Editing Haboush's theorem (section)

{{short description|Each semi-simple algebraic group is geometrically reductive}}
In [[mathematics]] '''Haboush's theorem''', often still referred to as the '''Mumford conjecture''', states that for any [[semisimple algebraic group]] ''G'' over a [[field (mathematics)|field]] ''K'', and for any linear representation ρ of ''G'' on a ''K''-[[vector space]] ''V'', given ''v''&nbsp;≠&nbsp;0 in ''V'' that is fixed by the action of ''G'', there is a [[G-invariant|''G''-invariant]] [[polynomial]] ''F'' on ''V'', without constant term,  such that

:''F''(''v'') ≠ 0.

The polynomial can be taken to be [[homogeneous polynomial|homogeneous]], in other words an element of a symmetric power of the dual of ''V'', and if the characteristic is ''p''>0 the degree of the polynomial can be taken to be a power of ''p''.
When ''K'' has characteristic 0 this was well known; in fact [[Weyl's theorem on complete reducibility|Weyl's theorem on the complete reducibility]] of the representations of ''G'' implies that ''F'' can even be taken to be linear. Mumford's conjecture about the extension to prime characteristic ''p'' was proved by W. J. {{harvtxt|Haboush|1975}}, about a decade after the problem had been posed by [[David Mumford]], in the introduction to the first edition of his book ''Geometric Invariant Theory''.