Let  be a semisimple linear algebraic group of inner type over a field , and let  be a projective homogeneous -variety such that  splits over the function field of . We introduce the -invariant of  which characterizes the motivic behavior of , and generalizes the -invariant defined by A. Vishik in the context of quadratic forms.
We use this -invariant to provide motivic decompositions of all generically split projective homogeneous -varieties, e.g. Severi-Brauer varieties, Pfister quadrics,...
                    
                 
                
                    
                
            
        
            
            
            
            
            
                
            
                
            
                
            
                
            
                
            
                
                    
                
            
                
            
                
             
            
            
                
            
            
            
                
                    
                
            
            
            
            
                
            
            
             
            
                
            
            
            
                
                
                
                    
                       
Let  be a split semisimple linear algebraic group over a field and let  be a split maximal torus of . Let  be an oriented cohomology (algebraic cobordism, connective -theory, Chow groups, Grothendieck’s , etc.) with formal group law . We construct a ring from  and the characters of , that we call a formal group ring, and we define a characteristic ring morphism  from this formal group ring to  where  is the variety of Borel subgroups of . Our main result says that when the torsion index...
                    
                 
                
                    
                
            
        
            
            
            
            
            
                
            
                
            
                
            
                
            
                
                    
                
            
                
            
                
             
            
            
                
            
            
            
                
                    
                
            
            
            
            
                
            
            
             
            
                
            
            
            
                
                
                
                
                    
                
            
        
        
        
            
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