# On some notations and notions of a paper on smoothness of Schubert varieties by Lakshmibai and Sandhya

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.

• I would guess that the P_i’s are maximal parabolics. May 6, 2019 at 20:09
• @NajibIdrissi In some versions of the English language (notably in India), "doubt" just means "question". May 6, 2019 at 20:09
• I think they’re the “standard” maximal parabolics corresponding to the nodes of the Dynkin diagram. May 6, 2019 at 20:16
• The survey paper Lakshmibai, Musili, and Shesadri - Geometry of $G/P$ is more careful. It says: (1) $P_i$ is the maximal parabolic corresponding to the fundamental weight $\pi_i$ (p. 433); (2) $W_Q = N_Q(T)/T$ (p. 432), as you guess. It does not mention equi-dimensionality, but an algebraic-geometer colleague says that the unusual useage probably suggests that there is some $d$ such that every component of every fibre has dimension $d$. May 7, 2019 at 21:25
• I think that $P_i$ should be the block upper-triangular matrices of the form $\begin{pmatrix} A & X \\ 0 & B \end{pmatrix}$, where $A$ is $i \times i$ and $B$ is $(n - i) \times (n - i)$. May 7, 2019 at 21:42

(3) My guess would be that $$W(Q_i)$$ is the Weyl group of the Levi part of $$Q_i$$.
• $\DeclareMathOperator\Int{Int}$Note that there is no difference between the Weyl group of a parabolic $Q$, in the sense proposed by @user102248, and the Weyl group of a Levi component. Indeed if $Q = M U$ is a Levi decomposition, and if $m \in M$ and $u \in U$ are such that $m u$ normalises $T$, then the image of $\Int(m u)T = T$ in $Q \to Q/U$ is $\Int(m)T = T$, so $m \in N_M(T)$; so then $u \in N_U(T)$, but, if $u \ne 1$, then there is some $t \in T$ that doesn't centralise $u$, so $\Int(u)t^{-1} = t^{-1}\cdot[t, u] \in T$ has non-trivial component $[t, u] \in U$, which is a contradiction. May 7, 2019 at 22:22