Haboush's theorem

In mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group G over a field K, and for any linear representation ρ of G on a K-vector space V, given v ≠ 0 in V that is fixed by the action of G, there is a G-invariant polynomial F on V, without constant term, such that

F(v) ≠ 0.

The polynomial can be taken to be 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 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. 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.

Applications

Haboush's theorem can be used to generalize results of geometric invariant theory from characteristic 0, where they were already known, to characteristic p>0. In particular Nagata's earlier results together with Haboush's theorem show that if a reductive group (over an algebraically closed field) acts on a finitely generated algebra then the fixed subalgebra is also finitely generated.

Haboush's theorem implies that if G is a reductive algebraic group acting regularly on an affine algebraic variety, then disjoint closed invariant sets X and Y can be separated by an invariant function f (this means that f is 0 on X and 1 on Y).

C.S. Seshadri (1977) extended Haboush's theorem to reductive groups over schemes.

It follows from the work of Nagata (1963), Haboush, and Popov that the following conditions are equivalent for an affine algebraic group G over a field K:

G is reductive (its unipotent radical is trivial).
For any non-zero invariant vector in a rational representation of G, there is an invariant homogeneous polynomial that does not vanish on it.
For any finitely generated K algebra on which G act rationally, the algebra of fixed elements is finitely generated.

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 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) = 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)^λ of elements transforming according to λ. So we can assume that V is contained in the T-invariant subspace A(G)^λ of A(G).
The representation A(G)^λ is an increasing union of subrepresentations of the form E_λ+nρ⊗E_nρ, where ρ is the Weyl vector for a choice of simple roots of T, n is a positive integer, and E_μ 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_nρ has dimension (n+1)^N 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_nρ has the same dimension as in characteristic 0.
If q=p^r for a positive integer r, and n=q−1, then E_nρ contains the Steinberg representation of G(F_q) of dimension q^N. (Here F_q ⊂ K is the finite field of order q.) The Steinberg representation is an irreducible representation of G(F_q) and therefore of G(K), and for r large enough it has the same dimension as E_nρ, so there are infinitely many values of n such that E_nρ is irreducible.
If E_nρ is irreducible it is isomorphic to its dual, so E_nρ⊗E_nρ is isomorphic to End(E_nρ). Therefore, the T-invariant subspace A(G)^λ of A(G) is an increasing union of subrepresentations of the form End(E) for representations E (of the form E_(q−1)ρ)). 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.

References

Demazure, Michel (1976), "Démonstration de la conjecture de Mumford (d'après W. Haboush)", Séminaire Bourbaki (1974/1975: Exposés Nos. 453--470), Lecture Notes in Mathematics, vol. 514, Berlin: Springer, pp. 138–144, doi:10.1007/BFb0080063, ISBN 978-3-540-07686-5, MR 0444786
Haboush, W. J. (1975), "Reductive groups are geometrically reductive", Annals of Mathematics, 102 (1): 67–83, doi:10.2307/1970974, JSTOR 1970974
Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994. xiv+292 pp. MR1304906 ISBN 3-540-56963-4
Nagata, Masayoshi (1963), "Invariants of a group in an affine ring", Journal of Mathematics of Kyoto University, 3 (3): 369–377, doi:10.1215/kjm/1250524787, ISSN 0023-608X, MR 0179268
Nagata, M.; Miyata, T. (1964). "Note on semi-reductive groups". Journal of Mathematics of Kyoto University. 3 (3): 379–382. doi:10.1215/kjm/1250524788.
Popov, V.L. (2001) [1994], "Mumford hypothesis", Encyclopedia of Mathematics, EMS Press
Seshadri, C.S. (1977). "Geometric reductivity over arbitrary base". Advances in Mathematics. 26 (3): 225–274. doi:10.1016/0001-8708(77)90041-x.