Representations of the Infinite-Dimensional Affine Group

We introduce an infinite-dimensional affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by the fact that the group does not act on the phase space. However it is possible to define its action on some classes of functions.


Introduction
Given a vector space V the affine group can be described concretely as the semidirect product of V by GL(V ), the general linear group of V : The action of GL(V ) on V is the natural one (linear transformations are automorphisms), so this defines a semidirect product.
Affine groups play important role in the geometry and its applications, see, e.g., [4,12]. Several recent papers [1,3,5,6,8,15] are devoted to representations of the real, complex and p-adic affine groups and their generalizations, as well as diverse applications, from wavelets and Toeplitz operators to non-Abelian pseudo-differential operators and p-adic quantum groups.
In the particular case of field V = R d the group Aff(R d ) defined as following. Consider a function b : R d → R d which is a step function on R d . Take another matrix valued function A : R d → L(R d ) s.t. A(x) = Id + A 0 (x), A(x) is invertible, A 0 is a matrix valued step function on R d . Introduce an infinite dimensional affine group Aff(R d )(R d ) that is the set of all pairs g = (A, b) with component satisfying assumptions above. Define the group operation The unity in this group is e = (Id, 0).
It is clear that for step mappings we use these definitions are correct. Our aim is to construct irreducible representations of Aff(R d )(R d ). As a rule, only special classes of irreducible representations can be constructed for infinite-dimensional groups. For various classes of such groups, special tools were invented; see [7,10] and references therein.
We will follow an approach by Vershik-Gefand -Graev [13] proposed in the case of the group of diffeomorphisms. A direct application of this approach meets certain difficulties related with the absence of the possibility to define the action of the group Aff(R d )(R d ) on a phase space similar to [13]. A method to overcome this problem is the main technical step in the present paper. We wold like to mention that a similar approach was already used in [9] for the construction of the representation for p-adic infinie dimensional affine group.

Infinite dimensional affine group
In our definitions and studies of vector and matrix valued functions on R d we will use as basic functional spaces collections of step mappings. It means that each such mapping is a finite sum of indicator functions with measurable bounded supports with constant vector/matrix coefficients. Such spaces of functions on R d are rather unusual in the framework of infinite dimensional groups but we will try to show that their use is natural for the study of affine groups. For It is an affine group with constant coefficients. Note that for a ball B N (0) ⊂ R d with the radius N centered at zero we have g(x) = (1, 0), x ∈ B c N (0). Define the action of g on a point x ∈ R d as Denote the orbit O x = {gx|g ∈ G x } ⊂ R d . Actually, as a set O x = R d but elements of this set are parametrized by g ∈ G x . For any element y ∈ O x and h ∈ G x we can define hy = h(gx) = (hg)x ∈ O x . It means that we have the group G x action on the orbit O x .
It gives (g 1 g 2 )(x)x = g 1 (x)(g 2 (x)x) that corresponds to the group multiplication considered in the given point x.
Remark 2.1. The situation we have is quite different w.r.t. the standard group of motions on a phase space. Namely, we have one fixed point x ∈ R d and the section group G x associated with this point. Then we have the motion of x under the action of G x . It gives the group action on the orbit O x .
We will use the configuration space Γ(R d ), i.e., the set of all locally finite subsets of R d . Each configuration may be identified with the measure We define the vague topology on Γ(R d ) as the weakest topology for which all mappings Here we have the group action of Aff(R d )(R d ) produced by individual transformations of points from the configuration. Again, as above, we move a fixed configuration using previously defined actions of G x on x ∈ γ. Note that gγ is not more a configuration. More precisely, for some B N (0) the set (gγ) N = gγ ∩ B c N (0) is a configuration in B c N (0) but the finite part of gγ may include multiple points.
For any f ∈ D(R d , C) we have corresponding cylinder function on Γ(R d ): Denote P cyl the set of all cylinder polynomials generated by such functions. More generally, consider functions of the form These functions form the set F b (Γ(R d )) of all bounded cylinder functions. For any clopen set Λ ∈ O b (R d ) (also called a finite volume) denote Γ(Λ) the set of all (with necessity finite) configurations in Λ. We have as before the vague topology on this space and the Borel σ-algebra B(Γ(Λ)) is generated by functions Such sets form a σ-algebra B Λ (Γ(R d )) of cylinder sets for the finite volume Λ. The set of bounded functions on Γ(R d ) measurable w.r.t. B Λ (Γ(R d )) we denote B Λ (Γ(R d )). That is a set of cylinder functions on Γ(R d ). As a generating family for this set we can use the functions of the form For so-called one-particle functions f : Then gf ∈ D(R d ). Thus, we have the group action Note that due to our definition, we have < f, gγ >=< gf, γ > and it is reasonable to define for cylinder functions (1) the action of the group Aff(R d ) as Denote m(dx) the Haar measure on R d . The dual transformation to one-particle motion is defined via the following relation Here as above B c R (0) = {x ∈ R d | |x| p ≥ R}. Proof. We have following representations for coefficients of g(x): where B k are certain balls in R d . Then Note that informally we can write (g * m)(dx) = dm(g −1 x).
Note that by the duality we have the group action on the Lebesgue measure. Namely, for f ∈ D(R d ) and g 1 , In particular (g −1 ) * (g * m) = m.
Proof. Due to the formula for the action V g F we need to analyze the support of functions and only for x + h ∈ Λ this value may be nonzero, i.e., supp gf j ⊂ Λ − h.
Denote π m the Poisson measure on Γ(R d ) with the intensity measure m.
Lemma 2.4. For all F ∈ P cyl or F ∈ F b (Γ(R d )) and g ∈ Aff(R d ) holds Proof. It is enough to show this equality for exponential functions F (γ) = e <f,γ> , f ∈ D(R d ).
We have Remark 2.5. For all functions F, G ∈ F (Γ(R d )) a similar calculation shows Let π m be the Poisson measure on Γ(R d ) with the intensity measure m. For any Λ ∈ O b (R d ) consider the distribution π Λ m of π m in Γ(Λ) corresponding the projection γ → γ Λ . It is again a Poisson measure π m Λ in Γ(Λ) with the intensity m Λ which is the restriction of m on Λ. Infinite divisibility of π m gives for F j ∈ B Λ j (Γ(R d )), j = 1, 2 with Proof. Due to our calculations above we have But we have shown Proof. By the definition, F j ∈ B Λ j (Γ(R d )), j = 1, 2 for some Λ 1 , Λ 2 ∈ O(R d ).
Let us take g = (1, h1 B ) with the following assumptions: Then accordingly to previous lemmas

Aff(R d ) and Poisson measures
For F ∈ P cyl or F ∈ F b (Γ(R d )), we consider the motion of F by g ∈ Aff(R d ) given by the operator V g . Operators V g have the group property defined point-wisely: for any γ ∈ Γ(R d ) This equality is the consequence of our definition of the group action of Aff(R d ) on cylinder functions.
As above, consider π m , the Poisson measure on Γ(R d ) with the intensity measure m. For the transformation V g the dual object is defined as the measure V * g π m on Γ(R d ) given by the relation where V * g π m = π g * m , see Lemma 2.4.
Corollary 3.1. For any g ∈ Aff(R d ) the Poisson measure V * g π m is absolutely continuous w.r.t. π m with the Radon-Nykodim derivative . Proof. Note that density ρ g = 1 B c R (0) + r 0 g , r 0 g ∈ D(R d , R + ) of g * m w.r.t. m may be equal zero on some part of R d and, therefore, the equivalence of of considered Poisson measures is absent. Due to [11], the Radon-Nykodim derivative Remark 3.2. As in the proof of Proposition 2.2 from [2] we have an explicit formula for R(g, γ): The point-wise existence of this expression is obvious.
This fact gives us the possibility to apply the Vershik-Gelfand-Graev approach realized by these authors for the case of diffeomorphism group.
Theorem 3.3. Operators U g , g ∈ Aff(R d ) are unitary in L 2 (Γ(R d ), π m ) and give an irreducible representation of Aff(R d ).
Proof. Let us check the isometry property of these operators. We have using Lemmas 2.4, 2.2 From Lemma 2.4 follows that U * g = U g −1 . We need only to check irreducibility that shall follow from the ergodicity of Poisson measures [13]. But to this end we need first of all to define the action of the group Aff(R d ) on sets from B(Γ(R d ). As we pointed out above, we can not define this action point-wisely. But we can define the action of operators V g on the indicators 1 A (γ) for A ∈ B(Γ(Q)). Namely, for given A we take a sequence of cylinder sets A n , n ∈ N such that π m (A∆A n ) → 0, n → ∞.
Therefore, G = 1 or G = 0 π m -a.s. We denote this function V g 1 A .
For the proof of the ergodicity of the measure π m w.r.t. Aff(R d ) we need to show the following fact: for any A ∈ B(Γ(R d )) such that ∀g ∈ Aff(R d ) V g 1 A = 1 A π m − a.s. holds π m (A) = 0 or π m (A) = 1.
Fist of all, we will show that for any pair of sets A 1 , A 2 ∈ B(Γ(Q)) with π m (A 1 ) > 0, π m (A 2 ) > 0 there exists g ∈ Aff(R d ) such that Because any Borel set may be approximated by cylinder sets, it is enough to show this fact for cylinder sets. But for such sets due to Lemma 2.7 we can choose g ∈ Aff(R d ) such that Then using an approximation we will have (2).
To finish the proof of the ergodicity, we consider any A ∈ B(Γ(R d ) such that ∀g ∈ Aff(R d ) V g 1 A = 1 A π m − a.s., π m (A) > 0.
We will show that then π m (A) = 1. Assume π m (Γ \ A) > 0. Due to the statement above, there exists g ∈ Aff(R d ) such that But due to the invariance of 1 A it means that is impossible.