Let U + be the plus part of the enveloping algebra of a Kac Moody Lie algebra g with a symmetric Cartan datum. In [L1] we have defined a canonical basis of U + under the assumption that the Cartan datum is of finite type; this was later generalized to Cartan data of possibly infinite type in [K, L3]. The basis of U + is obtained from a canonical basis of the quantized version of U + by specializing the quantum parameter to 1.
In [L4] we have constructed a basis of U + in terms of constructible functions on a Lagrangian variety, assuming that the Cartan datum is of affine type. (The same construction applies for Cartan data of finite type.) This basis will be called the semicanonical basis of U + . In this paper we extend the definition of semicanonical basis to include the case where the Cartan datum is not necessarily of affine or finite type. While the semicanonical basis is not necessarily the same as the canonical basis, we will show that the semicanonical basis has a number of properties in common with the canonical basis: compatibility with various filtrations of U + , compatibility with the canonical antiautomorphism of U + .
2000 Academic Press 1. THE VARIETY 4 V 1.1. We assume given a finite non-empty graph in which there is no edge joining a vertex with itself. This is the same as giving two finite sets I, H with I{<, two maps H Ä I denoted by h [ h$, h [ h" and a fixed point free involution h [ h of H, such that (h )$=h"{h$ for all h # H.
We fix a function =: H Ä C* such that =(h)+=(h )=0 for all h # H. For i, j # I we set i } j=&|(h # H | h$=i, h"= j)| if i{ j and i } j=2 if i= j. Then (i } j) is a symmetric Cartan datum.
Let U + be the Q-algebra defined by generators e i (i # I ) and the Serre relations : p, q # N; p+q=&i } j+1