Covariant projective representations of Hilbert–Lie groups

Lemma A.1 ([46, Lemma A.4])

Let ( V , ∥ ⋅ ∥ ) be a normed space and U ⊆ V a proper open convex subset with complement U c = V ∖ U . Then the distance function

is continuous, concave and d U − 1 : U → R is a continuous convex function on 𝑈 with

In particular, all the sets U c : = { x ∈ U : d U ( x ) ≥ c } with c > 0 are closed subsets of 𝑉.

If 𝐺 is a connected Hilbert–Lie group, then each non-empty open Ad ⁡ ( G ) -invariant convex subset U ⊆ g intersects the center.

Proof

If U = g , there is nothing to show because 0 ∈ U . We may therefore assume that U ≠ g . Then 𝑈 is a proper open convex subset and the group Ad ⁡ ( G ) acts isometrically on 𝔤 preserving 𝑈. Let d U : V → R denote the distance from ∂ U . Then Lemma A.1 shows that, for every c > 0 , U c : = { x ∈ U : d U ( x ) ≥ c } is a closed convex subset of 𝔤. Let c > 0 be such that U c ≠ ∅ . All these sets are Ad ⁡ ( G ) -invariant. Now U c is a Bruhat–Tits space with respect to the induced Hilbert metric from 𝔤, and since Ad ⁡ ( G ) acts on this space isometrically with bounded orbits, the Bruhat–Tits Fixed Point Theorem implies the existence of a fixed point z ∈ U c ⊆ U (see [25]). Now it only remains to observe that z ⁢ ( g ) is the set of fixed points for the action of Ad ⁡ ( G ) on 𝔤. ∎

(a) Let 𝑉 be a locally convex space. We consider the affine group Aff ⁡ ( V ) ≅ V ⋊ GL ⁡ ( V ) which acts on 𝑉 by ( x , g ) . v = g . v + x . On the space V ̃ : = V × R , the group Aff ⁡ ( V ) acts by linear maps

and we thus obtain a realization of Aff ⁡ ( V ) as a subgroup of GL ⁡ ( V ̃ ) . The corresponding Lie algebra is a ⁢ f ⁢ f ⁡ ( V ) ≅ V ⋊ g ⁢ l ⁡ ( V ) with the bracket

(b) A homomorphism ρ : g → a ⁢ f ⁢ f ⁡ ( V ) is therefore given by a pair ( ρ l , θ ) , consisting of a linear representation ρ l : g → g ⁢ l ⁡ ( V ) and map θ : g → V satisfying

A linear map θ : g → V satisfying (B.1) is called a 1-cocycle with values in the representation ( ρ l , V ) of 𝔤. We write Z 1 ⁢ ( g , V ) ρ l for the set of all such cocycles.

(c) On the group level, a homomorphism ρ : G → Aff ⁡ ( V ) is given by a pair ( ρ l , γ ) of a linear representation ρ l : G → GL ⁡ ( V ) and map γ : G → V satisfying

A map γ : G → V satisfying (B.2) is called a 1-cocycle with values in the representation ( ρ l , V ) of 𝐺. We write Z 1 ⁢ ( G , V ) ρ l for the set of all such smooth cocycles. Typical examples of 1-cocycles are the coboundaries γ ( g ) : = ρ l ( g ) v − v for some v ∈ V .

Let 𝐺 be a connected Lie group with Lie algebra 𝔤 and suppose that ω ∈ Z 2 ⁢ ( g , R ) is a continuous 2-cocycle. Then the following assertions hold.

1. θ ω ( x ) ( y ) : = ω ( x , y ) is a 1-cocycle on 𝔤 with values in the coadjoint representation ( ad ∗ , g ′ ) , where ad ∗ ⁡ ( x ) ⁢ β = − β ∘ ad ⁡ x .

2. If 𝐺 is simply connected and g ̂ = R ⊕ ω g the central extension defined by 𝜔, then there exists a unique smooth representation

   (B.3) Ad g ̂ : G → Aut ⁡ ( g ̂ ) , Ad g ̂ ⁡ ( g ) ⁢ ( z , y ) = ( z + Θ ω ⁢ ( g − 1 ) ⁢ ( y ) , Ad ⁡ ( g ) ⁢ y )

   whose derived representation is given by

   ad g ̂ ⁡ ( x ) ⁢ ( z , y ) = ( ω ⁢ ( x , y ) , [ x , y ] ) = [ ( 0 , x ) , ( z , y ) ] ,

   so that T e ⁢ ( Θ ω ) ⁢ ( x ) = − θ ω ⁢ ( x ) = ω ⁢ ( ⋅ , x ) . The corresponding dual representation of 𝐺 on g ̂ ′ ≅ R × g ′ is given by

   Ad g ̂ ∗ ⁡ ( g ) ⁢ ( z , α ) = ( z , Ad ∗ ⁡ ( g ) ⁢ α + z ⁢ Θ ω ⁢ ( g ) ) .

3. Identifying g ′ with the hyperplane { 1 } × g ′ in g ̂ ′ , we thus obtain an affine action

   Ad ω ∗ ⁡ ( g ) ⁢ α = Ad ∗ ⁡ ( g ) ⁢ α + Θ ω ⁢ ( g )

   of 𝐺 on g ′ .

Proof

(a) follows from an easy calculation [36, Lemma VI.4].

(b) The uniqueness of the affine representation follows from the general fact that, for a connected Lie group, smooth representations are uniquely determined by their derived representations [37, Remark II.3.7]. The existence follows form [33, Proposition 7.6] because ℝ is complete. In [33, Proposition 7.6], one finds more details on how to obtain the corresponding cocycle. For each x ∈ g , let f x ∈ C ∞ ⁢ ( G , R ) be the unique function with d ⁢ f x = i x r ⁢ Ω and f x ⁢ ( e ) = 0 , where Ω ∈ Ω 2 ⁢ ( G , R ) is the left invariant 2-form with Ω e = ω and x r ⁢ ( g ) = x . g is the right invariant vector field with x r ⁢ ( e ) = x . Then Θ ω ( g ) ( y ) : = f y ( g ) defines a smooth function G × g → R ,

where γ g is any piecewise smooth path in 𝐺 from 𝑒 to 𝑔. Note that

(c) is an immediate consequence of (b). ∎

We call a Banach–Lie algebra elliptic if the adjoint group

is bounded.

Suppose that 𝔤 is a Banach–Lie algebra and g ̂ = R ⊕ ω g is the central extension defined by ω ∈ Z 2 ⁢ ( g , R ) . Then g ̂ is elliptic if and only if 𝔤 is elliptic and Θ ω is bounded.

Proof

From (B.3), it follows that the adjoint group ⟨ e ad ⁡ g ̂ ⟩ = ⟨ e ad g ̂ ⁡ g ⟩ of g ̂ is bounded if and only if the adjoint group of 𝔤 is bounded and Θ ω ⁢ ( G ) is a bounded subset of g ′ . ∎

Let 𝑉 be a locally convex space. We call a subset X ⊆ V ′ semi-equicontinuous if its support functional

is bounded on some non-empty open subset of 𝑉.

Let 𝔤 be a Hilbert–Lie algebra, 𝐺 a corresponding simply connected Lie group, ω ∈ Z 2 ⁢ ( g , R ) a continuous cocycle, and Θ ω : G → g ′ the corresponding 1-cocycle with T e ⁢ ( Θ ω ) ⁢ x = ω ⁢ ( ⋅ , x ) for x ∈ g . On g ′ , we consider the affine action

Then the following are equivalent.

1. 𝜔 is a coboundary.

2. Θ ω is bounded.

3. Θ ω ⁢ ( G ) is semi-equicontinuous.

4. One orbit of the affine action of 𝐺 on g ′ is semi-equicontinuous.

5. All orbits of the affine action of 𝐺 on g ′ are bounded.

Proof

(a) ⇒ (b): If 𝜔 is a coboundary, then there exists an α ∈ g ′ with

and Θ ω = α − Ad ∗ ⁡ ( g ) ⁢ α follows from the equality of derivatives

Since the coadjoint action preserves the norm on g ′ , the boundedness of Θ ω follows.

(b) ⇒ (c) is trivial.

(c) ⇔ (d) follows from the fact that all orbits of the coadjoint action of 𝐺 on g ′ are bounded.

(b) ⇔ (e) follows with the same argument.

(c) ⇒ (b): Since Z ⁢ ( G ) 0 acts trivially on 𝔤, we obtain for z ∈ z : = z ( g ) the relation

Therefore, Θ ω ⁢ ( exp ⁡ z ) is a linear space. As it is semi-equicontinuous, it is trivial. This means that ω ⁢ ( z , g ) = { 0 } . Accordingly, Θ ω ⁢ ( G ) ⊆ z ⊥ ≅ [ g , g ] ′ .

The set 𝑊 of all points x 0 ∈ g for which the support functional

is bounded in a neighborhood of x 0 is a non-empty open invariant convex cone. We have seen in the preceding paragraph that W + z = W . Hence W ∩ [ g , g ] ≠ ∅ , and therefore [ g , g ] ⊆ W follows from Proposition A.2. Therefore, W = g , which implies that Θ ω ⁢ ( G ) is equicontinuous, i.e., that Θ ω is bounded.

(b) ⇒ (a): If Θ ω is bounded, then the orbit of 0 ∈ g ′ under the affine action defined by Θ ω is bounded. Hence the Bruhat–Tits Theorem [25] implies the existence of a fixed point 𝛼 of the corresponding affine isometric action on g ′ . This means that Θ ω ⁢ ( g ) = α − Ad ∗ ⁡ ( g ) ⁢ α for each g ∈ G , and by taking derivatives in 𝑒, we see that ω ⁢ ( y , x ) = α ⁢ ( [ x , y ] ) is a coboundary. ∎

Let g ♯ = R ⊕ ω g be the central Lie algebra extension corresponding to the cocycle ω ∈ Z 2 ⁢ ( g , R ) . Then we have a natural affine embedding g ′ ↪ ( g ♯ ) ′ , α ↦ ( 1 , α ) , and this embedding intertwines the coadjoint action of the group 𝐺 in Theorem B.6 with the affine action on g ′ , specified by ( g , α ) ↦ g ∗ α . Therefore, coadjoint orbits of g ♯ not vanishing on the central element ( 1 , 0 ) correspond to affine orbits in g ′ .

Let 𝖧 be a complex Hilbert space and g = u 2 ⁢ ( H ) .

(a) As we have seen in Section 4.2, each cocycle ω ∈ Z 2 ⁢ ( g , R ) can be written as

for some bounded skew hermitian operator d ∈ u ⁢ ( H ) (see [35, Proposition III.19 and proof]). Then θ ω ⁢ ( x ) = [ d , x ] if we identify 𝔤 and g ′ via the invariant scalar product defined by

From that, we easily derive that Θ ω : G = U 2 ⁡ ( H ) → g is given by

This is a cocycle for the adjoint action, which is an orthogonal representation of 𝐺 on the real Hilbert space 𝔤. In view of Theorem B.6, Θ ω is bounded if and only if 𝜔 is a coboundary, which is equivalent to d ∈ R ⁢ i ⁢ 1 + u 2 ⁢ ( H ) . In particular, the central extension defined by a non-trivial 2-cocycle 𝜔 is not an elliptic Banach–Lie algebra, hence in particular not Hilbert (Corollary B.4).

(b) The existence of unbounded cocycles 𝜃 for the adjoint action of U 2 ⁡ ( H ) also implies that we obtain interesting representations of this group by composing the Fock representations of the affine group Mot ⁢ ( u 2 ⁢ ( H ) ) ≅ u 2 ⁢ ( H ) ⋊ O ⁡ ( u 2 ⁢ ( H ) ) with the homomorphism

For details, we refer to [17].

We may now ask for the convexity properties of the affine orbit

resp., the coadjoint orbit of ( 1 , 0 ) in g ̂ ′ . Clearly, the convexity properties of this orbit will depend on the cocycle 𝜔 and not only on its cohomology class. For example, if the central extension is trivial, then the affine action on g ′ is equivalent to the coadjoint action of g ′ and different orbits have very different convexity properties.

(a) In view of ⟨ O ( 1 , 0 ) , ( t , x ) ⟩ = t + Θ ω ⁢ ( G ) ⁢ ( x ) , we have

where, for a subset C ⊆ g ′ , we put

Therefore, O ( 1 , 0 ) is (semi-)equicontinuous in g ̂ ′ if and only if Θ ω ⁢ ( G ) ⊆ g ′ has the corresponding property.

(b) The subspace

is an invariant ideal of g ̂ which intersects the central line R × { 0 } trivially. Its image in 𝔤 consists of all those elements x ∈ g for which the evaluation function

is constant on Θ ω ⁢ ( G ) . Since Θ ω is a 1-cocycle with values in g ′ , we have, for any g ∈ G ,

All these expressions vanish if and only if Ad ⁡ ( G ) ⁢ x is contained in the radical

Therefore, the image of O ( 1 , 0 ) ⊥ in 𝔤 is the largest Ad ⁡ ( G ) -invariant ideal of 𝔤 contained in rad ⁡ ( ω ) . The central extension splits on this ideal.