  Let $G$ be a simply connected semi-simple complex algebraic group. Fix a
maximal torus $T$ and a Borel subgroup $B$ such that $T\subset B\subset G$. Let
$W$ the Weyl group of $G$ relative to $T$. For any $w$ in $W$, let $X_w=\bar
{BwB/B}$ denote the Schubert variety corresponding to $w$. This talk is
concerned with the following problem : Is there a flat family over Spec${\bf
C}[t]$, such that the general fiber is $X_w$ and the special fiber is a toric
variety? Our approach of the problem is based on the canonical/global base of
Lusztig/Kashiwara and the so-called string parametrization of this base studied
by P. Littelmann and made precise by A. Berenstein and A. Zelevinsky. Fix $w$
in $W$ and let $P^+$ be the semigroup of dominant weights. For all $\lambda$ in
$P^+$, let ${\cal L}_\lambda$ be the line bundle on $G/B$ corresponding to
$\lambda$. Then, the direct sum of global sections $R_w:=\bigoplus_{\lambda\in
P^+}H^0(X_w,{\cal L}_\lambda)$ carries a natural structure of $P^+$-graded
${\bf C}$-algebra. Moreover, there exists a natural action of $T$ on $R_w$. Our
principal result can be stated as follows : There exists a filtration $({\cal
F}_m^w)_{m\in{\bf N}}$ of $R_w$ such that (i) for all $m$ in ${\bf N}$, ${\cal
F}_m^w$ is compatible with the $P^+$-grading of $R_w$, (ii) for all $m$ in
${\bf N}$, ${\cal F}_m^w$ is compatible with the action of $T$, (iii) the
associated graded algebra is the ${\bf C}$-algebra of the semigroup of integral
points in a rational convex polyhedral cone. Equations for this cone were
obtained by A. Berenstein and A. Zelevinski from $\tilde w_0$-trails in
fundamental Weyl modules of the Langlands dual of $G$. By standard arguments,
the previous theorem gives a positive answer to the Degeneration Problem.
