Let $G$ be a semisimple algebraic group over an algebraically closed
field $\Omega$, with opposite Borel subgroups $B$ and $B^-$ intersecting
in a maximal torus $T$, and let $W=N(T)/T$ be the Weyl group. Besides
the usual Bruhat cell decomposition $G/B=\bigcup_{w\in W}\,Bw·B$,
there is another disjoint union $G/B=\bigcup_{w\in W}B^-
w·B$. Motivated by the study of Schubert varieties (closures of
Bruhat cells), the author studies the various intersections
$By·B\cap B^{-}x·B$, where $x,y\in W$. When it is nonempty,
such an intersection is a disjoint union of sets, each of which looks
like the product of a certain number of copies of the additive group
$\Omega$ and a certain number of copies of the multiplicative group
$\Omega^*$. This union is indexed by the set ${\scr D}$ of
"distinguished subexpressions" (corresponding to $x$) of a fixed reduced
expression for $y$ as a product of simple reflections; necessarily
$x\leq y$ in the Bruhat order of $W$ \ref[cf. the author, same journal
39 (1977), no. 2, 187--198;
MR0435249 (55 \#8209)].
The proof is intricate,
relying on a close study of the unipotent part of $B$. But in case $y$ is a
Coxeter element, the result simplifies: $By·B\cap B^-x·B
\cong(\Omega^*)^{l(y)-l(x)}$, where $l$ is the length function on $W$.
So in this case the closure in $G/B$ gives a toroidal embedding. (Closures
in general are more complicated; the author intends to discuss this
elsewhere.)
The notion of distinguished subexpression makes sense for any
Coxeter group $W$, which allows the author to develop his combinatorial
machinery in this generality. As a very interesting by-\break product, he
produces an explicit closed formula for the auxiliary polynomials
$R_{x,y}(q)$ involved in the study of \n Kazhdan-Lusztig\en polynomials
and first defined recursively in the paper of \n D. A. Kazhdan\en
and \n G. Lusztig\en \ref[ibid. 53 (1979), no. 2, 165--184;
MR0560412 (81j:20066)].
Here $R_{x,y}(q)$ is expressed as the sum over
${\scr D}$ of terms $q^m(q-1)^n$, where the subexpression determines
$m$ and $n$ (in the Weyl group case, these are the numbers of copies
of $\Omega$ and $\Omega^*$ mentioned above). As a further by-product
of the combinatorics of subexpressions in the Coxeter group $W$, the author
recovers the key step in the proof that the Bruhat order is
"$L$-shellable", first observed by \n A. Björner\en and \n M. Wachs\en
\ref[Adv. in Math. 43 (1983), no. 1, 87--100;
MR0644668 (83i:20043)].
Reviewed by James E. Humphreys
