# CHAPTER 20

*GL*(*n*) × *GL*(*n*) × … × *GL*(*n*) **Examples**

In this chapter, we investigate the following question. Suppose we have a geometrically irreducible middle extension sheaf *G* on G_{m}/k which is pure of weight zero, such that the object *N* := *G*(1/2)[1] *∈ P*_{arith} has “dimension” *n* and has *G*_{geom,N} = *G*_{arith,N} = *GL*(*n*). Suppose in addition we are given *s* ≥ 2 distinct characters χ_{i} of *k*^{×}. We want criteria which insure that for the objects

*N*_{i} := *N* ⊗ *L*_{χi},

the direct sum ⊕_{i}N_{i} has *G*_{geom,⊕iNi} = *G*_{arith,⊕iNi} = ∏_{i} GL(*n*). Because we have a priori inclusions *G*_{geom,⊕iNi} ⊂ *G*_{arith,⊕iNi} ⊂ ∏_{i} GL(*n*), it suffices to prove that *G*_{geom,⊕iNi} = ∏_{i} GL(*n*). To show this, it suffices to show both of the following two statements.

(1) The determinants in the Tannakian sense “det”(