# Jacques Faraut, Soji Kaneyuki, Adam Koranyi, Qi-keng Lu, Guy's Analysis and Geometry on Complex Homogeneous Domains PDF By Jacques Faraut, Soji Kaneyuki, Adam Koranyi, Qi-keng Lu, Guy Roos

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Additional resources for Analysis and Geometry on Complex Homogeneous Domains

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P(i) = J(gexpiX) = o. 1. For -y = 9 exp iX we put 1i"(r) = 7r(g)Exp(id7r(X)). Then 111i"(r)II ~ 1, and 1i"(-y)* = 1i"(r#). In fact -y# = expiX. g-1 = g-1 exp(iAd(g)X), 42 III. Positive Unitary Representations and 1i'(-y). = Exp(id1l"(X))1I"(g-1), 1i'(-y*) = 1I"(g-1)ExP(id1l"(Ad(g)X)) = 1I"(g-1 )11" (g )Exp( id1l"(X) )1I"(g-1) = Exp( id1l"(X))1I"(g-1). We will prove first that 1i' is weakly holomorphic, then that 1i' is a representation. 1): there exists an open neighborhood U of e in GC, and a dense subspace 'Ho C 'H such that, for u E 'Ho, the map CPu : g ~ 11" (g)u has a holomorphic extension to U, CPu: U -t 'H.

For u E 'H, the linear form v ~ (vlu) is continuous on 'Ho. Therefore there exists a unique u* E 'Ho such that (vlu) = b(v,u*). One puts Au = u*. The operator A is continuous, injective, maps 'H into 'Ho, and (Aulv) = b(Au, Av) (u, v E 'H). • It follows that A is selfadjoint. 3. We put B(u, v) = b(u, v) Then B(u, u) ~ + (ulv) lIull 2 (u, v E Y). (u E Y). jB(u,u). Then Y c 'Ho c 'H, and the Hermitian form b extends continuously to 'Ho. One defines the selfadjoint operator A as in the Lemma, and shows that A commutes with the representation 7r, 7r(g)A = A7r(g) (g E G).

The operator A is continuous, injective, maps 'H into 'Ho, and (Aulv) = b(Au, Av) (u, v E 'H). • It follows that A is selfadjoint. 3. We put B(u, v) = b(u, v) Then B(u, u) ~ + (ulv) lIull 2 (u, v E Y). (u E Y). jB(u,u). Then Y c 'Ho c 'H, and the Hermitian form b extends continuously to 'Ho. One defines the selfadjoint operator A as in the Lemma, and shows that A commutes with the representation 7r, 7r(g)A = A7r(g) (g E G). 2 that A = p,l, therefore (u, v) = p,B(u, v). References [Cartier,1974]' [Nelson,1959j.