Editing Haboush's theorem (section)

==Proof==
The theorem is proved in several steps as follows:
*We can assume that the group is defined over an [[algebraically closed]] field ''K'' of characteristic ''p''>0.
*Finite groups are easy to deal with as one can just take a product over all elements, so one can reduce to the case of '''connected''' reductive groups (as the connected component has finite index). By taking a central extension which is harmless one can also assume the group ''G'' is '''simply connected'''.
*Let ''A''(''G'') be the coordinate ring of ''G''. This is a representation of ''G'' with ''G'' acting by left translations. Pick an element ''{{prime|v}}'' of the dual of ''V'' that has value 1 on the invariant vector ''v''. The map ''V'' to ''A''(''G'') by sending ''w''∈''V'' to the element ''a''∈''A''(''G'') with ''a''(''g'') = ''{{prime|v}}''(''g''(''w'')). This sends ''v'' to 1∈''A''(''G''), so we can assume that ''V''⊂''A''(''G'') and ''v''=1.
*The structure of the representation ''A''(''G'') is given as follows. Pick a maximal torus ''T'' of ''G'', and let it act on  ''A''(''G'') by right translations (so that it commutes with the action of ''G''). Then ''A''(''G'') splits as a sum over characters λ of ''T'' of the subrepresentations ''A''(''G'')<sup>λ</sup> of elements transforming according to λ. So we can assume that ''V'' is contained in the ''T''-invariant subspace ''A''(''G'')<sup>λ</sup> of ''A''(''G'').
*The representation ''A''(''G'')<sup>λ</sup> is an increasing union of subrepresentations of the form ''E''<sub>λ+''n''ρ</sub>⊗''E''<sub>''n''ρ</sub>, where ρ is the Weyl vector for a choice of simple roots of ''T'', ''n'' is a positive integer, and ''E''<sub>μ</sub> is the space of sections of the [[line bundle]] over ''G''/''B'' corresponding to a character μ of ''T'', where ''B'' is a [[Borel subgroup]] containing ''T''.
*If ''n'' is sufficiently large then ''E''<sub>''n''ρ</sub> has dimension (''n''+1)<sup>''N''</sup> where ''N'' is the number of positive roots. This is because in characteristic 0 the corresponding module has this dimension by the [[Weyl character formula]], and for ''n'' large enough that the line bundle over ''G''/''B'' is [[very ample]], ''E''<sub>''n''ρ</sub> has the same dimension as in characteristic 0.
*If ''q''=''p''<sup>''r''</sup>  for a positive integer ''r'', and ''n''=''q''&minus;1, then ''E''<sub>''n''ρ</sub> contains the [[Steinberg representation]] of ''G''('''F'''<sub>''q''</sub>) of dimension ''q''<sup>''N''</sup>. (Here '''F'''<sub>''q''</sub> ⊂ ''K'' is the finite field of order ''q''.) The Steinberg representation is an irreducible representation of ''G''('''F'''<sub>''q''</sub>) and therefore of ''G''(''K''), and for ''r'' large enough it has the same dimension as ''E''<sub>''n''ρ</sub>, so there are infinitely many values of ''n'' such that ''E''<sub>''n''ρ</sub> is irreducible.
*If ''E''<sub>''n''ρ</sub> is irreducible it is isomorphic to its dual, so ''E''<sub>''n''ρ</sub>⊗''E''<sub>''n''ρ</sub> is isomorphic to End(''E''<sub>''n''ρ</sub>). Therefore, the ''T''-invariant subspace ''A''(''G'')<sup>λ</sup> of ''A''(''G'') is an increasing union of subrepresentations of the form End(''E'') for representations ''E'' (of the form ''E''<sub>(''q''&minus;1)ρ</sub>)). However, for representations of the form End(''E'')  an invariant polynomial that separates 0 and 1 is given by the determinant. This completes the sketch of the proof of Haboush's theorem.