Let \(G\) be a semi-simple simply connected algebraic group (assumed, for convenience, to be defined over an algebraically closed field of characteristic 0) and let \(B\subset G\) be a Borel subgroup and \(T\subset B\) a maximal torus. Let \(W\) be the Weyl group associated to the pair \((G,T)\). For any \(w\in W\), let \(X(w)= \overline {BwB}/B \subset G/B\) be the associated Schubert variety. Also let \(V(\lambda)\) be the (finite dimensional) irreducible \(G\)-module with highest weight \(\lambda\), and (for \(w\in W)\) let \(V_w(\lambda)\) be the Demazure module defined as the smallest \(B\)-submodule of \(V(\lambda)\) containing the extremal weight vector \(wv_\lambda\) (where \(v_\lambda\) is a highest weight vector of \(V(\lambda))\). As usual, denote by \(\rho\) the half sum of the positive roots of \(G\), and the set of positive roots of \(G\) is denoted by \(R^+\).
For any \(v\leq w\in W\) (where \(\leq\) is the Bruhat-Chevalley partial order on \(W)\), \(v\in X(w)\), in particular, the identity element \(e\in X(w)\). Let \(T(w,e)\) denote the Zariski tangent space of \(X(w)\) at \(e\). Then \(T(w,e)\) can be canonically thought of as a \(T\)-stable subspace of \(u^-\) (where \(u^-\) is the span of all the negative root spaces). – The main result of the paper under review is the following:
Fix \(w\in W\) and let \(\beta\in R^+\). Then \(-\beta\) is a weight of \(T(w,e)\) iff \(\rho-\beta\) is not a weight of \({V(\rho) \over V_w (\rho)}\).
This result is obtained by using some results of P. Littelmann on the Lakshmibai-Seshadri paths. – For \(\beta \in R^+\), let \(F_\beta\) denote a root vector corresponding to the negative root \(-\beta\).
As an immediate consequence of her result together with a result of Polo, the author obtains that for any \(\beta \in R^+\), \(F_\beta v_\rho \in V_w(\rho)\) iff for all (not necessarily distinct) \(\beta_1, \dots, \beta_\ell \in R^+\) with \(\sum \beta_i = \beta\), \(F_{\beta_1} \cdots F_{\beta_\ell} v_\rho \in V_w(\rho)\).
As another consequence of her result, she explicitly determines the tangent space \(T(w,e)\) for any \(w\in W\) and any classical \(G\) (i.e. \(G\) of type \(A, B, C, D)\), refining her earlier works (partly obtained with Seshadri and Rajeswari).