aa r X i v : . [ m a t h . R T ] J a n L´EVY PROCESSES ON THE LORENTZ-LIE ALGEBRA
AMEUR DHAHRI AND UWE FRANZ
Abstract.
L´evy processes in the sense of Sch¨urmann on the Lie algebra ofthe Lorentz grouop are studied. It is known that only one of the irreducibleunitary representations of the Lorentz group admits a non-trivial one-cocycle.A Sch¨urmann triple is constructed for this cocycle and the properties of theassociated L´evy process are investigated. The decommpositions of the restric-tions of this triple to the Lie subalgebras so (3) and so (2 ,
1) are described.
Introduction
Factorisable representations of current groups and current algebras have a longhistory, cf. [4, 17, 12], and they played an important role in the development ofquantum stochastic calculus, see [21] and the references therein.Factorisable representations of current groups of a Lie group G can be viewedas L´evy processes in the sense of Sch¨urmann [19] on the level of the associateduniversal enveloping algebra U ( g ) or the Lie algebra g , see [8, 7]. We followed thisapproach in [3], where we associated classical L´evy processes to the representationintroduced in [23] and studied, e.g., their marginal distributions. Furthermore,in [1, 2], this approach was used to define “quadratic” exponential vectors and a“quadratic” second quantization functor.Interesting factorisable representations of current groups of a Lie group existonly if the Lie group has a representation which admit a non-trivial cocycle, see[22, 25, 10]. If we restrict our attention to unitary representations and simple Liegroups, then this leaves only the two series. G = SO ( n,
1) and G = SU ( n, so (2 , ∼ = su (1 , ∼ = sl (2 , R ) of the unique simple Lie algebra of rank one. Inthe present paper we study the Lie algebra so (3 ,
1) of the Lorentz group.In Section 1, we recall the definition of the Lorentz group and its Lie algebra andwe introduce some notations which we shall use in this paper. For the purpose ofself-containedness we also recall several facts about their unitary representations.In Section 2, we recall the definitions of L´evy processes and Sch¨urmann tripleson real Lie algebras. We also construct a Sch¨urmann triple on so (3 , SO (3 ,
1) studied in [10, 24].
In the remaining sections we study restrictions of this Sch¨urmann triple to Liesubalgebras of so (3 ,
1) In Section 3, we obtain the decomposition of the restrictionto so (3). In Section 4, we study the restriction to so (2 , The Lorentz group and its Lie algebra
The
Lorentz group O (3 ,
1) is the the group of all isometries of Minkowski space-time, i.e., it is the group of all 4 × * txyz , t ′ x ′ y ′ z ′ + = tt ′ − xx ′ − yy ′ − zz ′ . The identity component SO (3 , + of O (3 ,
1) is called the restricted Lorentz group .It consists of the 4 × A = ( a jk ) ∈ O (3 ,
1) with det( A ) = +1 and a ≥ SO (3 , + is isomorphic to the projective speciallinear group (or M¨obius group) P SL (2 , C ).The Lie algebra so (3 , ∼ = sl (2 , C ) of the Lorentz group O (3 ,
1) has as basis { H , H , H , F , F , F } with the relations[ H j , H k ] = iǫ jkℓ H ℓ . [ F j , F k ] = − iǫ jkℓ F ℓ , [ F j , H k ] = iǫ jkℓ H ℓ , for j, k, ℓ ∈ { , , } , where ǫ jkℓ is the Levi-Civita symbol, ǫ j,k,ℓ = +1 if ( jkℓ ) = (1 , , , (2 , ,
1) or (3 , , , − jkℓ ) = (2 , , , (3 , ,
1) or (1 , , , . We shall consider so (3 ,
1) with the involution which makes these six elementshermitian. Then the infinitesimals of unitary representations of SO (3 , + are*-representations of so (3 , { A , A , A , B , B , B } and { H , H − , H + , K , K − , K + } with A j = 12 ( H j + iF j ) , B j = 12 ( H j − iF j ) , j = 1 , , ,H , H ± = H ± iH , F , K ± = K ± iK . In terms of these bases the relations become[ A j , A k ] = iǫ jkℓ A ℓ , [ B j , B k ] = iǫ jkℓ B ℓ , [ A j , B k ] = 0[ H + , H − ] = 2 H , [ F + , F − ] = − H , [ H , H ± ] = ± H ± , [ F , F ± ] = ∓ H ± , [ H , F ± ] = ± F ± , [ F , H ± ] = ± F ± , [ H ± , F ∓ ] = ± F , [ H ± , F ± ] = 0 , [ F , H ] = 0 , and A ∗ j = B j , H ∗± = H ∓ , F ∗± = F ∓ . The representation theory of SO (3 , + can be found in the monographs [9, 15, 18].Here we are only interested in unitary representations. On the level of the Lie ´EVY PROCESSES ON THE LORENTZ-LIE ALGEBRA 3 algebra so (3 ,
1) they can be described as follows. For ℓ ∈ Z , ℓ ≥
0, and ℓ ∈ C ,let D ℓ ℓ = span { ξ ℓm ; ℓ = ℓ , ℓ + 1 , . . . , m = − ℓ, − ℓ + 1 , . . . , ℓ } and set C ℓ = ( i √ ( ℓ − ℓ )( ℓ − ℓ ) ℓ √ ℓ − if ℓ ≥ , ℓ = 0 , ,A ℓ = (cid:26) iℓ ℓ ℓ ( ℓ +1) if ℓ > , ℓ = 0 , (note that ℓ = 0 or ℓ = can only occur if ℓ = 0 or ℓ = , resp.).We define an action of so (3 ,
1) on D ℓ ℓ by ρ ℓ ℓ ( H ) ξ ℓm = mξ ℓm , ρ ℓ ℓ ( H ± ) ξ ℓm = p ( ℓ ∓ m )( ℓ ± m + 1) ξ ℓ,m ± ,ρ ℓ ℓ ( F ) ξ ℓm = C ℓ p ℓ − m ξ ℓ − ,m − mA ℓ ξ ℓm − C ℓ +1 p ( ℓ + 1) − m ξ ℓ +1 ,m ,ρ ℓ ℓ ( F ± ) ξ ℓm = ± C ℓ p ( ℓ ∓ m )( ℓ ∓ m − ξ ℓ − ,m ± − A ℓ p ( ℓ ∓ m )( ℓ ± m + 1) ξ ℓ,m ± ± C ℓ +1 p ( ℓ ± m + 1)( ℓ ± m + 2) ξ ℓ +1 ,m ± (we use the same basis as [9] and [15], note that [5] uses ψ ℓm = i m − ℓ ξ ℓm instead).As was observed in [5], “formally” there exists also a basis { φ m ,m } on which A , A ± = A ± iA , B , and B ± = B ± iB act as A φ m ,m = m φ m ,m , A ± = p ( j ∓ m )( j ± m + 1) φ m ± ,m ,B φ m ,m = m φ m ,m , B ± = p ( j ∓ m )( j ± m + 1) φ m ,m ± , where j = j = ( ℓ + ℓ − D ℓ ℓ s.t. the family { ξ ℓm } is an orthonormal systemand denote by H ℓ ℓ the completion of D ℓ ℓ .The representation ρ ℓ ℓ is the infinitesimal representation of an irreducible uni-tary representation of SO (3 , + on H ℓ ℓ in the following two cases: a): ℓ is purely imaginary (and no restriction on ℓ ∈ Z + ), this is the prin-cipal series . b): ℓ = 0 and 0 ≤ ℓ <
1, this is the supplementary series .The representations ρ ,ℓ and ρ , − ℓ with ℓ purely imaginary are easily seen to beequal, since only the square of ℓ occurs in their definition. The remaining repre-sentations are all inequivalent. Together with the trivial representation ε , whichsends so (3 ,
1) identically to 0, these two families exhaust all irreducible unitaryrepresentations of SO (3 , + . Note that the representation ρ is not irreducible.It can be decomposed as ρ ∼ = ε ⊕ ρ on D = span { ξ } ⊕ span { ξ ℓm ; ℓ ≥ , m = − ℓ, . . . , ℓ } . The elements C A = 2 A + A + A − + A − A + and C B = 2 B + B + B − + B − B + generate the center of the universal enveloping algebra U (cid:0) so (3 , (cid:1) and satisfy C ∗ A = C B . The Casimir invariants J = C A + C B = 2 H + H + H − + H − H + − F − F + F − − F − F + ,J = − i ( C A − C B ) / H + F − + H − F + + F + H − + F − H + + 4 F H , AMEUR DHAHRI AND UWE FRANZ are hermitian and also generate the center of U (cid:0) so (3 , (cid:1) . In the irreducible unitaryrepresentations defined above they act as ρ ℓ ℓ ( J ) = ( ℓ + ℓ − D ℓ ℓ and ρ ℓ ℓ ( J ) = iℓ ℓ id D ℓ ℓ . In this paper we will work with (in general) unbounded involutive representations ofinvolutive complex Lie algebras ( g , ∗ ) or their universal envelopping algebras U ( g )acting on some pre-Hilbert space D . See [16] for necessary and sufficient conditionsfor such a representationation to be the infinitesimal representation associated toa unitary representation H = D of the connected simply connected Lig group G associated to the real Lie algebra g R = { X ∈ g ; X ∗ = − X } .2. Sch¨urmann triples on so (3 , and their L´evy processes We start by recalling the definition of Sch¨urmann triples and L´evy processes onreal Lie algebras. Let g be a real Lie algebra, ( g C , ∗ ) its complexification equippedwith the involution that makes the elements of g anti-hermitian, and U ( g ) itsuniversal enveloping algebra, without unit, but with the involution induced from( g C , ∗ )For D a complex pre-Hilbert space with, we let L ( D ) be algebra of linear op-erators on D having an adjoint defined everywhere on D , and L AH ( D ) the anti-Hermitian linear operators on D . Definition 2.1. [7, Definition 8.1.1] Let D be a pre-Hilbert space and ω ∈ D aunit vector. A family (cid:0) j st : g → L AH ( D ) (cid:1) ≤ s ≤ t of representations of g is a L´evy process on g over ( D, ω ) if a): (increment property) we have j st ( X ) + j tu ( X ) = j su ( X )for all 0 ≤ s ≤ t ≤ u and all X ∈ g ; b): (independence) we have[ j st ( X ) , j s ′ t ′ ( Y )] = 0 , X, Y ∈ g , ≤ s ≤ t ≤ s ′ ≤ t ′ , and h ω, j s t ( X ) k · · · j s n t n ( X n ) k n ω i = h ω, j s t ( X ) k ω i · · · h ω, j s n t n ( X n ) k n ω i , for n, k , . . . , k n ∈ N , 0 ≤ s ≤ t ≤ s ≤ · · · ≤ t n , X , . . . , X n ∈ g ; c): (stationarity) for n ∈ N , X ∈ g , 0 ≤ s ≤ t , h ω, j st ( X ) n ω i = h ω, j ,t − s ( X ) n ω i i.e., the moments depend only on the difference t − s ; d): (pointwise continuity) we havelim t ց s h ω, j st ( X ) n ω i = 0 , n ∈ N , X ∈ g . To a L´evy process we can associate the states ϕ t = h ω, j ,t ( a ) n ω i for a ∈ U ( g ), t ≥
0, and the functional L ( a ) = dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ϕ t ( a ) , a ∈ U ( g . ´EVY PROCESSES ON THE LORENTZ-LIE ALGEBRA 5 The functional L is a generating functional in the sense of the following definition:A linear functional L : U ( g ) → C on the non-unital *-algebra U ( g ) is called a generating functional on g if a): L is Hermitian, i.e., L ( u ∗ ) = L ( u ) for u ∈ U ( g ); b): L is positive, i.e., L ( u ∗ u ) ≥ u ∈ U ( g ). Definition 2.2. [7, Definition 8.2.1] Let D be a pre-Hilbert space. A Sch¨urmanntriple on g over D is a triple ( ρ, η, ψ ), where a): ρ : g → L AH ( D ) is a representation of g on D , i.e., ρ (cid:0) [ X, Y ] (cid:1) = ρ ( X ) ρ ( Y ) − ρ ( Y ) ρ ( X )for X, Y ∈ g ; b): η : g → D is a ρ -1-cocycle, i.e. it satisfies η (cid:0) [ X, Y ]) = ρ ( X ) η ( Y ) − ρ ( X ) η ( Y ) , X, Y ∈ g ; c): ψ : g → C is a linear functional with imaginary values s.t. the bilinear map( X, Y ) η ( X ) , η ( Y ) i is the 2- ε - ε -coboundary of ψ (where ε denotes thetrivial representation), i.e., ψ (cid:0) [ X, Y ] (cid:1) = h η ( Y ) , η ( X ) i − h η ( X ) , η ( Y ) i , X, Y ∈ g . See [11] for more information on the cohomology of Lie algebras and Lie groups.The Sch¨urmann triple and in particular the linear functional ψ in a Sch¨urmanntriple have unique extensions to g C (by linearity) and to U ( g ) (as representation,cocycle and coboundary, resp.) and the extension of the functional is a generatingfunctional. Therefore it corresponds to a L´evy process on g (which is unique indistribution). This L´evy process can be realised on the symmetric Fock spaceΓ (cid:0) L ( R + , D ) (cid:1) = L ∞ n =0 L ( R + , D ) ⊗ s n over L ( R + , D ) as j st ( X ) = Λ st (cid:0) ρ ( X ) (cid:1) + A + st (cid:0) η ( X ) (cid:1) + A − st (cid:0) η ( X ∗ ) (cid:1) + ψ ( X )( t − s )id , X ∈ g C , cf. [19]. Here Λ st ( M ) = Λ( M ⊗ [ s,t ] ), A + st ( v ) = A ( v ⊗ [ s,t ] ), and A − st ( v ) = A ( v ⊗ [ s,t ] ), with M ∈ L ( D ), v ∈ D , denote the conservation, creation, andannihilation operators, see, e.g., [7], Chapter 5].If the cocycle η is a coboundary, then the associated L´evy process is unitarilyequivalent to the second quantisation of ρ , cf. [7, Proposition 8.2.7], and the L´evyprocesses of cocycles that differ only by a coboundary are also unitarily equivalent.Therefore it is most interesting to study the processes associated to non-trivialcocycles. The L´evy processes associated to reducible Sch¨urmann triples can beconstructed as tensor products of the L´evy processes of their irreducible compo-nents, for this reason we shall study only irreducible representations.If the Casimir invariants are invertible in some representation, then all 1-cocylesare coboundaries, cf. [3, Lemma 2.2]. Therefore the only non-trivial irreducibleunitary representation that can have a non-trivial cocycle is ρ . Since so (3 ,
1) issimple, it is clear that the trivial representation has no non-zero cocycles at all, cf.[3, Lemma 2.1].From [6, 13], we know that only one irreducible unitary representation of SO (3 , ρ . We will describe a non-trivial 1-cocycle of this representation below, after recalling a useful lemma. AMEUR DHAHRI AND UWE FRANZ
Lemma 2.3. (Raabe-Duhamel test) Let ( u n ) n ∈ N be a sequence of strictly positivereal numbers such that u n +1 u n = 1 − αn + o (cid:18) n (cid:19) . If α < , then the series P n ∈ N u n diverges, if α > , then the series P n ∈ N u n converges (nothing can be concluded in the case α = 1 ). We can now construct a non-trivial 1-cocycle for ρ . Proposition 2.4.
There exists a - ρ -cocycle c with c ( F + ) = ξ . The cocycle c isnot a coboundary and every other non-trivial - ρ -cocycle is a linear combinationof c and some - ρ -coboundary.Proof. The cocycle c can not be a coboundary, since the vector ξ is not in theimage of ρ ( F + ). Indeed, assume there exists a vector ζ = P x ℓm ξ ℓm ∈ H s.t. ρ ( F + ) ζ = ξ . Then we have x ℓm = 0 for all m = 0 and for pairs with m = 0 and ℓ odd. For m = 0 and even ℓ we find the recurrence relation x ℓ, = − C ℓ − C ℓ x ℓ − , with the initial condition x = − i q . We have | x ℓ, | | x ℓ − , | = (cid:0) ( ℓ − − (cid:1) (4 ℓ − ℓ − (cid:0) ℓ − − (cid:1) = ( ℓ − ℓ )(4 ℓ − ℓ − ℓ − ℓ + 3)= 4 ℓ − ℓ − ℓ + 2 ℓ ℓ − ℓ − ℓ + 8 ℓ + 3 = 1 + o (cid:18) ℓ (cid:19) i.e., α = 0 <
1. So the Raabe-Duhamel test implies that the series P | x ℓm | diverges and therefore there exists no such vector ζ in H .It was shown in [6, 13] that the first cohomology group of ρ has dimension one,this implies the uniqueness. To prove existence, one checks that c ( H ) = c ( H ± ) = 0 c ( F ) = − √ ξ , c ( F ± ) = ± ξ , ± defines indeed a 1- ρ -cocycle. (cid:3) Remark . Let ℓ = 0. The 1- ρ ,ℓ -coboundary of ξ is given by ∂ξ ( H ) = 0 = ∂ξ ( H ± ) ,∂ξ ( F ) = − iC (0 , ℓ ) ξ , ∂ξ ( F ± ) = ± iC (0 , ℓ ) √ ξ , ± , with C (0 , ℓ ) = √ − ℓ √ . We see that c is formally the limit of the 1- ρ ,ℓ -coboundaries iC (0 ,ℓ ) √ ∂ξ as ℓ tends to 1. Proposition 2.6.
There exists a unique Sch¨urmann triple ( ρ , c, ψ ) containingthe representation ρ and the cocycle c .Proof. This is a consequence of the fact that so (3 ,
1) is simple and that thereforethe second cohomology group of the trivial representation is trivial. Any element ´EVY PROCESSES ON THE LORENTZ-LIE ALGEBRA 7 in so (3 ,
1) can be written as a commutator and so Condition c) in Definition 2.2allows to deduce the values of ψ from the values of c . We have, e.g.,2 ψ ( F ) = ψ (cid:0) [ F + , H − ] (cid:1) = D c (cid:0) ( F + ) ∗ (cid:1) , c ( H − ) E − D c (cid:0) ( H − ) ∗ (cid:1) , c ( F + ) E = 0 . It turns out that ψ is identically equal to 0 on su (3 , (cid:3) We would like to characterise the L´evy process associated to the Sch¨urmanntriple ( ρ , c, ψ ). For this purpose one could compute the action of the Casimirinvariants on the vacuum vector Ω ∈ Γ (cid:0) L ( R + , D ) (cid:1) .Since ψ vanishes on g , we have a simple formula for the action of Lie algebraelements on the vacuum vector, j st ( X )Ω = c ( X ) ⊗ [ s,t ] , X ∈ g C , ≤ s ≤ t, so we have j st ( H )Ω = 0 = j st ( H ± )Ω ,j st ( F )Ω = − √ ξ ⊗ [ s,t ] , j st ( F ± )Ω = ± ξ , ± ⊗ [ s,t ] . For the Casimir invariants we get j st ( J )Ω = j st ( H + ) (cid:0) − ξ , − ⊗ [ s,t ] (cid:1) + j st ( H − ) (cid:0) ξ , ⊗ [ s,t ] (cid:1) = 0 . and j st ( J )Ω = − j st ( F ) (cid:18) − √ ξ ⊗ [ s,t ] (cid:19) − j st ( F + ) (cid:0) − ξ , − ⊗ [ s,t ] (cid:1) − j st ( F − ) (cid:0) ξ ⊗ [ s,t ] (cid:1) = − ( t − s )Ω + √ (cid:0) ξ , ⊗ [ s,t ] (cid:1) ⊗ (cid:0) ξ , − ⊗ [ s,t ] (cid:1) + √ (cid:0) ξ , − ⊗ [ s,t ] (cid:1) ⊗ (cid:0) ξ , ⊗ [ s,t ] (cid:1) −√ (cid:0) ξ ⊗ [ s,t ] (cid:1) ⊗ (cid:0) ξ ⊗ [ s,t ] (cid:1) . . The action of j st ( J ) on the vacuum vector shows that j st ( J ) is not a multipleof the identity, which implies that the representatoin j st restricted to the subspacegenerated from the vacuum vector can not be irreducible.To get a better understanding of the representations j st , 0 ≤ s ≤ t , we will nowconsider the restrictions of ρ to Lie subalgebras of so (3 , Restriction to the Lie sub algebra so (3)The basis we used to describe the representations of so (3 ,
1) is already adapted tothe subalgebra so (3) = span { H , H + , H − } , so it is easy to decompose the restrictionof representations of so (3 ,
1) to its Lie subalgebra so (3) into its irreducible parts.The representation ρ restricted to so (3) = span { H , H + , H + } decomposes intoa direct sum of finite-dimensional irreducible representations. Recall that the irre-ducible representations of so (3) are all unitarily equivalent to one of the following.Let s ∈ Z , set E s = span { e − s , e − s +1 , . . . , e s } , where e − s , . . . , e s form an orthonor-mal basis, and set π s ( H ) e m = me m , π s ( H ± ) e m = p ( s ∓ m )( s ± s + 1) e m ± , AMEUR DHAHRI AND UWE FRANZ for m = − s, . . . , s . It is not difficult to check that we have( D , ρ | so (3) ) ∼ = ∞ M s =3 ( E s , π s ) . Restriction to Lie sub algebra so (2 , H , F + , F − span a Lie sub algebra of so (3 ,
1) that is isomor-phic to the non-compact form sl (2; R ) ∼ = su (1 , ∼ = so (2 ,
1) of the three-dimensionalsimple Lie algebra sl (2). We will now describe the restriction of our L´evy processeson so (3 ,
1) to this Lie sub algebra.Recall that so (2 ,
1) admits the highest and lowest weight representations π + t and π − t , with t >
0, acting on D ± t = span { e n ; n ∈ N } (where ( e n ) are an orthonormalbasis) as π ± t ( H ) e n = ± ( n + t ) e n ,π + t ( F + ) e n = p ( n + 1)( n + 2 t ) e n +1 , π − t ( F + ) e n = p n ( n + 2 t − e n − π + t ( F − ) e n = p n ( n + 2 t − e n − , π − t ( F − ) e n = p ( n + 1)( n + 2 t ) e n +1 , cf. [3]. There is also a third family π c,µ acting on D c,µ = span { f n , n ∈ Z } (where( f n ) is an orthonormal basis) as π c,µ ( H ) f n = ( n − µ ) f n ,π c,µ ( F + ) f n = p n + (1 − µ ) n + µ ( µ − − cf n +1 ,π c,µ ( F − ) f n = p ( n − + (1 − µ )( n −
1) + µ ( µ − − cf n − , with 0 ≤ µ < c < µ ( µ − π + t and π − t are called the positive and the negative discrete series.Our third family contains both the principal unitary series and the complementaryunitary series. See, e.g., [26], Section 6.4] for more information on the representationtheory of SU (1 , K = H − ( F + F − + F − F + ) = H ( H − − F − F + the Casimirelement of so (2 , π ± t ( K ) = 2 t ( t −
1) id and π c,µ ( K ) = 2 c id . The subrepresentations π + t and π − t can be detected by their cyclic vector e which is characterised by the equations π + t ( F − ) e = 0 , π + t ( H ) e = te (or π − t ( F + ) e = 0 , π + t ( H ) e = − te resp.) . Proposition 4.1.
If we restrict the representation ρ of so (3 , to the Lie subal-gebra so (2 , , then it decomposes as ( D , ρ | so (2 , ) ∼ = ( D +1 , π +1 ) ⊕ ( D − , π − ) ⊕ ( D R , π R ) where the ”rest” ( π R , D R ) is a direct sum of unitary irreducible representations ( π c, , D c, ) belonging to the third family.Proof. We need to determine all eigenvectors of ρ ( H ) that are annihilated by ρ ( F − ). A non-zero vector ξ = P ∞ ℓ =1 P ℓm = − ℓ x ℓm ξ ℓm is an eigenvector of ρ ( H ),if and only if there exists an integer m ∈ Z s.t. x ℓm = 0 for m = m . And thisinteger is then its eigenvalue. ´EVY PROCESSES ON THE LORENTZ-LIE ALGEBRA 9 Let us now study, when such a vector is annihilated by ρ ( F − ). We considerfirst m ∈ {− , , } . For ξ = P ∞ ℓ =1 x ℓ ξ ℓm ∈ D we have ρ ( F − ) ξ = − ∞ X ℓ =2 (cid:16) x ℓ +1 C ℓ +1 p ( ℓ + m + 1)( ℓ + m )(1) + x ℓ − C ℓ p ( ℓ − m )( ℓ − m + 1) (cid:17) ξ ℓm , (2)If this vector vanishes, then the coefficients x ℓ +1 , ℓ ∈ N , are determined by x viathe recurrence relation(3) x ℓ +1 = − C ℓ p ( ℓ − m )( ℓ − m + 1) C ℓ +1 p ( ℓ + m + 1)( ℓ + m ) x ℓ − . We get | x ℓ +1 || x ℓ − | = ( ℓ − (cid:0) ℓ + 1) − (cid:1) ( ℓ − m )( ℓ − m + 1)(4 ℓ − (cid:0) ( ℓ + 1) − (cid:1) ( ℓ + m + 1)( ℓ + m )= 1 − m ℓ + o (cid:18) ℓ (cid:19) and the Raabe-Duhamel test shows that for x = 0 this series can only converge if m = 1.Furthermore, in the case m = 1 we get x = 0 from the coefficient of ξ in (1),and then x ℓ = 0 for all ℓ > x = 1, x = 0 and let ( x ℓ ) ℓ ≥ denote the solution of the recurrencerelation (3). Then ξ + = P ∞ ℓ =1 x ℓ ξ ℓ is a non-zero vector s.t. ρ ( H ) ξ + = ξ + , ρ ( F − ) ξ + = 0 , it is therefore the cyclic vector of a subrespresentation of ( D , ρ | so (2 , ) that isunitarily equivalent to ( D +1 , π +1 ).A careful study of the equation ρ ( F − ) ξ = 0 shows that all solutions are of theform λξ + for some λ ∈ C .Indeed, the discussion above shows this already for m ∈ {− , , } . Further-more, there are no solution with m >
1, because the condition ρ ( F − ) ξ = 0implies immediately x = 0 = x . And the Raabe-Duhamel test allows to showthat there are no solutions with m <
1, either.The discussion of the condition ρ ( F + ) ξ = 0, which leads to subrepresentationsthat are unitarily equivalent to a representation of the form ( D − t , π − t ) is similary.Set y = 1, y = 0 and let ( y , ℓ ) ℓ ≥ be the sequence determined from these valuesvia the recurrence relation y ℓ +1 = C ℓ p ( ℓ + m )( ℓ + m + 1) C ℓ +1 p ( ℓ − m )( ℓ − m + 1)with m = 1. Set ξ − = P ∞ ℓ =1 y ℓ ξ ℓ, − . Then we have { ξ ∈ D ; ρ ( F + ) ξ = 0 } = C ξ − , which implies that ( D , ρ | so (2 , ) contains a unique subrepresentation that isunitarily equivalent to ( D + − , π + − ).Since the spectrum of ρ ( H ) is equal to Z , it follows that the remaining sub-representations have to belong to the family ( D c, , π c, ), c < (cid:3) Conclusion
We have identified the Sch¨urmann triple underlying the factorizable current rep-resentations of the Lorentz group in [10, 24].The decomposition in Proposition 4.1 can be used to compute the classical dis-tribution of elements of j st ( F + + F − + λH ), since the distributions of elements ofthis form are known for the irreducible unitary representations of so (2 , O ( n, U ( n,
1) in the future.
Acknowledgements
UF was supported by the French ‘Investissements d’Avenir’ program, projectISITE-BFC (contract ANR-15-IDEX-03), and by an ANR project (No./ ANR-19-CE40-0002).This work was presented at the “International Conference on Infinite Dimen-sional Analysis, Quantum Probability and Related Topics, QP38” held at TokyoUniversity of Science in October 2017, and has been submitted for publication inthe proceedings of that meeting. We thank the organizers Noboru Watanabe andSi Si for their hospitality.
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