2. The Nonlinear Representation T and Spaces of Differentiable Vectors In this chapter we prove properties of Ty, Y G U(p), and of spaces Ei, i 0, in order be able to use an implicit functions theorem in Frechet spaces. To begin with, we introduce the linear representation of V0, with differential T 1 , given by equation (1.5). We shall denote in the same way the operators Tj- G Ep, X G ?, and its closure. It follows that Tp are skew-adjoint, so exp(tTp ), t G R, is unitary on Ep. Let 6M (resp. /?) be the solutions of equation D6M = 0 (resp. (i^d^ + ra)/3 = 0) with initial conditions a^,dM (resp. a), where (a, a, a) G Ep. We define the representation U1 of P 0 by C7^(a,d,a) = (ag,ag,ag), ( a , a , a ) e £ p , # = (n,L) G R4xSX(2,C), and a,(y) = A L 6(A^((0,y) - n ) ) , (2.1a) «9(V) = ^{^b(Kl\{y\y) - n))y0=Q, (2.1b) a9(y)=T(L)(3(Al1(0,y)-n), (2.1c) where L i— A^ is the canonical projection of 5L(2,C) onto 50(3,1) and T is the Dirac spinor representation of 5L(2, C). We define Ag = A^. As T ^ , 1 i 3, is not in general skew-adjoint on Ep, we state the following result of which we omit the proof as it is straightforward by using Fourier decomposition: Lemma 2.1. g i— » Ug is a strongly continuous linear representation of V0 on Ep, with p - 1 / 2 , and g »- (A" 1 0 A" 1 0 I)Ug is unitary in Exl2. In order prove properties of the space of differentiable vectors E^ of C/1, it will be useful to know that the Fourier transform of U1 is a unitary representation after a multiplication by a C°°-multiplier. Theorem 2.2. Let hg(p0,p) = (po/iAg^o)-^1/2, (p0,p) eR4, ge V0 and u = (a, a, a) G Ep, p - 1 / 2 . Let, Vgu=(A-1hg(\Vl-iV)ag,Ag1hg(\Vl-iV)dgjag), where (ag,agiag) UgU. Then g i— » Vg is a unitary representation of V0 on Ep.The representations (A _ 1 0A"" 1 01)U l on E1/2 andV on Ep are unit arily equivalent. Moreover the representations U1 on Ep and V on Ep have the same Hilbert space of Cn-vectors, namely E^, n 0. Proof. The map g i- Hg from V0 to L00(M3,G?L(4,R)), defined by Hg(p) = A^hg^p) is C°°. It follows then from the definition of Ep that the map g \-+ Vgu is Cn if and only if this is the case for the map g i-» UgU. By construction, the space of Cn-vectors for U1 is Ep. This proves the last part of the proposition. By direct calculation one finds that for u G S(R 3 ,R 4 ) 0 S(R 3 ,R 4 ) 0 5(R 3 ,C 4 ) Vgu = ( | V | ^ + 1 / 2 0 | V | ^ + 1 / 2 0 I^Ag1 0 A 1 01)U l g {\V\ p - 1/ 2 0 |V|^" 1 / 2 0 1 ) u . 20
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