Overgroups Of Tensor Product Generated By Special Linear Groups

Let K be a skew field.F is the center of K,and d=dimFk<∞,n=r≥3. N=SL(n,KR)(?) SL(r,KL)≤X≤GL(nrd,F).Then one of the following results is true: (1)N(?)X≤Γ1; (2)N(?)SL(nrd,F). HereΓ=(GL(n,KR)(?) GL(r,EL))(GalK/F),Γ1=Γ∪Γ(τJ)normalize N,J is an F-antiautomorphism over K,andτis the transpose mapping of matrix over K.