I am reading the paper Criterion for smoothness of Schubert varieties in $\mathrm{Sl}(n)/B$ by V Lakshmibai and B Sandhya; Proc. Indian Acad. Sci. (Math. Sci.), Vol. 100, No. 1, April 1990, pp. 45-52. https://www.ias.ac.in/article/fulltext/pmsc/100/01/0045-0052.

There are a few notions and notations in the paper that I don't understand. My questions are :

(1) In page 3 of the paper (printed page 47), in Section 2, the authors define certain parabolic subgroups $Q_i$ inductively as $Q_1=P_1$ and $Q_{i+1}=Q_i \cap P_{i+1}$ . What I don't understand is: What are these $P_i$ s ?

(2) In Theorem 2.1, they use the notion of "equidimensionality" of the restriction of projection maps. What does this being equidimensional mean ?

(3) In Theorem 2.1, what does $W(Q_i)$ mean ? Is it the Weyl subgroup corresponding to $Q$ i.e. is $W(Q_i) \cong N_{Q_i} (T)/T$ ?

It would be highly appreciated if someone could clarify my doubts. Thanks.

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