One of the observations of [L4] has been that the canonical basis [L1] of the + part of a quantized enveloping algebra attached to a Cartan matrix of finite type is naturally in bijection with a collection of subsets of the set of ``totally positive'' elements in G(R((=))) where G is a semisimple simply connected algebraic group attached to the dual Cartan matrix and = is an indeterminate. (See also [L5].)This collection of subsets is not defined in algebraic-geometric terms since its definition involves positive real numbers. This paper arose from the desire to replace this collection of subsets by some algebraic-geometric objects which make sense over any field and still are in bijection with the canonical basis. For simplicity we will restrict ourselves to the simply laced case. (The general case can be easily reduced to it.) We replace
) where s is some large integer; this is regarded as an algebraic group over C of dimension s dim G. We replace the collections of subsets above by a finite collection of closed, irreducible algebraic subvarieties of G s . This collection of subvarieties of G s is naturally in bijection with a large subset of the canonical basis which becomes the entire canonical basis as s ร .The bijection is established at the combinatorial level. I do not know how to relate directly the canonical basis with the subvarieties above.