Journal of Lie Theory Volume 12 (2002) 31–39 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

On Kazhdan’s Property (T) for Sp_{2} (k)

M. B. Bekka and M. Neuhauser^{∗}

Communicated by A. Valette

Abstract. The aim of this note is to give a new and elementary proof of Kazhdan’s Property (T) for Sp_{2} (k) , the symplectic group on 4 variables, for any local field k. The crucial step is the proof that the Dirac measure δ_{0} at 0 is the unique mean on the Borel subsets of the second symmetric power S^{2}(k^{2}) of k^{2} which is invariant under the natural action of SL_{2} (k) . In the case where k has characteristic 2, we observe that this is no longer true if S^{2}(k^{2}) is replaced by its dual, the space of the symmetric bilinear forms on k^{2}.

1. Introduction

A locally compact group G has Kazhdan’s Property (T) ([5]) if whenever a strongly continuous unitary representation π of G on a Hilbert space H has almost invariant vectors, then it actually has a nonzero fixed vector. Recall that π has almost invariant vectors if, for any ε > 0 and any compact subset K of G, there exists a unit vector ξ in H with ∥π(g)ξ − ξ∥ < ε for all g in K. Kazhdan’s Property (T) is a powerful tool and has remarkable applications. For an excellent account, see [3].

Let k be a local field (that is, a non-discrete locally compact field), and let G be a connected almost simple algebraic k-group with k-rank ≥ 2. Then, as is well known, G(k), the group of the k-rational points in G, has Property (T). This central result is quickly deduced, using Howe-Moore’s theorem and root theory, from the fact that SL_{3} (k) and Sp_{2} (k) , the symplectic group on 4 variables, have Property (T) (see [6, Chap. III, (5.3) Theorem]). The aim of this note is to give an elementary proof of Property (T) for Sp_{2} (k). A proof in this spirit was given in [1] for SL_{3} (k). Property (T) for Sp_{2} (k) was first established in [2] and [8].

Theorem 1.1. ([2], [8]) Sp_{2} (k) has Property (T), for any local field k.

The group Sp_{2} (k) contains a copy of the semi-direct product SL_{2} (k) ⋉ S^{2∗}(k^{2}), for the standard action of SL_{2} (k) on the space S^{2∗}(k^{2}) of the symmetric bilinear forms on k^{2} (see Section 4).

Theorem 1.1 is a consequence of the following fact.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Theorem 1.2. The pair (SL_{2} (k) ⋉ S^{2∗}(k^{2}), S^{2∗}(k^{2})) has the relative Property (T), for any local field k. Recall that a pair (G, H) of a locally compact group G and a closed subgroup H has the relative Property (T), if whenever a unitary representation π of G on a Hilbert space H has almost invariant vectors, then it has a nonzero H -fixed vector.

∗The second author acknowledges the financial support he received from the Bayerisch-Französischen Hochschulzentrum

Our proof of Theorem 1.2 is based on the next result and avoids the usual Mackey type analysis of the irreducible unitary representations of SL_{2} (k)⋉S^{2∗}(k^{2}), used in [6, Chap. III, (5.1) Lemma], in [10, 7.4.2 Theorem] or in [2] and [8] (see also [3] for a sketch of proof of Theorem 1.2 based on the so-called Furstenberg Lemma). The dual group of the abelian group S^{2∗}(k^{2}) may be identified with the second symmetric power S^{2}(k^{2}) of k^{2}.

Theorem 1.3. Let k be a local field. The Dirac measure δ_{0} at 0 is the unique mean on the Borel subsets of S^{2}(k^{2}) which is invariant under the natural action of SL_{2} (k) on S^{2}(k^{2}).

Recall that a mean on a ring of subsets of a set X is a finitely additive positive measure m on this ring with total mass 1 (that is, with m(X) = 1). Theorem 1.3 has a completely elementary proof. Theorem 1.2 is a consequence of Theorem 1.3, via the following general fact which was observed in [7, Theorem 5.5] and for which we give a short and different proof.

Proposition 1.4. ([7]) Let G be a locally compact group and N a normal abelian subgroup. Let N̂ be the dual group of N. Assume that the Dirac measure at the trivial character of N is the unique mean on the Borel subsets of N̂ which is invariant under the dual action of G on N̂ given by conjugation. Then (G, N) has the relative Property (T).

The two natural representations of SL_{2} (k) on S^{2}(k^{2}) and S^{2∗}(k^{2}) (which are contragredient to each other) are equivalent representations, but only if char(k), the characteristic of k, is different from 2. In fact, in case char(k) = 2, one has the following somewhat surprising result.

Theorem 1.5. The pair (SL_{2} (k) ⋉ S^{2}(k^{2}), S^{2}(k^{2})) does not have the relative Property (T) if k is a local field of characteristic 2. The paper is organized as follows. In Section 2, we give the proof of Theorem 1.3. Proposition 1.4 and Theorem 1.2 are proved in Section 3 and Theorem 1.1 and Theorem 1.5 in Section 4.

2. Proof of Theorem 1.3

Denote by ρ the standard representation of SL_{2} (k) on the second symmetric power S^{2}(k^{2}) of k^{2}. Fix a basis {e_{1}, e_{2}} of k^{2}. Identify S^{2}(k^{2}) with k^{3} by

means of the basis \{e_{1}^{2}, e_{1}⊗e_{2}, e_{2}^{2}\}. The matrix of ρ(g) for g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2} (k)

is then

ρ(g) = ( \begin{matrix}a^{2} & ab & b^{2} \\ 2ac & ad + bc & 2bd \\ c^{2} & cd & d^{2} \end{matrix} ).

So, for u_{b} = ( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ), one has

ρ(u_{b}) = ( \begin{matrix}1 & b & b^{2} \\ 0 & 1 & 2b \\ 0 & 0 & 1 \end{matrix} ) , ∀b ∈ k.

Hence,

ρ(u_{b}) ( \begin{matrix}x \\ y \\ z \end{matrix} ) = ( \begin{matrix}x + by + b^{2}z \\ y + 2bz \\ z \end{matrix} ) , ∀ ( \begin{matrix}x \\ y \\ z \end{matrix} ) ∈ k^{3}.

Assume first that char(k) ≠ 2. For every c ∈ R, c > 0, consider the following Borel subset of k^{3}

Ω_{c} = \{( \begin{matrix}x \\ y \\ z \end{matrix} ) ∈ k^{3} : |z| > c |y|\},

where |·| denotes the absolute value of k. One has, for every b ∈ k and every

( \begin{matrix}x \\ y \\ z \end{matrix} ) ∈ Ω_{c},

(|2b| − \frac{1}{c}) |z| < |y + 2bz| < (|2b| + \frac{1}{c}) |z|.  (1)

Let μ be an invariant mean on B(k^{3}), the Borel subsets of k^{3}. For every fixed c > 0, choose a sequence (b_{n})_{n} in k such that

|b_{n+1}| > |b_{n}| + \frac{2}{|2| c}, ∀n ∈ N.

Then, by (1),

ρ(u_{b_{n}})Ω_{c} ∩ ρ(u_{b_{m}})Ω_{c} = Ø,

for all n ≠ m. Since μ(k^{3}) < ∞, this implies that

μ (Ω_{c}) = 0, ∀c > 0.

For ω = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ∈ SL_{2} (k), one has

ρ(ω) = ( \begin{matrix}0 & 0 & 1 \\ 0 & −1 & 0 \\ 1 & 0 & 0 \end{matrix} ).

So,

ρ(ω) ( \begin{matrix}x \\ y \\ z \end{matrix} ) = ( \begin{matrix}z \\ −y \\ x \end{matrix} ) , ∀ ( \begin{matrix}x \\ y \\ z \end{matrix} ) ∈ k^{3},

34 Bekka and Neuhauser and hence,

ρ(ω)Ω_{c} = \{( \begin{matrix}x \\ y \\ z \end{matrix} ) ∈ k^{3} : |x| > c |y|\}

Observe that μ (ρ(ω)Ω_{c}) = 0, by invariance of μ. For r = ( \begin{matrix}2^{−1} & 2^{−1} \\ −1 & 1 \end{matrix} ) ∈ SL_{2} (k), one has

ρ(r) = ( \begin{matrix}4^{−1} & 4^{−1} & 4^{−1} \\ −1 & 0 & 1 \\ 1 & −1 & 1 \end{matrix} ).

So,

ρ(r) ( \begin{matrix}x \\ y \\ z \end{matrix} ) = ( \begin{matrix}4^{−1} (x + y + z) \\ −x + z \\ x − y + z \end{matrix} ) , ∀ ( \begin{matrix}x \\ y \\ z \end{matrix} ) ∈ k^{3}.

Let ( \begin{matrix}x \\ y \\ z \end{matrix} ) ∈ k^{3} with |y| ≥ |x| and |y| ≥ |z| and (x, z) ≠ (0, 0). Then

|x + y + z| + |x − y + z| = |x + y + z| + |−x + y − z| ≥ |2y| = |2| |y|
≥ \frac{|2|}{2} (|x| + |z|)
> \frac{|2|}{4} (|−x + z|) .

Hence, for c_{0} = |2| /4 (|4| + 1), one has

either |4^{−1} (x + y + z)| > c_{0} (|−x + z|) or |x − y + z| > c_{0} (|−x + z|) . Therefore,

ρ(r) ( \begin{matrix}x \\ y \\ z \end{matrix} ) ∈ Ω_{c_{0}} ∪ ρ(ω)Ω_{c_{0}},

for all ( \begin{matrix}x \\ y \\ z \end{matrix} ) ∈ k^{3} \ (Ω_{1} ∪ ρ(ω)Ω_{1}) with (x, z) ≠ (0, 0). Since

ρ(r) ( \begin{matrix}0 \\ y \\ 0 \end{matrix} ) = ( \begin{matrix}y \\ 0 \\ −y \end{matrix} ),

this also holds if (x, z) = (0, 0) and y ≠ 0. The above shows that

k^{3} \ {0} = Ω_{1} ∪ ρ(ω)Ω_{1} ∪ ρ(r^{−1})Ω_{c_{0}} ∪ ρ(r^{−1}ω)Ω_{c_{0}}. Since

μ (Ω_{1}) = μ (ρ(r^{−1})Ω_{c_{0}}) = μ (ρ(r^{−1}ω)Ω_{c_{0}}) = μ (ρ(ω)Ω_{1}) = 0,

this implies that μ (k^{3} \ {0}) = 0 and concludes the proof of the theorem in the case where char(k) ≠ 2.

Assume now that char(k) = 2. Let

Ω = \{( \begin{matrix}x \\ y \\ z \end{matrix} ) ∈ k^{3} \ \{0\} : |x| ≤ |z| and |y| ≤ |z|\}.

One has, with the above notation,

ρ(u_{b}) ( \begin{matrix}x \\ y \\ z \end{matrix} ) = ( \begin{matrix}x + by + b^{2}z \\ y \\ z \end{matrix} ) , ∀ ( \begin{matrix}x \\ y \\ z \end{matrix} ) ∈ k^{3}

and |x| < |b^{2}z| and |by| < |b^{2}z| if |b| > 1. Choose a sequence (b_{n})_{n} in k with |b_{n+1}| > |b_{n}| > 1. Then

ρ(u_{b_{n}})Ω ∩ ρ(u_{b_{m}})Ω = Ø ∀n ≠ m,

since |x + by + b^{2}z| = |b^{2}z|, if |x| < |b^{2}z| and |by| < |b^{2}z|. Indeed, the absolute value on k being non-archimedean satisfies

|x + by| ≤ max {|x| , |by|} < |b^{2}z|

and, hence, |x + by + b^{2}z| ≤ |b^{2}z| as well as

|b^{2}z| ≤ max {|x + by + b^{2}z| , |x + by|} ≤ |b^{2}z|

Let μ be an invariant mean on B(k^{3}). By the above, μ (Ω) = 0.

Clearly,

ρ(ω)Ω = \{( \begin{matrix}x \\ y \\ z \end{matrix} ) ∈ k^{3} \ \{0\} : |z| ≤ |x| and |y| ≤ |x|\}.

So, k^{3} \ {0} = Ω ∪ ρ(ω)Ω ∪ Ω′, where

Ω′ = \{( \begin{matrix}x \\ y \\ z \end{matrix} ) ∈ k^{3} : |x| < |y| and |z| < |y|\}.

Moreover,

ρ(u_{1}) ( \begin{matrix}x \\ y \\ z \end{matrix} ) = ( \begin{matrix}x + y + z \\ y \\ z \end{matrix} ),

and |x + y + z| = |y| when |x| < |y| and |z| < |y|. Hence, ρ(u_{1})Ω′ is contained in ρ(ω)Ω. Therefore, μ (Ω′) = 0. This implies μ (k^{3} \ {0}) = 0. So, μ has to be δ_{0}.

3. Proofs of Proposition 1.4 and Theorem 1.2

The proof of Proposition 1.4 is based on the following elementary lemma from [1, Lemma 2.1] Lemma 3.1. Let G be a locally compact group and let (π, H) be a unitary representation of G with almost invariant vectors. Then there exists an Ad Ginvariant state on the C*-algebra L(H) of all bounded operators on H, that is, a positive linear form φ on L(H) with φ (Id) = 1 and φ (π(x)T π(x)^{−1}) = φ(T) for all x ∈ G and T ∈ L(H).

Proof of Proposition 1.4. Let (π, H) be a unitary representation of G on H, with almost invariant vectors. Let

P : B(N̂) → L(H), E ↦ P (E)

be the projection valued measure on N̂ associated with the unitary representation π|_{N} of the abelian group N. One has

π(g)P (E)π(g)^{−1} = P (gE), ∀E ∈ B(N̂), g ∈ G,

where

gλ(n) = λ(g^{−1}ng), n ∈ N, λ ∈ N̂

is the dual action of g ∈ G on N̂. By the above lemma, there exists an Ad Ginvariant state φ on the C*-algebra of all bounded linear operators on H. Define

μ(E) = φ(P (E)), ∀E ∈ B(N̂).

Then μ is a G-invariant mean on B(N̂). So, μ is the Dirac measure at the trivial character λ_{0} of N. Hence, P ({λ_{0}}) ≠ 0. This shows that π|_{N} has a non-zero fixed vector and (G, N) has the relative Property (T).

Proof of Theorem 1.2. The dual vector space of the space S^{2∗}(k^{2}) of the bilinear forms on k^{2}, may be identified with the second symmetric power S^{2}(k^{2}) of k^{2}, by means of the duality formula

(B, ∑_{i} x_{i} ⊗ y_{i}) ↦ ∑_{i} B(x_{i}, y_{i}), ∀B ∈ S^{2∗}(k^{2}) and ∑_{i} x_{i} ⊗ y_{i} ∈ S^{2}(k^{2}).

Under this identification, the dual action of the natural action of SL_{2} (k) on S^{2∗}(k^{2}) corresponds to the inverse transpose of the representation ρ of SL_{2} (k) on S^{2}(k^{2}) considered in Section 2, that is, to the representation ρ̃ of SL_{2} (k) on S^{2}(k^{2}) defined by ρ̃(g) = ρ (^{t}g^{−1}), where ^{t}g is the transpose of g.

Observe that ρ and ρ̃ are equivalent (for any value of char(k)). Indeed, for

ω = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , one has ^{t}g^{−1} = ωgω^{−1} for g ∈ SL_{2} (k).

The dual group of S^{2∗}(k^{2}) may be identified with S^{2}(k^{2}) as follows (see [9, Chap. II, section 5, Theorem 3]). Fix a non-trivial character λ of the additive group of k. Then, for any X ∈ S^{2}(k^{2}), the formula

λ_{X}(Y) = λ (Y (X)) , ∀Y ∈ S^{2∗}(k^{2})

defines a character on S^{2∗}(k^{2}) and the mapping X → λ_{X} is an isomorphism between the additive group S^{2}(k^{2}) and the dual group of S^{2∗}(k^{2}). Theorem 1.2 follows now from Theorem 1.3.

4. Proofs of Theorem 1.1 and Theorem 1.5

For the proof of Theorem 1.1, we shall need the two following well-known lemmas, the first of which is called Mautner’s lemma (for the elementary proofs, see [6, Chap. II, (3.2) Lemma and (3.4) Lemma]).

Lemma 4.1. Let G be a locally compact group, (π, H) a unitary representation of G and x, y ∈ G such that lim_{n→∞} x^{n}yx^{−n} = e. If π (x) ξ = ξ for some ξ ∈ H, then π (y) ξ = ξ.

Lemma 4.2. Let (π, H) be a unitary representation of SL_{2} (k). Let ξ be a vector in H which is invariant under the subgroup

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

Then ξ is invariant under SL_{2} (k).

Recall that Sp_{2} (k) is the group of all matrices g ∈ GL_{4} (k) with ^{t}gJg = J, where ^{t}g is the transpose of g and

J = ( \begin{matrix}0 & I_{2} \\ −I_{2} & 0 \end{matrix} ).

Proof of Theorem 1.1. Consider the following subgroups of Sp_{2} (k):

G = \{( \begin{matrix}A & B \\ 0 & ^{t}A^{−1} \end{matrix} ) : A ∈ SL_{2} (k) , A^{t}B = B^{t}A\}
N = \{( \begin{matrix}I_{2} & B \\ 0 & I_{2} \end{matrix} ) : ^{t}B = B\}.

Since

( \begin{matrix}A & 0 \\ 0 & ^{t}A^{−1} \end{matrix} ) ( \begin{matrix}I_{2} & B \\ 0 & I_{2} \end{matrix} ) ( \begin{matrix}A & 0 \\ 0 & ^{t}A^{−1} \end{matrix} )^{−1} = ( \begin{matrix}I_{2} & AB^{t}A \\ 0 & I_{2} \end{matrix} )

and N ≅ S^{2∗}(k^{2}), the group G is isomorphic to SL_{2} (k) ⋉ S^{2∗}(k^{2}).

Let (π, H) be a unitary representation of Sp_{2} (k) with almost invariant vectors. Then, by Theorem 1.2, there exists a vector ξ ≠ 0 in H which is fixed by N.

Consider the following subgroups of Sp_{2} (k) :

H_{1} = \{( \begin{matrix}a & 0 & b & 0 \\ 0 & 1 & 0 & 0 \\ c & 0 & d & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) : ad − bc = 1\} ≅ SL_{2} (k)

and

H_{2} = \{( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & a & 0 & b \\ 0 & 0 & 1 & 0 \\ 0 & c & 0 & d \end{matrix} ) : ad − bc = 1\} ≅ SL_{2} (k) .

Since ξ is fixed by the subgroups

N_{1} = \{( \begin{matrix}I_{2} & \begin{matrix}b & 0 \\ 0 & 0 \end{matrix} \\ 0 & I_{2} \end{matrix} ) : b ∈ k\} and N_{2} = \{( \begin{matrix}I_{2} & \begin{matrix}0 & 0 \\ 0 & b \end{matrix} \\ 0 & I_{2} \end{matrix} ) : b ∈ k\}

of H_{1} and H_{2}, it is fixed by H_{1} and H_{2} (by Lemma 4.2). In particular, ξ is fixed by the matrices

( \begin{matrix}λ & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & λ^{−1} & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) and ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & λ & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & λ^{−1} \end{matrix} ) , λ ∈ k.

Now choosing λ ∈ k with |λ| < 1 or |λ^{−1}| < 1, it follows from Mautner’s Lemma above that ξ is fixed by the subgroups

L_{1} = \{( \begin{matrix}I_{2} & 0 \\ B & I_{2} \end{matrix} ) : ^{t}B = B\} ,
L_{2} = \{( \begin{matrix}A & 0 \\ 0 & ^{t}A^{−1} \end{matrix} ) : A ∈ GL_{2} (k)\}.

Since L_{1} ∪ L_{2} ∪ N generates Sp_{4} (k) (see [4, Section 6.9]), the vector ξ is fixed under Sp_{2} (k).

Proof of Theorem 1.5. Suppose that the characteristic of k is 2. Set N = S^{2}(k^{2}). The dual group of N may be identified with the space S^{2∗}(k^{2}) of symmetric bilinear forms on k^{2}, with compatible SL_{2} (k)-actions (see proof of Theorem 1.2 above). Now, SL_{2} (k) has non-zero fixed points in S^{2∗}(k^{2}). Indeed, since char(k) = 2,

( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}0 & s \\ s & 0 \end{matrix} )^{t} ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}0 & s \\ s & 0 \end{matrix} ) , ∀ ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2} (k) ,

and the bilinear forms B_{s} defined by the symmetric matrices ( \begin{matrix}0 & s \\ s & 0 \end{matrix} ) , s ∈ k,

are fixed under SL_{2} (k). The characters λ_{s} of N corresponding to the B_{s}’s extend to characters λ̃_{s} of G = SL_{2} (k) ⋉ N defined by

λ̃_{s}(A, X) = λ_{s}(X), ∀A ∈ SL_{2} (k) , X ∈ N.

Now, lim_{s→0} λ̃_{s} = 1_{G} uniformly on compact subsets of G and λ_{s} ≠ 1_{N} for s ≠ 0. Hence, (G, N) does not have the relative Property (T).

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M. B. Bekka Université de Metz Département de Mathématiques Ile du Saulcy F- 57045 Metz, France bekka@poncelet.univ-metz.fr M. Neuhauser Technische Universität München Zentrum Mathematik Arcisstr. 21 D-80290 München, Germany neuhausm@mathematik.tu-muenchen.de

Received September 1, 2000 and in final form October 6, 2000