the vector space $\C[N]$. Let $\d = (d_i)$ be a dimension vector. The variety $\overline{E}_\d$ of representations of $\C\overline{Q}$ with dimension vector $\d$ is a vector space with a natural symplectic structure. Lusztig [@Lu0] has shown that $\L_\d$ is a Lagrangian subvariety of $\overline{E}_\d$, whose number of irreducible components is equal to the dimension of the degree $\d$ homogeneous component of $U(\n)$. Let $Z$ be an irreducible component of $\L_\d$. Since $\vph \colon X \mapsto \vph_X$ is a constructible map on $\L_\d$, it is constant on a Zariski dense open subset of $Z$. Let $\vph_Z$ denote this generic value of $\vph$ on $Z$. Then, if we denote by $\ZZ$ the collection of all irreducible components of all varieties $\L_\d$, one can easily check that $$\SC^* := \{\vph_Z \mid Z \in \ZZ\} \subset \C[N]$$ is dual to the basis $\SC = \{f_Z \mid Z \in \ZZ\}$ of $\M \cong U(\n)$ constructed by Lusztig in [@Lu1], and called by him the semicanonical basis.

For example, suppose that $X\in\nil(\L)$ is *rigid*, that $\Ext^1_\L(X,X) = 0$. Then $X$ is a generic point of the unique irreducible component $Z$ on which it sits, that is, $\vph_X = \vph_Z$ belongs to $\SC^*$.

Categories attached to Weyl 