International Congress of Mathematicians

Representations of Algebraic Groups and Principal Bundles 631Here we introduce the basic notion of a low height representation in characteristicp. Yet f : G —¥ SL(n) = SL(V) be a representation of G in char p, Gbeing reductive. Fix a Borei B and a Torus T in G. Let L(A»), 1 < i < m, bethe simple G-modules occurring in the Jordan-Holder filtration of V. Write eachA, as yjgyOij, where {aj} is the system of simple roots corresponding to B andjqij £ Q Vi, j. Define htXi = /Jftj. Then one has the basic [9,20]:jDefinition 2.1: f is a low-height representation ofG, or V is a low-height moduleover G, if 2ht(Xi) < p Vi.Remark 2.2: If 2ht(Xi) < p Vi, then it easily follows that V is a completelyreducibleG-module. In fact for any subgroup F of G, V is completely reducibleover F 44> F itself is completely reducible in G. By definition, an abstract subgroupF of G is completely reducible in G 44> for any parabolic P of G, if F is containedin P then F is contained in a Levi component L of P. These results were provedby Serre[20] using the notion of a saturated subgroup of G.In general, denote sup (2ht A,) by htaV. If V is the standard SL(n) module,then htsL(n)^l(V)= i(n — i),l < i < n — 1. Aloregenerally, hta(Vi®V 2 ) = ht(}Yi +ht(}V2- The following theorem is the key link between low-height representationsand semistability of induced bundles [9]:Theorem 2.3: Let E —t X be a semistable G-bundle, where G is semisimple andthe base X is a normal projective variety. Let f : G —¥ SL(n) be a low-heightrepresentation. Then the induced bundle E(SL(nj) is again semistable.The proof is an interplay between the results of Bogomolov, Kempf, Rousseauand Kirwan in G.I.T. on one hand and the results of Serre mentioned earlier on theother. The group scheme E(G) over X acts on E(SL(n)/P) and assume that a isa section of the latter. Consider the generic point if of X and its algebraic closureK. Then E(G)-% acts on E(SL(n)/P)-^-, and a is a if-rational point of the latter.There are 2 possibilities:1) a is G.I.T semistable. In this case, one can easily prove that deg CT # T^ > 0.2) a is G.I.T. unstable, i.e., not semistable. Let P(a) be the Kempf-Rousseauparabolic for a, which is defined over K. For deg CT # T^ to be > 0 it is sufficientthat P(a) is defined over K. Note that since V is a low-height representationof G, one has p> h. One then has ([20]).Proposition 2.4: If p > h, there is a unique G-invariant isomorphism log:G u —¥ g , where G u is the unipotent variety of G and g is the nilpotent varietyof g = Lie G.Proposition 2.4 is used inProposition 2.5: Let H be any semisimple group and W a low-height representationof H. Let Wi C W and assume that 3X £ Lie H, X nilpotent such that X £Lie (Stab (Wij). Then in fact one has X £ Lie [Stab (Wi) re d]-Along with some facts from G.I.T, Proposition 2.5 enables us to prove thatP(a) is in fact defined over K, thus finishing the sketch of the proof of Theorem 2.3.See also Ramanathan-Ramanan [19]. One application of low-height representations