EXPONENTIATION OF COMMUTING NILPOTENT VARIETIES

Abstract. Let H be a linear algebraic group over an algebraically closed field of characteristic p> 0. We prove that any “exponential map ” for H induces a bijection between the variety of r-tuples of commuting [p]-nilpotent elements in Lie(H) and the variety of height r infinitesimal one-parameter subgroups of H. In particular, we show that for a connected reductive group G in pretty good characteristic, there is a canonical exponential map for G and hence a canonical bijection between the aforementioned varieties, answering in this case questions raised both implicitly and explicitly by Suslin, Friedlander, and Bendel. Let H be a linear algebraic group over an algebraically closed field k of char-acteristic p> 0. Let H(r) denote the r-th Frobenius kernel of H, and h its Lie algebra. There is a [p]-mapping on h which sends x 7 → x[p], and we set N1(h) to be the restricted nullcone of h, which consists of those x for which x[p] = 0. Let Cr(N1(h)) denote the variety of r-tuples of commuting elements in N1(h), while Homgs/k(Ga(r), H) is the affine variety of height r infinitesimal one-parameter sub