Adapted algebras and standard monomials (Q1878436)

Let \(G\) be a semisimple simply connected algebraic group over an algebraically closed field. Let \(B \subseteq G\) be a Borel subgroup, \(U \subseteq B\) its unipotent radical, \(T \subseteq B\) a maximal torus and \(\mathcal W\) the Weyl group with respect to \(T\). Let also \(Q\) be a parabolic subgroup which normalizes a highest weight vector in the Weyl module associated to a dominant weight \(\lambda\), and \(\mathcal L_{\lambda}\) the corresponding ample line bundle on \(G/Q\). Adapted algebras, introduced by \textit{P. Caldero} in [Transform. Groups 8, No. 1, 37--50 (2003; Zbl 1044.17007)], are maximal subalgebras of \(C_q[U^-]\) spanned by a subset \(S \subseteq \mathcal B^*\), such that \(bb' \in S\) for all \(b, b' \in S\), where \(\mathcal B^*\) denotes the dual basis of the canonical basis \(\mathcal B\) of \(U_q(n^-)\), with respect to the canonical pairing. A basis for \(H^0(G/Q, \mathcal L_{\lambda})\) consisting of path vectors was constructed by \textit{P. Littelmann} in [J. Am. Math. Soc. 11, No. 3, 551--567 (1998; Zbl 0915.20022)]. In this article the authors prove that in the \(q\)-commutative parts of \(\mathcal B^*\) given by adapted algebras \(A_{\widetilde{\omega_0}}\), associated to a reduced expression \(\widetilde{\omega_0}\) of the longest word \(\omega_0 \in \mathcal W\), the two basis coincide up to multiplication by a root of unity and vice-versa.