Journal of Lie Theory Volume 10 (2000) 93–105 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag Berlin

On Kazhdan’s property (T) and Kazhdan constants associated to a Laplacian for SL(3,R)

M. E. B. Bekka and M. Mayer

Communicated by D. Poguntke

Abstract. The first purpose of this paper is to give a very elementary proof of Property (T) for SL(3, k) over any local field k. Here we use a modification of an argument due to Burger. The second approach to Property (T) is based on spectral properties of a Laplacian in the enveloping algebra. It is shown that for a connected Lie group G Property (T) can be characterized by a spectral property of a Laplacian on the space of smooth K -finite vectors, where K is a compact subgroup of G.

1. Introduction

A locally compact group has Kazhdan’s Property (T) if the following holds: whenever a (strongly continuous) unitary representation of G has almost invariant vectors, then actually it has a nonzero fixed vector. Recall that a representation (π, H) has almost invariant vectors if, for any ε > 0 and for any compact set K of G , there exists a unit vector ξ ∈ H with ∥π(g)ξ − ξ∥ < ε for all g ∈ K .

Property (T), discovered in 1967 by D. Kazhdan [11], is a powerful tool, with applications, for instance, in rigidity, geometry, graph theory and operator algebras (see [14, 20, 12, 7]). Most semisimple Lie groups have Property (T). More precisely, all simple Lie groups, except those which are locally isomorphic to SO(n, 1) and SU(n, 1) , have Property (T). It is an important fact that Property (T) is inherited by lattices. So, for instance, SL(n, ℤ) has Property (T) for n ≥ 3 .

The main step in establishing Property (T) for simple Lie groups of ℝ– rank ≥ 2 is the proof for SL(3, ℝ) and Sp(2, ℝ) of this property. The usual proofs for SL(3, ℝ) (and for Sp(2, ℝ) ) are based on the study - by means of Mackey’s theory - of the irreducible unitary representations of a copy of the semi-direct product SL(2, ℝ) ⋉ ℝ^{2} inside SL(3, ℝ) (or the semi-direct product of SL(2, ℝ) by the space of symmetric 2 × 2–matrices), see [14, 20, 12]. Another argument, due to M. Burger, based on the so-called Furstenberg lemma, appears in [9]. In [10], an alternative proof is given using estimates of independent interest for matrix coefficients of unitary representations.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

The first purpose of this paper is to give a very elementary proof of Property (T) for SL(3, k) over any local field k . It is a modification of Burger’s argument mentioned above. Instead of Furstenberg lemma, it uses the fact that there is no SL(2, k)–invariant mean on the Borel sets in k^{2} \ {0} (see 2.2 below). In case k = ℝ, a new proof is given in terms of the Lie algebra for the following well–known but crucial fact (Lemma 2.4): If a vector in a unitary representation of SL(2, ℝ) is invariant under the upper triangular unipotent matrices then it is invariant under SL(2, ℝ) . In fact, our proof shows that the same result is true for the universal covering group { \atop SL(2, ℝ)} .

Our second approach to Property (T) for SL(3, ℝ) is of a quantitative nature and is based on spectral properties of a Laplacian in the enveloping algebra U(𝔰𝔩(3, ℝ)) . In [3], Kazhdan’s Property (T) for a connected Lie group G is characterized as follows: Let 𝔤 be the Lie algebra of G , let X_{1}, X_{2}, …, X_{n} be a basis of 𝔤, and let Δ := −(X_{1}^{2} + X_{2}^{2} + … + X_{n}^{2}) be the associated Laplacian in U(𝔤) . Then G has Property (T) if and only if there exists ε > 0 such that inf sp(\overline{dπ(Δ)}) ≥ ε for any unitary representation (π, H) of G without nonzero fixed vector, where dπ denotes the derived representation of π in the space H^{∞} of C^{∞}–vectors in H and sp the spectrum. As this is more convenient for computations, we first show that one may equally consider the smaller space of K –finite vectors for a compact subgroup K of G . Recall that a vector ξ ∈ H is K –finite if the linear span of π(K)ξ has finite dimension.

Theorem 1.1. The connected Lie group G has Property (T) if and only if there exists a constant ε > 0 such that

inf{⟨dπ(Δ)ξ, ξ⟩, ξ ∈ H^{∞,K}, ∥ξ∥ = 1} ≥ ε

for any unitary representation (π, H) of G without nonzero fixed vector, where H^{∞,K} is the space of all K –finite C^{∞}–vectors in H for a compact subgroup K of G . Our main result gives a bound for the constant ε appearing above.

Theorem 1.2. Let K := SO(3, ℝ) , and let X_{1}, X_{2}, …, X_{8} be the following basis of the Lie algebra 𝔰l(3, ℝ) :

X_{1} := ( \begin{matrix}0 & 1 & 0 \\ −1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) , X_{2} := ( \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 0 \end{matrix} ) , X_{3} := ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) ,

X_{4} := ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & −1 & 0 \end{matrix} ) , X_{5} := ( \begin{matrix}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ) , X_{6} := ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) ,

X_{7} := ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ −1 & 0 & 0 \end{matrix} ) , X_{8} := ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} ) ,

and let

Δ := −(X_{1}^{2} + X_{2}^{2} + … + X_{8}^{2}).

Then, for any unitary representation (π, H) of SL(3, ℝ) without nonzero fixed vector,

inf{⟨dπ(Δ)ξ, ξ⟩; ξ ∈ H^{∞,K}, ∥ξ∥ = 1} ≥ α ≈ 0.4613 ,

where α is the maximal value of the function \frac{2 sin^{2} θ}{πθ} on ℝ. The reason for the choice of the above basis is that Δ = −C + 2C_{K}, where C and C_{K} are the Casimir operators of SL(3, ℝ) and SO(3, ℝ), respectively. The proof of the above estimate is inspired by some ideas due to Howe and Tan [10], Chapter V.3.3.

Kazhdan’s constants depending on a generating set may also be defined at the group level (see [9]). Such constants have been studied, for instance, in [1, 4, 5, 6, 8, 15, 17]. The paper is organized as follows. In Section 2, we give the proof of Kazhdan’s Property (T) for SL(3, k) . Section 3 is devoted to the proof of Theorem 1.1, and Theorem 1.2 is proved in Section 4.

2. Kazhdan Property (T) for SL(3, k)

The proof depends on the following three lemmas. The first lemma says that the representation (π, H) is amenable in the sense of [2], where more general results are proved. We thank S. Popa for the following direct and simple proof.

Lemma 2.1. Let G be a locally compact group, and let (π, H) be a unitary representation of G with almost invariant vectors. Then there is an Ad(G)– invariant state φ on the C^{∗}–algebra L(H) of all bounded operators on H, that is, a positive linear form φ : L(H) → ℂ with φ(I) = 1 and φ(π(x)T π(x)^{−1}) = φ(T) for all x ∈ G, T ∈ L(H) .

Proof. Let {ξ_{n}}_{n} ⊆ H , ∥ξ_{n}∥ = 1 , with lim_{n→∞} ∥π(x)ξ_{n} − ξ_{n}∥ = 0 for all x ∈ G . Define states φ_{n} on L(H) by

φ_{n}(T) := ⟨T ξ_{n}, ξ_{n}⟩, T ∈ L(H) .

Since the set of states on L(H) is a weak–∗–compact subset of the unit ball of L(H)^{∗} , we may assume (upon passing to a subnet) that there exists a state φ on L(H) with

lim_{n→∞} φ_{n}(T) = φ(T) ∀T ∈ L(H) .

Then φ is Ad(G)–invariant as, for any x ∈ G, T ∈ L(H) ,

|φ_{n}(π(x)T π(x)^{−1}) − φ_{n}(T)| = |⟨T π(x)ξ_{n}, π(x)ξ_{n}⟩ − ⟨T ξ_{n}, ξ_{n}⟩|
≤ 2∥T ∥∥π(x)ξ_{n} − ξ_{n}∥

and hence

|φ(π(x)T π(x)^{−1}) − φ(T)| = lim_{n→∞} |φ_{n}(π(x)T π(x)^{−1}) − φ_{n}(T)| = 0 .

Recall that a local field is a locally compact, nondiscrete field (archimedean or nonarchimedean), and that the topology of such a field is defined by an absolute value. In fact, any such field is isomorphic either to ℝ, to ℂ, to a finite extension of the p–adic numbers or to the Laurent series in one variable over a finite field (see [19, Chap. I,A73]).

Lemma 2.2. Let k be a local field . Let SL(2, k) act on k^{2} in the natural way. Then the Dirac measure at {0} is the only finitely additive SL(2, k)–invariant probability measure on the Borel sets of k^{2}. Equivalently, there is no finitely additive, SL(2, k)–invariant probability measure on the Borel sets of k^{2} \ {0}.

Proof. Let

μ : B(k^{2}) → ℝ^{+}

be a finitely additive, SL(2, k)–invariant probability measure on the Borel sets B(k^{2}) of k^{2} . Let | · | be an absolute value on k . Let

Ω := \{( \begin{matrix}x \\ y \end{matrix} ) ∈ k^{2} \ \{0\}; |y| ≥ |x|\} .

Take a sequence {λ_{n}}_{n} ⊆ k with |λ_{n+1}| > |λ_{n}| + 2 for all n ∈ ℕ , and let

g_{n} := ( \begin{matrix}1 & λ_{n} \\ 0 & 1 \end{matrix} ) ∈ SL(2, k) .

Then

Ω_{n} := g_{n}Ω = \{( \begin{matrix}x \\ y \end{matrix} ) ∈ k^{2} \ \{0\}; \frac{|x|}{|λ_{n}| + 1} ≤ |y| ≤ \frac{|x|}{|λ_{n}| − 1}\} .

Indeed, for ( \begin{matrix}x \\ y \end{matrix} ) ∈ Ω ,

g_{n} ( \begin{matrix}x \\ y \end{matrix} ) = ( \begin{matrix}x + λ_{n}y \\ y \end{matrix} )

and

|x + λ_{n}y| ≥ |λ_{n}y| − |x| ≥ (|λ_{n}| − 1)|y| ,
|x + λ_{n}y| ≤ |x| + |λ_{n}y| ≤ (|λ_{n}| + 1)|y| .

Clearly the sets Ω_{n} are pairwise disjoint, as

\frac{1}{|λ_{n}| − 1} < \frac{1}{|λ_{m}| + 1} for n > m .

Hence ∑_{i=1}^{n} μ(Ω_{i}) ≤ μ(k^{2}\{0}) ≤ 1 for all n ∈ ℕ . Since μ(Ω_{i}) = μ(g_{i}Ω) = μ(Ω) , this shows that μ(Ω) = 0 . Now, let

Ω′ := ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) · Ω = \{( \begin{matrix}x \\ y \end{matrix} ) ∈ k^{2} \ \{0\}; |x| ≥ |y|\} .

Then μ(Ω′) = μ(Ω) = 0 . Since Ω ∪ Ω′ = k^{2} \ {0}, μ(k^{2} \ {0}) = 0 . So, μ is the Dirac measure at 0.

Remark 2.3. As the proof shows, the above lemma applies to other groups than SL(2, k) , for instance to SL(2, ℤ) when char k = 0. The conclusion of the lemma is certainly known to several people. A. Valette showed us a proof of the lemma, in the case k = ℝ , using only two matrices from SL(2, ℤ). The last ingredient is the following well-known lemma (see [9, 14, 20, 12]) for which we give a new proof based on consideration of the Lie algebra, in the case where k = ℝ.

Lemma 2.4. Let (π, H) be a unitary representation of SL(2, k). Assume that

N := \{( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) , x ∈ k\}

has a nonzero fixed vector ξ ∈ H . Then ξ is fixed by SL(2, k) .

Proof. (k = ℝ) Let

H := ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) X := ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) Y := ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )

be the standard basis of 𝔰𝔩(2, ℝ) with usual commutator relations

[X, Y] = H, [H, X] = 2X, [H, Y] = −2Y .

The space H^{N} of the N –invariant vectors is invariant under the subgroup A generated by H . Hence, H^{N} contains a dense subspace D of C^{∞}–vectors under the action of AN . Clearly, X = 0 on D and D is H –invariant. We first show that H = 0 on D . We write W ξ instead of dπ(W)ξ for W ∈ 𝔰𝔩(2, ℝ), ξ ∈ H , whenever this makes sense. Recall that each W is skew-symmetric on H^{∞} . In fact, it is well-known that iW is essentially selfadjoint on H^{∞} for any W ∈ 𝔰𝔩(2, ℝ) . Consider

C := H^{2} + 2(XY + Y X) = H^{2} + 4XY − 2H ,

the Casimir operator of 𝔰𝔩(2, ℝ) . We have, for any ξ ∈ D, η ∈ H^{∞} :

⟨ξ, Cη⟩ = ⟨ξ, H^{2}η⟩ + 4⟨ξ, XY η⟩ − 2⟨ξ, Hη⟩
= ⟨ξ, (H^{2} − 2H)η⟩ = ⟨(H^{2} + 2H) ξ, η⟩ .

Thus, D is contained in the domain of C^{∗} and C^{∗} = H^{2} + 2H on D . As is well-known (see [18, p.269, Ex.(3)], for instance), C is essentially selfadjoint on H^{∞} . Whence C|_{D} = C^{∗}|_{D} or (H^{2} + 2H)^{∗} = H^{2} − 2H on D . Thus H = 0 on D . Now, fix ξ ∈ D . Then , for any η ∈ H^{∞} ,

⟨ξ, Y Hη⟩ = 2⟨ξ, Y η⟩ ,

as [H, Y] = −2Y and Hξ = 0 . On the other hand, the range of H − 2I as an operator on H^{∞} is dense in H . Indeed, this follows from the fact that iH is essentially selfadjoint on H . Thus,

⟨ξ, Y η⟩ = 0 for all η ∈ D′ ,

where D′ = (H − 2I)H^{∞} . This shows that ξ is in the domain of Y^{∗} = −\overline{Y} and that \overline{Y} ξ = 0 . Hence, by Stone’s theorem,

exp(tY)ξ = exp(t\overline{Y})ξ = ξ

for all t ∈ ℝ . Thus ξ is fixed by the subgroup \overline{N} , generated by Y . Hence, any ξ ∈ D is fixed by SL(2, ℝ) . By density, this is true for any N –fixed vector in H .

Remark 2.5. The above proof is somewhat involved because it is not a priori clear whether the space of N –fixed vectors contains any nonzero C^{∞}–vector. The arguments above become much shorter in the case of an N –invariant C^{∞}–vector ξ as the reader may wish to verify. On the other hand, because it relies on Lie algebra considerations, the proof works for any covering group of SL(2, ℝ) (where N has to be taken as the one–parameter subgroup generated by X).

Theorem 2.6. [11] SL(3, k) has Kazhdan’s Property (T).

Proof. Let (π, H) be a unitary representation of SL(3, k) with almost invariant vectors. Let

H := \{( \begin{matrix}a & b & x \\ c & d & y \\ 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, k), ( \begin{matrix}x \\ y \end{matrix} ) ∈ k^{2}\} ≃ SL(2, k) ⋉ k^{2},

and let

V := \{( \begin{matrix}1 & 0 & x \\ 0 & 1 & y \\ 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}x \\ y \end{matrix} ) ∈ k^{2}\} ≃ k^{2} .

Let

P : B(V̂) → L(H), E ↦ P (E)

be the projection valued measure associated with the unitary representation π|_{V} of the abelian group V , see e. g. , [13]. Clearly

π(g)P (E)π(g)^{−1} = P (g · E) ∀g ∈ SL(2, k), E ∈ B(V̂) .  (1)

Here

g · γ ( \begin{matrix}x \\ y \end{matrix} ) := γ (g^{−1} ( \begin{matrix}x \\ y \end{matrix} )) , g ∈ SL(2, k), γ ∈ V̂

is the dual action. By Lemma 2.1, there exists an Ad(SL(3, k))–invariant state φ on L(H) . Define

m(E) := φ(P (E)) ∀E ∈ B(k^{2}) .

Then, m is a finitely additive probability measure on B(V̂) . Moreover, m is SL(2, k)–invariant, by (1). Now k̂ identifies with k in such a way that the action of SL(2, k) corresponds to the transpose of the action of SL(2, k) on k^{2} (see [19]). In this way, m becomes an invariant finitely additive probability measure on the Borel sets of k^{2}. So, by Lemma 2.2, m is the Dirac measure at 0. In particular, P ({0}) ≠ 0 . This shows that π|_{V} has a nonzero invariant vector ξ. By Lemma 2.4, ξ is invariant under the following two copies of SL(2, k)

( \begin{matrix}∗ & 0 & ∗ \\ 0 & 1 & 0 \\ ∗ & 0 & ∗ \end{matrix} ) and ( \begin{matrix}1 & 0 & 0 \\ 0 & ∗ & ∗ \\ 0 & ∗ & ∗ \end{matrix} );

which generate together SL(3, k) . This concludes the proof.

3. Kazhdan Constants and K-Finite Vectors

In this section, we prove Theorem 1.1. Let (π, H) be a (strongly continuous) unitary representation of a connected Lie group G and denote by dπ the derived representation of the Lie algebra 𝔤 . Let (X_{1}, X_{2}, …, X_{n}) be a basis of 𝔤 and Δ := − ∑_{j=1}^{n} X_{i}^{2} ∈ U(𝔤) be the associated Laplacian. Then dπ(Δ) is defined on H^{∞} , positive and essentially selfadjoint. Let \overline{dπ(Δ)} be its closure.

In [3] the operators dπ(Δ) and \overline{dπ(Δ)} are used in order to decide whether or not (π, H) contains weakly the trivial representation. The main result of that paper is

Theorem 3.1. Let (π, H) be a unitary representation of a connected Lie group G . Then the following are equivalent:

(i) (π, H) contains weakly the trivial representation.

(ii) 0 is an approximative eigenvalue of dπ(Δ) .

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

We are going to see that one may also consider the restriction of the Laplacian to smaller subspaces where the computations become easier. Recall that ξ ∈ H is called analytic for π if the mapping

g ↦ ⟨π(g)ξ, η⟩

is analytic on G for all η ∈ H . The space of analytic vectors is denoted by H^{ω} . Finally, let T be an operator in H . A vector ξ ∈ H is called analytic for T if the series

∑_{j=1}^{∞} \frac{∥T^{j}ξ∥}{j!}t^{j}

has positive convergence radius. By the spectral theorem, the analytic vectors of a selfadjoint operator are dense in H . The following result of R. Goodman relates these notions of analyticity. Define

B_{π} := (I + \overline{dπ(Δ)})^{\frac{1}{2}} .

B_{π} is a selfadjoint, positive operator, B_{π} ≥ I . Then ξ is an analytic vector for B_{π} if and only if ξ is an analytic vector for π and this is the case if and only if ∥dπ(Δ)^{m}ξ∥ ≤ (2m)!M^{m}∥ξ∥ , for all m ∈ ℕ_{0} and a suitable M > 0 . For all these results, see [18, Chapter 4.4].

Proposition 3.2. With the above notations, let D be a subspace of H . Assume that D ∩ H^{ω} is dense in H . Then the following are equivalent:

(i) (π, H) contains weakly the trivial representation.

(ii) inf{⟨dπ(Δ)ξ, ξ⟩; ξ ∈ D, ∥ξ∥ = 1} = 0 .

Proof. By 3.1 it is enough to prove that (i) implies (ii). Since D ∩ H^{ω} is dense in H , Goodman’s result implies that D contains a dense set of vectors analytic for B_{π}|_{D} , where B_{π} is defined as above. Since B_{π}|_{D} is symmetric, Nelson’s theorem (see e.g., [16, X.39]) shows that B_{π}|_{D} is essentially selfadjoint, whence \overline{B_{π}|_{D}} = B_{π} . By functional calculus, there is a sequence \{ξ_{n}\}_{n} of unit vectors in the domain of B_{π} , satisfying ∥B_{π}(ξ_{n}) − ξ_{n}∥ < \frac{1}{2n} . Let \{η_{n}\}_{n} be sequence in D with ∥B_{π}ξ_{n} − B_{π}η_{n}∥ + ∥ξ_{n} − η_{n}∥ < \frac{1}{2n} . Then ∥B_{π}(η_{n}) − η_{n}∥ < \frac{1}{n} . Let ψ_{n} := \frac{η_{n}}{∥η_{n}∥} ∈ D. Then lim_{n→∞}⟨(B_{π} − I)ψ_{n}, ψ_{n}⟩ = 0 , by Cauchy–Schwarz inequality and, hence,

lim_{n→∞}⟨dπ(Δ)ψ_{n}, ψ_{n}⟩ = 0 = lim_{n→∞}⟨(B_{π} − I)^{2}ψ_{n}, ψ_{n}⟩ + 2 lim_{n→∞}⟨(B_{π} − I)ψ_{n}, ψ_{n}⟩ = 0

as desired.

An often useful choice of D is the following: Let K be a compact subgroup of G. By the Peter–Weyl theorem, we may decompose (π|_{K}, H) in K -isotypic components H(σ)

H = ∑_{σ∈K̂} H(σ) .

Then the subspaces (algebraic sum)

H^{∞,K} := ∑_{σ∈K̂} (H(σ) ∩ H^{∞}) and H^{ω,K} := ∑_{σ∈K̂} (H(σ) ∩ H^{ω})

are dense in H [18, 4.4.3.1,4.4.5.16].

Corollary 3.3. With the above notations, the following are equivalent:

(i) (π, H) contains weakly the trivial representation.

(ii) κ_{K}(dπ(Δ), G) := inf{⟨dπ(Δ)ξ, ξ⟩, ξ ∈ H_{K}^{∞}, ∥ξ∥ = 1} = 0 .

We define the infinitesimal Kazhdan constant κ_{K}(Δ, G) to be the least upper bound of all{κ_{K}(dπ(Δ), G) where π ranges through all unitary representations which do not contain the trivial representation}.

Corollary 3.4. Let G be a connected Lie group. Then G has property (T) if and only if there exists ε > 0 such that

κ_{K}(Δ, G) > ε .

Proof. Using 3.2, this is similar to the proof of [3, 3.10].

4. Kazhdan Constants for SL(3, ℝ)

We now give a bound for the above Kazhdan’s constant in the case of SL(3, ℝ) . Consider the subgroup

H := \{( \begin{matrix}a & b & x \\ c & d & y \\ 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℝ), x, y ∈ ℝ\} ≃ SL(2, ℝ) ⋉ ℝ^{2} .

Observe that if (π, H) is a unitary representation of SL(3, ℝ) and if ξ is fixed under the action of the subgroup

ℝ^{2} ≃ V := \{( \begin{matrix}1 & 0 & x \\ 0 & 1 & y \\ 0 & 0 & 1 \end{matrix} ) , x, y ∈ ℝ\} ,

then ξ is a fixed vector under SL(3, ℝ) (see Lemma 2.5 above, and the proof of 2.6). So, we may assume that (π, H) is a unitary representation of SL(3, ℝ) without V –fixed vectors. As in the proof of 2.6, there exists a projection valued measure P on the Borel sets of V̂ ≃ V such that

π|_{V} = ∫V̂ γdP (γ) and P ({0}) = 0 ,

and we have for all Borel sets E ⊆ V̂ :

P (g · E) = π(g)P (E)π(g)^{−1}

(compare with (1)) for all g ∈ SL(2, ℝ) . Next, choose as a basis of 𝔰𝔩(2, ℝ):

K := ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , A := ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , B := ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ,

If we embed SL(2, ℝ) in SL(3, ℝ) as above, K, A, B correspond to X_{1}, X_{2}, and X_{3} in Theorem 1.2. We have the commutator relations

[K, A] = −2B, [K, B] = 2A, [A, B] = 2K .

For an angle 0 ≤ θ ≤ π, let S(θ) be the sector

S(θ) := \{v ∈ V̂ , arg v ∈ [−\frac{θ}{2}, \frac{θ}{2}[\} .

S(θ)

The first crucial fact is that, for any unit K –eigenvector v, we have

∥P (S(θ))v∥^{2} = \frac{θ}{2π} .  (2)

This is an easy computation (see [10, p.223]). Indeed, since V has no nonzero fixed vector, P (ℝ^{2} \{0}) = I. For an arc E on the unit circle, set μ(E) = ∥P (S(E))v∥^{2}, where S(E) is the sector defined by E. The usual properties of a spectral measure show that μ is a measure on the circle. Further, μ is rotation invariant by the above relation (1). Hence, μ is the normalized arc lengh on the unit circle proving (2).

Set a_{t} := exp(tA), and let θ_{t} be the angle corresponding to the sector a_{t} · S(θ) . One easily computes that

θ_{t} = 2arctan(e^{2t} tan \frac{θ}{2}) .  (3)

For a C^{∞}– vector v , we have

∥P (S(θ_{t}))v∥^{2} = ∥π(a_{t})P (S(θ))π(a_{−t})v∥^{2} = ∥P (S(θ))π(a_{−t})v∥^{2} ,

by (1), and hence, differentiating at t = 0,

\frac{d}{dt}|_{t=0}∥P (S(θ_{t}))v∥^{2} = \frac{d}{dt}|_{t=0}∥P (S(θ))π(a_{−t})v∥^{2}
= \frac{d}{dt}|_{t=0}⟨P (S(θ))π(a_{−t})v, π(a_{−t})v⟩
= −⟨dπ(A)v, P (S(θ))v⟩ − ⟨P (S(θ))v, dπ(A)v⟩ .

Let v be a C^{∞}– vector which is a unit K –eigenvector. Then, (2) and (3) show that

\frac{d}{dt}|_{t=0}∥P (S(θ_{t}))v∥^{2} = \frac{1}{π} \frac{d}{dt}|_{t=0}arctan(e^{2t} tan \frac{θ}{2}) = \frac{1}{π} \frac{2 tan \frac{θ}{2}}{1 + (tan \frac{θ}{2})^{2}}
= \frac{1}{π} sin θ .

Hence,

−⟨dπ(A)v, P (S(θ))v⟩ − ⟨P (S(θ))v, dπ(A)v⟩ = \frac{1}{π} sin θ .

Using Cauchy-Schwarz inequality, we find

2∥dπ(A)v∥\sqrt{\frac{θ}{2π}} = 2∥dπ(A)v∥∥P (S(θ))v∥ ≥ \frac{1}{π} sin θ ,

and

∥dπ(A)v∥ ≥ \frac{1}{\sqrt{2π}} \frac{sin θ}{\sqrt{θ}} ,

for all smooth unit K –eigenvectors v . Observe that the same inequality holds for B , since B is conjugate to A under SO(2, ℝ).

Now, we consider a second copy of SL(2, ℝ) ⋉ ℝ^{2} in SL(3, ℝ), namely

H′ := \{( \begin{matrix}1 & 0 & 0 \\ x & a & b \\ y & c & d \end{matrix} ) , ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℝ), x, y ∈ ℝ\} .

The same computation as above shows that, for j ∈ {2, 3} (respectively j ∈ {5, 6}) :

∥dπ(X_{j})v∥ ≥ \frac{1}{\sqrt{2π}} \frac{sin θ}{\sqrt{θ}} ,  (4)

if v is a smooth unit X_{1}–eigenvector (respectively, if v is a smooth unit X_{4}– eigenvector).

Consider the Cartan decomposition 𝔰𝔩(3, ℝ) = 𝔰𝔬(3, ℝ) ⊕ 𝔭 , where 𝔭 are the symmetric matrices in 𝔰𝔩(3, ℝ) . Then X_{1}, X_{4}, X_{7} is an orthonormal basis of 𝔰𝔬(3, ℝ) and X_{2}, X_{3}, X_{5}, X_{6}, X_{8} is one of 𝔭 , with respect to \frac{1}{6}K(X, Y), where K(X, Y) is the Killing form on SL(3, ℝ). Hence, the Casimir operator C on SL(3, ℝ) is

C = (X_{2}^{2} + X_{3}^{2} + X_{5}^{2} + X_{6}^{2} + X_{8}^{2}) − (X_{1}^{2} + X_{4}^{2} + X_{7}^{2}),

the Casimir operator on SO(3, ℝ) is − (X_{1}^{2} + X_{4}^{2} + X_{7}^{2}), and

Δ = −2(X_{1}^{2} + X_{4}^{2} + X_{7}^{2}) − C ,  (5)

Now, let w be a smooth SO(3, ℝ)–finite unit vector. Then w = ∑_{j=1}^{k} v_{j}, where the v_{j} belong to the isotypic components of pairwise inequivalent representations (in particular, they are pairwise orthogonal). By (5), dπ(Δ) preserves the isotypic components. Hence,

⟨Δw, w⟩ = ∑_{i,j=1}^{k}⟨dπ(Δ)v_{i}, v_{j}⟩ = ∑_{j=1}^{k}⟨dπ(Δ)v_{j}, v_{j}⟩

≥ ∑_{j=1}^{k} (⟨dπ(Δ_{1})v_{j}, v_{j}⟩ + ⟨dπ(Δ_{2})v_{j}, v_{j}⟩) ,

where Δ_{1} := − ∑_{ν=1}^{3} X_{ν}^{2} and Δ_{2} := − ∑_{ν=4}^{6} X_{ν}^{2} . Moreover, observe that Δ_{1} = −□_{1} − 2X_{1}^{2} and Δ_{2} = −□_{2} − 2X_{4}^{2} , where □_{i}, i = 1, 2 , are the respective Casimir operators in the copies of 𝔰𝔩(2, ℝ) . Hence, if v_{j} = ∑_{ℓ=1}^{r_{j}} u_{ℓ}^{j} is the orthogonal decomposition of the X_{1}–finite vector v_{j} in X_{1}–eigenvectors, equation (4) yields

⟨dπ(Δ_{1})v_{j}, v_{j}⟩ ≥ ∑_{ℓ=1}^{r_{j}} 2 (\frac{1}{\sqrt{2π}} \frac{sin θ}{\sqrt{θ}})^{2} ∥u_{ℓ}^{j}∥^{2} = \frac{sin^{2} θ}{πθ}∥v_{j}∥^{2}

Decomposing v_{j} into X_{4}–eigenvectors yields the same inequality, with Δ_{1} replaced by Δ_{2}. Thus,

⟨Δw, w⟩ ≥ \frac{2 sin^{2} θ}{πθ}  (6)

for any smooth SO(3, ℝ)–finite unit vector w .

Numerical computations show that the function f(θ) := \frac{2 sin^{2} θ}{πθ} assumes its maximal value of ≈ 0.4613 at θ ≈ 1.1656 .

0.1 0.2 0.3 0.4 0.5 0 0.5 1 1.5 2 2.5 3

Thus

⟨Δw, w⟩ ≥ 0.4613 ,

for all unit SO(3, ℝ)–finite vectors w .

References

[1] Bacher, R., and P. de la Harpe, Exact Values of Kazhdan’s Constants for Some Finite Groups, J. of Algebra 163 (1994), 495–515.

[2] Bekka, M. E. B., Amenable Unitary Representations of Locally Compact Groups, Invent. Math. 100 (1990), 383–401.

[3] Bekka, M. E. B., P.-A. Cherixand, and P. Jolissaint, Kazhdan Constants Associated with the Laplacian of Connected Lie Group, J. Lie Theory 8(1998), 95–110.

[4] Burger, M., Kazhdan’s Constants for SL(3, ℤ), J. reine ang. Math. 43 (1991), 36–67.

[5] Cherix, P.-A., Property (T) and Expanding Constants for Semi–Direct Products, Linear and Multilinear Alg. 39 (1995), 153–160.

[6] Cartwright, D. I., W. Mlotkowski, and T.Steger, Property (T) and Ã_{2}– Groups, Ann. Inst. Fourier Grenoble 44 (1993), 213–248.

[7] Connes, A., and V. Jones, Property T for von Neumann Algebras, Bull. London Math. Soc. 17 (1985), 57–62.

[8] Deutsch, A., and A. Valette, On Diameters of Orbits of Compact Groups in Unitary Representations, J. Austr. Math. Soc. 59 (1995), 308–312.

[9] de la Harpe, P., and A. Valette, “La Propriété (T) pour les Groupes Localement Compacts,” Astérisque 175, Société Mathématique de France, 1989.

[10] Howe, R., and Eng Che Tan, “Non-Abelian Harmonic Analysis,” Springer-Verlag, Berlin Heidelberg New York, 1992.

[11] Kazhdan D., Connection of the Dual Space of a Group with the Structure of its Closed Subgroups, Funct. Anal. Appl. 1 (1967), 63–65.

[12] Lubotzky, A., “Discrete Groups, Expanding Graphs and Invariant Measures,” Birkhäuser Verlag, Boston Basel Berlin, 1994.

[13] Mackey, G. W., “The Theory of Unitary Group Representations,” University of Chicago Press, 1976.

[14] Margulis, G. A., “Discrete Subgroups of Semisimple Lie Groups,” Springer Verlag, Berlin Heidelberg New York, 1991.

[15] —, Explicit constructions of concentrators, Problems Inform. Transmission 9 (1973), 325–332.

[16] Reed, M., and B. Simon, “Fourier Analysis, Selfadjointness, ” Academic Press, 1975.

[17] Shalom, Y., Explicit Kazhdan Constants for Representations of Semisimple Groups and their Lattices, Preprint 1997.

[18] Warner G., “Harmonic Analysis on Semi-Simple Lie Groups I,” Springer-Verlag, Berlin Heidelberg New York, 1972.

[19] Weil, A., “Basic Number Theory,” Springer-Verlag, Berlin Heidelberg New York, 1973.

[20] Zimmer, R. J., “Ergodic Theory and Semisimple Lie Groups,” Birkhäuser Verlag, Boston Basel Stuttgart, 1984.

M. E. B. Bekka Université de Metz Ile du Saulcy 57045 Metz Cedex 01 bekka@poncelet.univ-metz.fr Matthias Mayer Zentrum Mathematik Technische Universität München D-80290 München mayerm@mathematik.tu-muenchen.de

Received August 24, 1998 and in final form October 5, 1998