TY - JOUR

T1 - Unipotent elements and generalized exponential maps

AU - Sobaje, Paul

N1 - Publisher Copyright:
© 2018 Elsevier Inc.

PY - 2018/7/31

Y1 - 2018/7/31

N2 - Let G be a simple and simply connected algebraic group over an algebraically closed field k of characteristic p>0. Assume that p is good for the root system of G and that the covering map Gsc→G is separable. In previous work we proved the existence of a (not necessarily unique) Springer isomorphism for G that behaved like the exponential map on the restricted nullcone of G. In the present paper we give a formal definition of these maps, which we call ‘generalized exponential maps.’ We provide an explicit and uniform construction of such maps for all root systems, demonstrate their existence over Z(p), and give a complete parameterization of all such maps. One application is that this gives a uniform approach to dealing with the “saturation problem” for a unipotent element u in G, providing a new proof of the known result that u lies inside a subgroup of CG(u) that is isomorphic to a truncated Witt group. We also develop a number of other explicit and new computations for g and for G. This paper grew out of an attempt to answer a series of questions posed to us by P. Deligne, who also contributed several of the new ideas that appear here.

AB - Let G be a simple and simply connected algebraic group over an algebraically closed field k of characteristic p>0. Assume that p is good for the root system of G and that the covering map Gsc→G is separable. In previous work we proved the existence of a (not necessarily unique) Springer isomorphism for G that behaved like the exponential map on the restricted nullcone of G. In the present paper we give a formal definition of these maps, which we call ‘generalized exponential maps.’ We provide an explicit and uniform construction of such maps for all root systems, demonstrate their existence over Z(p), and give a complete parameterization of all such maps. One application is that this gives a uniform approach to dealing with the “saturation problem” for a unipotent element u in G, providing a new proof of the known result that u lies inside a subgroup of CG(u) that is isomorphic to a truncated Witt group. We also develop a number of other explicit and new computations for g and for G. This paper grew out of an attempt to answer a series of questions posed to us by P. Deligne, who also contributed several of the new ideas that appear here.

KW - Springer isomorphisms

KW - Unipotent elements

KW - Witt groups

UR - http://www.scopus.com/inward/record.url?scp=85047783135&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2018.05.015

DO - 10.1016/j.aim.2018.05.015

M3 - Article

AN - SCOPUS:85047783135

SN - 0001-8708

VL - 333

SP - 463

EP - 496

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -