In the paper: Pattern Avoidance and Rational Smoothness of
Schubert Varieties, Sara C. Billey, Advances in Mathematics 139, 141-156(1998), <https://www.sciencedirect.com/science/article/pii/S0001870898917443/pdf?md5=43f7ddfffa6e4e285eec1f7183f736c8&pid=1-s2.0-S0001870898917443-main.pdf>,  

 in Theorem 1.1 about rational-smoothness, the Author takes $W$ to be Weyl subgroup of any Semisimple Lie group, not necessarily simply-connected, and in fact in the next page, they do consider Schubert varieties of Type B as subvarieties of flag varieties $SO\_{2n+1}(\mathbb C)/ B(\mathbb C)$, where $SO\_{2n+1}$ are known to be **not simply connected**. 


However, chasing the reference the author provides for Theorem 1.1, we get to the paper: The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties, J.B. Carrell, Proc. Symposia Pure Math. 56 (Part 1) (1994), 53-61, <https://books.google.com/books?id=YZ0ECAAAQBAJ&lpg=PA53&ots=m3XRziUL2r&lr&pg=PA58#v=onepage&q&f=false>, the author of that article does start with a **simply connected, simple algebraic group** (see Section 3, page 57 and 58). 


Can someone please explain to me then how does this apparent ambiguity be taken care of then ? How does the situation of the two papers reconcile ? 


Further questions, as originated from the discussion in comments with Sam Hopkins below: If $W$ is a Weyl group of classical Type B or D, and we consider Schubert varieties corresponding to elements $w\in W$, does the question of rational-smoothness depend on whether our ambient group is $SO(n)$ or Spin group ? Does the question of smoothness depend on the ambient group ?


(I am not really bothered about Type A or C because they are pretty much always taken to be $Sl(n)$ and $Sp\_{2n}$ respectively, both of which are simply connected) 


Thanks 



On a criterion for rational-smoothness of Schubert varieties and an ambiguity of the taking the ambient Algebraic group to be simply connected or not
Expanded with lots of references at @JimHumphreys's [request](https://mathoverflow.net/questions/330999/on-a-criterion-for-rational-smoothness-of-schubert-varieties-and-an-ambiguity-of/331004#comment825502_331004). Borel is [Borel - Linear algebraic groups (2nd edition)](https://link.springer.com/book/10.1007/978-1-4612-0941-6). Springer is [Springer - Linear algebraic groups (2nd edition)](https://link.springer.com/book/10.1007/978-0-8176-4840-4).


Let $G$ be reductive. A maximal torus $T$ of $G$ is connected soluble, hence contained in a maximal such subgroup $B$, i.e., a Borel subgroup; but, since all Borel subgroups are conjugate (Springer Theorem 6.2.7(iii)), *every* Borel subgroup contains a maximal torus of $G$. Since every maximal torus of $G$ contains the centre $Z(G)$ of $G$ (Springer Proposition 8.1.8(i)), so too does every Borel subgroup of $G$. Thus the flag variety $G/B$ of $G$ equals $G\_{\text{ad}}/B\_{\text{ad}}$, where $G\_{\text{ad}} = G/Z(G)$ is the adjoint quotient of $G$ and $B\_{\text{ad}}$ is the image of $B$ in the adjoint quotient, which is also a Borel subgroup of $G\_{\text{ad}}$ (Springer Corollary 6.2.8); that is, the flag variety of $G$ equals that of $G\_{\text{ad}}$. Similarly, the Weyl group $\operatorname W(G, T) = \operatorname N\_G(T)/T$ equals $\operatorname N\_{G\_{\text{ad}}}(T\_{\text{ad}})/T\_{\text{ad}}$, where $T\_{\text{ad}}$ is the image of $T$ in $G\_{\text{ad}}$, which is a maximal torus in $G\_{\text{ad}}$ (Borel Proposition 11.14(1)); that is, the Weyl group of $G$ equals that of $G\_{\text{ad}}$. The same argument shows that, for any $w \in \operatorname W(G, T) = \operatorname W(G\_{\text{ad}}, T\_{\text{ad}})$, the image of $B w B$ in $G/B = G\_{\text{ad}}/B\_{\text{ad}}$ (a Schubert cell) is the same as that of $B\_{\text{ad}}w B\_{\text{ad}}$, so that their closures (Schubert varieties) are also equal.


Since $G$ and its derived group have the same adjoint quotient (Borel Proposition 14.2), this shows that all of the above objects are the same for $G$ as for its derived group. Since the simply connected cover of the derived group of $G$ is a central extension of the derived group (Springer Exercise 10.1.4(1)), hence also has the same adjoint quotient, all of the above objects are the same for the simply connected cover as for $G$.


