A COUNTEREXAMPLE TO A CONJUGACY CONJECTURE OF STEINBERG
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Springer Science+Business Media New York (2019)
A COUNTEREXAMPLE TO A CONJUGACY CONJECTURE OF STEINBERG MIKKO KORHONEN School of Mathematics The University of Manchester Manchester M13 9PL, UK korhonen [email protected]
Abstract. Let G be a semisimple algebraic group over an algebraically closed field of characteristic p ≥ 0. At the 1966 International Congress of Mathematicians in Moscow, Robert Steinberg conjectured that two elements a, a0 ∈ G are conjugate in G if and only if f (a) and f (a0 ) are conjugate in GL(V ) for every rational irreducible representation f : G → GL(V ). Steinberg showed that the conjecture holds if a and a0 are semisimple, and also proved the conjecture when p = 0. In this paper, we give a counterexample to Steinberg’s conjecture. Specifically, we show that when p = 2 and G is simple of type C5 , there exist two non-conjugate unipotent elements u, u0 ∈ G such that f (u) and f (u0 ) are conjugate in GL(V ) for every rational irreducible representation f : G → GL(V ).
1. Introduction Let G be a semisimple algebraic group over an algebraically closed field K of characteristic p ≥ 0. At the 1966 International Congress of Mathematicians in Moscow, Steinberg proposed the following conjecture [Ste68, Problem (4)]. Conjecture 1.1 (Steinberg). Two elements a and a0 of G are conjugate in G if and only if f (a) and f (a0 ) are conjugate in GL(V ) for every rational irreducible representation f : G → GL(V ). One motivation for the conjecture, observed by Steinberg in [Ste68], is that a positive answer to the conjecture would imply that G has only a finite number of unipotent conjugacy classes. When Steinberg posed Conjecture 1.1, the finiteness of the number of unipotent conjugacy classes was known in good characteristic [Ric67, Prop. 5.2], and it was eventually shown to be true in general by Lusztig [Lus76]. Lusztig’s proof, based on the theory of Deligne–Lusztig characters of finite groups of Lie type, still remains the only known uniform proof of the result in characteristic p > 0. The main purpose of this paper is to provide a counterexample to Conjecture 1.1. Specifically, our main result is the following, which is given by Theorem 5.3: Theorem 1.2. Let p = 2 and let G be simple of type C5 . Then there exist two nonconjugate unipotent elements u, u0 ∈ G such that f (u) and f (u0 ) are conjugate in DOI: 10.1007/S00031-019-09538-3 Received June 26, 2018. Accepted December 12, 2018. Corresponding Author: Mikko Korhonen, e-mail: korhonen [email protected]
MIKKO KORHONEN
GL(V ) for every rational irreducible representation f : G → GL(V ). We will also show that in the case of unipotent elements, our counterexample is minimal in the sense that up to isogenies, there are no other examples for simple groups of rank at most 5 (Theorem 5.6). It remains an open question to determine when exactly the conclusion of Conjecture 1.1 holds. Steinberg proved Conjecture 1.1 in the case where a and a0 are both semisimple [Ste65, 6.6], and also in the case where p = 0 [Ste78, Thm. 3]. Currently it is not kn
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