By lemma 3.2 and corollary 2.4,

lim_{s→0^{+}} ⟨\frac{S(s)S(t)ξ − S(t)ξ}{s} |η⟩ = ∫ (−Δp_{t})(g)φ_{ξ,η}(g)dg
= − ⟨S(t)\overline{dπ(Δ)}ξ|η⟩ .

By theorem 1.3, p.43 of [14], S(t)ξ belongs to D(A) and AS(t)ξ = −S(t)\overline{dπ(Δ)}ξ. This proves the claim. But ∥S(t)ξ − ξ ∥ ⟶_{t→0} 0 and

∥AS(t)ξ − (−\overline{dπ(Δ)}ξ)∥=∥−S(t)\overline{dπ(Δ)}ξ + \overline{dπ(Δ)}ξ ∥ ⟶_{t→0} 0.

Since A is a closed operator, ξ belongs to D(A) and Aξ = −\overline{dπ(Δ)}ξ.
So −\overline{dπ(Δ)} ⊂ A, and as they are selfadjoint, we have −\overline{dπ(Δ)} = A.

Lemma 3.4. If (π, H_{π}) is a unitary representation of a locally compact group G with almost invariant vectors and if μ is a probability measure over G, the spectrum of the operator π(μ) = ∫ π(g)dμ(g) contains 1.

Proof. Let (ξ_{k})_{k≥1} a sequence of unit vectors in H_{π} such that φ_{ξ_{k},ξ_{k}} tends to 1 uniformly over every compact set of G. We show that

lim_{n→∞} ∥π(μ)ξ_{k} − ξ_{k} ∥= 0.

Let ϵ > 0; there exists a compact set K in G such that μ(G − K) ≤ ϵ. So we have

∥π(μ)ξ_{k} − ξ_{k} ∥≤ ∫_{K} ∥π(g)ξ_{k} − ξ_{k} ∥ dμ(g) + 2ϵ < 3ϵ

for k sufficiently large.

As we now show, the converse is true under some restrictions on μ. Although we shall not use this result, we give the proof as we think it is of interest.

Lemma 3.5. Let G be a locally compact group and let μ be a absolutely continuous probability measure with respect to the Haar measure and such that the support generates topologically G. Let π be a unitary representation on a Hilbert space H_{π}. Then π has almost invariant vectors, if the spectrum of the operator π(μ) = ∫ π(g)dμ(g) contains 1.

Proof. By replacing μ by \frac{1}{2}(μ + μ̌), we can assume that μ is symmetric. Then π(μ) is selfadjoint and there exist unit vectors ξ_{n} such that ∥π(μ)ξ_{n} − ξ_{n} ∥→ 0.

So we have :

lim_{n→∞} ∫_{G}(1 − Re ⟨π(x)ξ_{n}|ξ_{n}⟩)dμ(x) = 0.

As μ ≥ 0 and 1 − Re ⟨π(x)ξ_{n}|ξ_{n}⟩ ≥ 0, there exist a subsequence of {ξ_{n}}_{n≥0} (that we will also denote by {ξ_{n}}_{n≥0}), such that

lim_{n→∞} 1 − Re ⟨π(x)ξ_{n}|ξ_{n}⟩ ≥ 0

for μ-almost every x ∈ G.

On an other hand, by compactness of the unit ball of L^{∞} gifted with the weak^{∗} topology, there exists a subsequence of {ξ_{n}}_{n≥0} (still denoted by {ξ_{n}}_{n≥0}) and a positive type function φ on G such that

lim_{n→∞} φ(x) − Re ⟨π(x)ξ_{n}|ξ_{n}⟩ ≥ 0

almost everywhere with respect to the Haar measure (this is true first for the weak^{∗} topology σ(L^{∞}, L^{1}) and the claim follows by the same arguments as before).

As μ is absolutely continuous, we have φ = 1 μ-almost everywhere. As φ is a measurable, positive definite function, φ is continuous. Hence φ = 1 on the support of μ.

Therefore, φ = 1 on the closed subgroup generated by Supp(μ) which is G, by assumption.

So lim_{n→∞} Re ⟨π(x)ξ_{n}|ξ_{n}⟩ = 1 almost everywhere on G and by Lebesgue’s dominated convergence, this is also true in the σ(L^{∞}, L^{1}) topology. By [2], Theorem 13.5.2, it follows that lim_{n→∞} Re ⟨π(x)ξ_{n}|ξ_{n}⟩ = 1 uniformly on every compact subset of G and, hence, lim_{n→∞} ∥π(x)ξ_{n} − ξ_{n} ∥= 0 uniformly on compact subsets of G. This shows that π almost has invariant vectors.

Now we are able to finish the proof of Theorem 3.1.

If π almost has invariant vectors, by lemma 3.4, the spectrum of π(p_{t}) contains 1, for every t > 0.

Now −\overline{dπ(Δ)} is the infinitesimal generator of the semi-group (π(p_{t}))_{t≥0} so that 0 is in the spectrum of \overline{dπ(Δ)} by functional calculus and lemma 3.3.

One may wonder whether it is necessary to use, as we did, arguments involving the heat kernel to prove 1) ⇒ 3). For instance, one might think that if, for a compact neighbourhood V of e in G, {ξ_{n}}_{n≥0} is a family of (1/n, V)- invariant C^{∞} vectors, then ∥dπ(Δ)ξ_{n} ∥ should tend to 0. The following example shows that this is not always true.

Example 3.6. Let G = ℝ be the real line. We define first, the following family of unitary representations of degree 2 :

s ↦ π_{n}(s) = ( \begin{matrix}exp(is/n) & 0 \\ 0 & exp(isn) \end{matrix} ) .

We consider the unitary representation π defined by s ↦ ⊕_{n≥0} π_{n}(s).

We also define ξ_{n} = (0, …, 0, \sqrt{\frac{n−1}{n}}, \sqrt{\frac{1}{n}}, 0, …) where the nonzero components of ξ_{n} are in 2n^{th} and (2n + 1)^{th} places.

By construction the ξ_{n} have norm 1. Let I be a compact subset of ℝ. Then, for every s in I, we have :

∥π(s)ξ_{n} − ξ_{n} ∥^{2} = |(exp(is/n) − 1)\sqrt{\frac{n − 1}{n}} |^{2} + |(exp(isn) − 1)\sqrt{\frac{1}{n}} |^{2}
= |(exp(is/n) − 1)|^{2} \frac{n − 1}{n} + | exp(isn) − 1|^{2} \frac{1}{n} .

As | exp(is/n) − 1|^{2} ⟶_{n→∞} 0 uniformly for s in I and as | exp(isn) − 1|^{2} ≤ 4, the ξ_{n} are (ϵ_{n}, I)-invariant vectors for a sequence {ϵ_{n}}_{n≥0} with ϵ_{n} > 0 and lim ϵ_{n} = 0.

However this family ξ_{n} does not satisfy ∥dπ(Δ)ξ_{n} ∥ ⟶_{n→∞} 0.

In fact,

dπ(Δ)ξ_{n} = (0, …, 0, \frac{1}{n^{2}} \sqrt{\frac{n − 1}{n}}, n^{2}\sqrt{\frac{1}{n}}, 0, …).

Thus, ∥dπ(Δ)ξ_{n} ∥≈ n^{3/2}; so it does not tend to 0.

Lemma 3.7. Let h be a selfadjoint operator on a Hilbert space H, with domain D(h), and such that its spectrum is bounded from below. Then

min(Sp h) = inf_{ξ∈D(H)^{1}} ⟨hξ|ξ⟩ , where D(H)^{1} = {ξ ∈ D(H)| ∥ξ ∥= 1}.

Proof. As h is selfadjoint, its residual spectrum is empty. So every spectral value is an approximate eigenvalue. For every spectral value λ, there exists a sequence {ξ_{n}}_{n≥0} in D(h)^{1} such that ⟨hξ_{n}|ξ_{n}⟩ → λ.

Hence,

inf_{ξ∈D(h)^{1}} ⟨hξ|ξ⟩ ≤ λ .

Let λ_{0} = min(Sp(h)) (the minimum exists since the spectrum of h is real, closed and bounded below). If we apply the last inequality to λ_{0}, we get :

inf_{ξ∈D(h)^{1}} ⟨hξ|ξ⟩ ≤ λ_{0} .

As for the other inequality, let h = ∫ λdE(λ) be the spectral decomposition of h; we have ⟨hξ|ξ⟩ = ∫ λ ⟨dE(λ)ξ|ξ⟩ and ⟨ξ|ξ⟩ = ∫ ⟨dE(λ)ξ|ξ⟩ for every fixed ξ in D(h). Then, for ξ in D(h)^{1},

⟨hξ|ξ⟩ = ∫_{spec(h)} λ ⟨dE(λ)ξ|ξ⟩ ≥ λ_{0} ∫_{spec(h)} ⟨dE(λ)ξ|ξ⟩ = λ_{0} .

This finishes the proof.

Definition 3.8. For a unitary representation π of G, we define the constant k(\overline{dπ(Δ)}, G) by

k(\overline{dπ(Δ)}, G) = inf_{ξ∈D(\overline{dπ(Δ)})^{1}} ⟨\overline{dπ(Δ)}ξ|ξ⟩ .

Corollary 3.9. The following holds :

k(\overline{dπ(Δ)}, G) = min Sp(\overline{dπ(Δ)}) = inf_{ξ∈[C^{∞}(Hπ)]^{1}} ∑_{i=1}^{n} ∥dπ(X_{i})ξ ∥^{2}

Proof. The first equality comes from the preceding lemma. On the other hand, it is clear that

k(\overline{dπ(Δ)}, G) ≤ inf_{ξ∈[C^{∞}(Hπ)]^{1}} ⟨\overline{dπ(Δ)}ξ|ξ⟩ = inf_{ξ∈[C^{∞}(Hπ)]^{1}} ∑_{i=1}^{n} ∥dπ(X_{i})ξ ∥^{2} .

To obtain the reverse inequality, it suffices to show that, for ϵ > 0, there exists a C^{∞} vector η of norm 1 such that

|k(\overline{dπ(Δ)}, G) − ⟨dπ(Δ)η|η⟩ | < ϵ .

By definition, there exists ξ in D(\overline{dπ(Δ)}) of norm 1 such that

0 ≤ ⟨\overline{dπ(Δ)}ξ|ξ⟩ − k(\overline{dπ(Δ)}, G) < ϵ/3 .

As \overline{dπ(Δ)} is the closure of dπ(Δ), there exists a C^{∞} vector η of norm 1 which is arbitrarily close to ξ with respect to the graph norm. As,

|k(\overline{dπ(Δ)}, G) − ⟨dπ(Δ)η|η⟩ | ≤ |k(\overline{dπ(Δ)}, G) − ⟨dπ(Δ)ξ|ξ⟩ | +
| ⟨\overline{dπ(Δ)}ξ|ξ⟩ − ⟨\overline{dπ(Δ)}ξ|η⟩ | +
| ⟨\overline{dπ(Δ)}ξ|η⟩ − ⟨dπ(Δ)η|η⟩ |

this proves the claim.

Theorem 3.10. G has property (T) if and only if there exists an ϵ > 0 such that k(\overline{dπ(Δ)}, G) ≥ ϵ for every unitary representation π of G without non zero fixed vectors.

Proof. ⇐) If π has almost invariant vectors, by thm 3.1, k(\overline{dπ(Δ)}, G) = min Sp(\overline{dπ(Δ)}) = 0. The assumption then implies that π has non zero fixed vectors, i.e. G has property (T). ⇒) Assume by contradiction that there exists a sequence of unitary representations \{π_{n}\}_{n≥0} without non zero fixed vector such that k(\overline{dπ_{n}(Δ)}, G) → 0. We claim that the representation σ = ⨁_{n≥0} π_{n} satisfies k(\overline{dσ(Δ)}, G) = 0. By the assumption, there exists, for every n, a vector ξ_{n} ∈ C^{∞}(H_{π_{n}})^{1} such that ⟨dπ_{n}(h)ξ_{n} | ξ_{n}⟩ < 1/n + k(\overline{dπ_{n}(Δ)}, G).

The vector η_{n} defined by η_{n} = (0, ⋯, 0, ξ_{n}, 0 ⋯), with ξ_{n} at the n^{th} place, is a C^{∞} vector in H_{σ}^{1} and

⟨\overline{dσ(Δ)}η_{n} | η_{n}⟩ = ⟨\overline{dπ_{n}(Δ)}ξ_{n} | ξ_{n}⟩ < 1/n + k(\overline{dπ_{n}(Δ)}, G).

Hence, k(\overline{dσ(Δ)}, G) = 0 and 0 is in the spectrum of \overline{dσ(Δ)}. By Theorem 3.1, this implies that σ has almost invariant vectors. Since G has property T, σ has a nonzero fixed vector {β_{n}}_{n≥0}. Choose n so that β_{n} ≠ 0. Then β_{n} is a non zero fixed vector for π_{n}, contradicting the assumption.

We can define K(Δ, G) = inf_{π∈_{G^{∗}}} k(\overline{dπ(Δ)}, G), the Laplacian Kazhdan constant.

Corollary 3.11. G has property (T) if and only if K(Δ, G) > 0.

This corollary is a direct consequence of the preceding proposition.

Remark 3.12. The above results (Theorem 3.1, Theorem 3.10 and Corollary 3.11) remain true if the vectors X_{1}, …, X_{n} are only assumed to generate 𝔤 as a Lie algebra. The relevant facts about the heat kernel associated with the sub-Laplacian Δ = − ∑_{i=1}^{n} X_{i}^{2} hold in this more general situation.

4. Comparison with the classical constants

Proposition 4.1. Let G be a connected Lie group and let {X_{i}}_{i=1,...,n} be a basis of its Lie algebra. Fix ϵ > 0 and set S = {exp(tX_{i}) | t ∈ [0, ϵ], i ∈ {1, …, n}}. Then, for any unitary representation π of G, one has :

κ(G, S, π) ≤ ϵ\sqrt{k(\overline{dπ(Δ)}, G)}.

In particular κ(G, S) ≤ ϵ\sqrt{K(Δ, G)}.

Proof. Let π be a unitary representation of G and let ξ ∈ C^{∞}(H_{π}).

Then sup_{s∈S} ∥π(s)ξ − ξ ∥ = sup_{t∈[0,ϵ]} max_{i∈{1...n}} ∥π(e^{tX_{i}})ξ − ξ ∥
= sup_{t∈[0,ϵ]} max_{i∈{1...n}} ∥ ∫_{0}^{t} π(e^{sX_{i}})dπ(X_{i})ξds∥
≤ sup_{t∈[0,ϵ]} max_{i∈{1...n}} ∫_{0}^{t} ∥π(e^{sX_{i}})dπ(X_{i})ξ ∥ ds
≤ ϵ max_{i∈{1...n}} ∥dπ(X_{i})ξ ∥
≤ ϵ\sqrt{∑_{i=1}^{n} ∥dπ(X_{i})ξ ∥^{2}} .

So

κ(G, S, π) = inf_{ξ∈H^{1}_{π}} sup_{s∈S} ∥π(s)ξ − ξ ∥
≤ inf_{ξ∈(C^{∞}(Hπ))^{1}} sup_{s∈S} ∥π(s)ξ − ξ ∥
≤ ϵ inf_{ξ∈(C^{∞}(Hπ))^{1}} \sqrt{∑_{i=1}^{n} ∥dπ(X_{i})ξ ∥^{2}}
= ϵ\sqrt{k(\overline{dπ(Δ)}, G)}

This ends the proof.

Remark 4.2. (i) 4.1 gives an alternative proof of implication (⇒) in 3.10. (ii) Proposition 4.1 states that :

κ(G, S, π) ≤ ϵ\sqrt{k(\overline{dπ(Δ)}, G)},

for every unitary representation π of G. However the converse fails very strongly : there exists no continuous function f : ℝ_{+} → ℝ_{+}, such that f(0) = 0 and

k(\overline{dπ(Δ)}, G) ≤ f(κ(G, S, π)),

for every connected Lie group G and for every unitary representation π of G. Indeed, κ(G, S, π) ≤ 2 for any representation π, and we shall exhibit a sequence (π_{k})_{k≥1} of unitary representations of ℝ such that k(\overline{dπ_{k}(Δ)}, G) = k^{2} for every k ≥ 1.

Let H_{k} = L^{2}([k, +∞)) et T_{k} be the selfadjoint operator on H_{k} defined by :

D(T_{k}) = {ξ ∈ H_{k} | ∫_{k}^{∞} λ^{2}|ξ(λ)|^{2}dλ < ∞} and (T_{k}ξ)(λ) = λ ξ(λ).

If π_{k}(t) = exp(itT_{k}), we have :

k(\overline{dπ_{k}(Δ)}, ℝ) = inf_{ξ∈D(T_{k})^{1}} ⟨T_{k}^{2}ξ | ξ⟩ = k^{2}.

Example 4.3. 1. Let G = SL(2, ℝ). Take, as basis of the Lie algebra 𝔰𝔩_{2}(ℝ), the matrices

H = \frac{1}{2} ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , V = \frac{1}{2} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , W = \frac{1}{2} ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) .

Then Δ = −(H^{2} + V^{2} + W^{2}) is equal to \frac{1}{4}(ω + 2W^{2}) with ω the Casimir operator in U(𝔰𝔩_{2}(ℝ)).

Let us use the coordinates (x, y, θ) on G, where a group element is expressed as

g = ( \begin{matrix}\sqrt{y} & x \\ 0 & \frac{1}{\sqrt{y}} \end{matrix} ) ( \begin{matrix}cos θ & sin θ \\ − sin θ & cos θ \end{matrix} ) , (x, θ ∈ ℝ, y > 0).

Then, in these coordinates, we find, viewing H, V, W as left invariant vector fields on G :

Δ = −y^{2}(\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) + y \frac{∂^{2}}{∂x∂θ} + \frac{1}{2} \frac{∂^{2}}{∂θ^{2}}

(see S. Lang, [9], Chap. X, §1 and §2).

Let ℍ ≅ G/K, K = SO(2), be the Poincaré upper half space

ℍ = {z = x + iy | x, y ∈ ℝ, y > 0},

with invariant measure y^{−2}dxdy.

Let π be the left regular representation of G on L^{2} (ℍ). As the functions on ℍ are independent of θ on the right, we have

dπ(Δ) = −y^{2}(\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} )

which is the Laplace-Beltrami operator on ℍ .

It is well known that Sp(dπ(Δ)) ⊆ [1/4; +∞[. Here is a very elementary argument for this due to Mac Kean [12]: Let f be a real smooth function on ℍ with compact support. Then

\frac{1}{4} (∫_{0}^{∞} f(x, y)^{2}y^{−2}dy)^{2} = (∫_{0}^{∞} f(x, y)\frac{∂f}{∂y} (x, y)y^{−1}dy)^{2}
≤ ∫_{0}^{∞} f(x, y)^{2}y^{−2}dy ∫_{0}^{∞} (\frac{∂f}{∂y} (x, y))^{2} dy

So

\frac{1}{4} ∫_{0}^{∞} f(x, y)^{2}y^{−2}dy ≤ ∫_{0}^{∞} (\frac{∂f}{∂y} (x, y))^{2} dy.

Hence

\frac{1}{4} ∥f ∥_{L^{2}(ℍ)}^{2} ≤ ∫_{−∞}^{+∞} dx ∫_{0}^{∞} (\frac{∂f}{∂y} (x, y))^{2} dy

≤ ∫_{−∞}^{+∞} dx ∫_{0}^{∞} [(\frac{∂f}{∂x}(x, y))^{2} + (\frac{∂f}{∂y} (x, y))^{2}] dy

= − ∫_{−∞}^{+∞} dx ∫_{0}^{∞} f(x, y) (\frac{∂^{2}f}{∂x^{2}} (x, y) + \frac{∂^{2}f}{∂y^{2}} (x, y)) dy
= ⟨dπ(Δ)f|f⟩ .

This implies that λ ≥ 1/4 for any λ ∈ Sp(π(Δ)).

It is actually known that inf Sp(\overline{dπ(Δ)}) = 1/4, that is, k(\overline{dπ(Δ)}, G) = 1/4 (see [9]).

2. Let G be the (three-dimensional) Heisenberg group. Thus G = ℝ^{3} with group law

(p, q, t)(p′, q′, t′) = (p + p′, q + q′, t + t′ + \frac{1}{2}(pq′ − qp′)).

The left invariant vectors fields on G corresponding to the coordinates (p, q, t) are :

P = \frac{∂}{∂p} − \frac{1}{2}q \frac{∂}{∂t}, Q = \frac{∂}{∂q} + \frac{1}{2}p\frac{∂}{∂t}, T = \frac{∂}{∂t}.

As is well known, G has for each h ∈ ℝ, h ≠ 0, an infinite dimensional unitary representation ρ_{h} on L^{2}(ℝ) so that

dρ_{h}(P) = hD, dρ_{h}(Q) = M, dρ_{h}(T) = \frac{h}{2πi}𝕀,

where D = \frac{1}{2πi} \frac{∂}{∂x} and M is the multiplication operator by x on L^{2}(ℝ).

A version of the Heisenberg Uncertainty Principle states that :

∥Mu∥_{2}^{2} + ∥Du∥_{2}^{2}≥ \frac{1}{2π} ∥u∥_{2}^{2}, ∀u ∈ L^{2}(ℝ)

with equality if and only if u is a multiple of the Gaussian x ↦ exp(−πx^{2}) (see [3], Corollary (1.37)). Replacing in this inequality u by the function x ↦ |h|^{1/4}u(h^{1/2}x) yields

∥Mu∥_{2}^{2} + ∥hDu∥_{2}^{2}≥ \frac{|h|}{2π} ∥u∥_{2}^{2}

with equality if u(x) = |h|^{1/4} exp(−π|h|x^{2}).

Thus, with Δ = −(P^{2} + Q^{2} + T^{2}), we see that

⟨dρ_{h}(Δ)u|u⟩ =∥Mu∥_{2}^{2} + ∥hDu∥_{2}^{2} +\frac{|h|^{2}}{4π^{2}} ∥u∥_{2}^{2}≥ (\frac{|h|}{2π} + \frac{|h|^{2}}{4π^{2}} ) ∥u∥_{2}^{2}

with equality for the above Gaussian function.

Hence, we obtain the exact value for the Kazhdan constant

k(\overline{dρ_{h}(Δ)}, G) = inf\{λ | λ ∈ Sp(dρ_{h}(Δ))\}
= (\frac{|h|}{2π} + \frac{|h|^{2}}{4π^{2}} )

3. The following well-known example was pointed out to us by A. Valette. Let G be a connected compact semi-simple Lie group. Let {X_{i}}_{i=1,···,n} be a basis of its Lie algebra 𝔤, which is orthogonal relatively to the Killing form.

Then Δ = − ∑_{i=1}^{n} X_{i}^{2} is the Casimir operator of G. As Δ ∈ Z(U(𝔤)), the

Schur lemma insures that, for π ∈ Ĝ, dπ(Δ) = c 𝕀. This constant c can be determined in the following way : if λ is the highest weight of π and ρ is the half sum of the positive roots, as we can see in [7], p. 247, we have

k(dπ(Δ), G) = ⟨λ + 2ρ | λ⟩ =∥λ + ρ∥^{2} − ∥ρ∥^{2} .

Remark 4.4. Let G be a simply connected, connected nilpotent Lie group, and let π be a unitary irreducible representation of G. In [6], a bound is given for k(\overline{dπ(Δ)}, G) in terms of the distance from 0 of the Kirillov orbit associated to π.

References

[1] Colin de Verdière, Y., Le trou spectral des graphes et leur propriété d’expansion, Séminaire de théorie spectrale et géométrie, Grenoble, (1993–1994), 51–63.

[2] Dixmier, J., “Les C^{∗}-algèbres et leurs représentations, ” Cahiers Scientifiques 29, Gauthier-Villars, 1964.

[3] Folland, G., “Harmonic Analysis on Phase Space,” Princeton University Press, 1989.

[4] de la Harpe, P., and A. Valette, “La propriété (T) pour les groupes localement compacts, ” Astérisque 175, Société Mathématique de France, 1989.

[5] Hochschild G., “La Structure des Groupes de Lie,” Monographies Universitaires de Mathématiques 27, Dunod, 1968.

[6] Hulanicki, A., J. Jenkins, and J. Ludwig, Minimum eigenvalues for positive Rockland operators, Proc. Amer. Math. Soc. 94 (1985), 718–720.

[7]  Jacobson, N., “Lie Algebras,” Interscience, 1962.

[8] Jorgensen, P., “Operators and Representation Theory,” Mathematics Studies 147, North-Holland, 1988.

[9]  Lang, S., “SL_{2}(ℝ),” Addison Wesley, 1975.

[10] Lubotzky, A., “Discrete Groups, Expanding Graphs and Invariant Measures,” Progress in Mathematics 125, Birkhäuser, 1994.

[11] Margulis, G., Explicit constructions of concentrators, Problems Inform. Transmission 9-4 (1973), 325–332.

[12] Mc Kean, h., Selberg trace formula as applied to a compact Riemann surface, Comm. Pure Appl. Math. 25 (1972), 225–246.

[13] Nelson, E., Analytic vectors, Annals of Mathematics 70 (1959), 572–615.

[14] Pazy, A., “Semigroups of Linear Operators and Applications to Partial Differential Equations,” Applied Mathematical Sciences 44, Springer, 1983.

[15] Poulsen, N., On C^{∞}-vectors and intertwining bilinear forms for representations of Lie groups, J. of Functional Analysis 9 (1972), 87–120.

[16] Yosida K., “Functional Analysis,” Die Grundlehren der mathematischen Wissenschaft 123, Springer, 1965.

URA CNRS 399 Département de mathématiques Université de Metz Ile de Saulcy F-57045 Metz, France bekka@poncelet.univmetz.fr School of mathematics, University of New South Wales Sydney, 2052, Australia pacherix@maths.unsw.edu.au Institut de mathématiques, Université de Neuchâtel Rue Emile-Argand 11 CH-2007 Neuchâtel Switzerland Jolissaint@maths.unine.ch

Received April 16, 1997 and in final form June 11, 1997