Toric degenerations of Schubert varieties

Abstract

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 $X_w=\overline{BwB/B}$ denote the Schubert variety corresponding to $w$. This talk is concerned with the following problem : Is there a flat family over Spec$\mathbf{C}[t]$, 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 ${\mathcal{L}}_{\lambda }$ be the line bundle on $G/B$ corresponding to $\lambda$. Then, the direct sum of global sections $R_w:={⨁}_{\lambda \in P^{+}}{H}^0(X_w,{\mathcal{L}}_{\lambda })$ carries a natural structure of $P^{+}$-graded $\mathbf{C}$-algebra. Moreover, there exists a natural action of $T$ on $R_w$. Our principal result can be stated as follows : There exists a filtration $({\mathcal{F}}_m^w{)}_{m\in \mathbf{N}}$ of $R_w$ such that (i) for all $m$ in $\mathbf{N}$, ${\mathcal{F}}_m^w$ is compatible with the $P^{+}$-grading of $R_w$, (ii) for all $m$ in $\mathbf{N}$, ${\mathcal{F}}_m^w$ is compatible with the action of $T$, (iii) the associated graded algebra is the $\mathbf{C}$-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.

12 pages