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\newtheorem{theorem}{Theorem}
\newcommand\Poin{\mathrm{Poin}}

\title{Note on a theorem of Eng}
\author[Victor Reiner]{Victor Reiner}
\address{School of Mathematics\\
         University of Minnesota\\
         Minneapolis, MN 55455}
\email{reiner@math.umn.edu}

\maketitle

In \cite[Theorem 1]{Eng}, O. Eng proved the following theorem.
Let $(W,S)$ be a finite Coxeter system,
$J$ a subset of $S$, $W^J$ the distinguished left coset representatives
for the parabolic subgroup $W_J$ in $W$, and $w_0$ the longest element of
$W$, which is always an involution.

\begin{theorem}
$$
\left[ \sum_{w \in W^J} q^{l(w)} \right]_{q=-1} =
 \#\{w \in W^J: w_0 w W_J = w W_J \}. 
$$
\end{theorem}

\noindent
Eng proved this by appeal to the classification of finite irreducible
Coxeter systems, and asked for a case-free proof.  
We give such a proof via geometry in the case where $(W,S)$
is a {\it Weyl group}. 

Associated to a Weyl group $(W,S)$ is a complex semisimple
algebraic group $G$ with a choice of a Borel subgroup $B$.  The choice
of $J \subseteq S$ gives rise to a parabolic subgroup $P$ containing $B$,
with the following well-known property (see e.g. \cite{Hiller}):
the smooth complex projective variety $G/P$ has a Schubert cell decomposition
indexed by $W^J$ (or the cosets $W/W_J$)
which gives rise to the following expression for
its Poincare polynomial:
$$
\sum_i \beta_{2i}(G/P) \,\, q^i = \sum_{w \in W^J}  q^{l(w)} 
$$
where $\beta_k$ denotes the $k^{th}$ Betti number.
Since $G/P$ is smooth and projective, it is a K\"ahler manifold,
and consequently its {\it signature} or {\it index} $I(G/P)$ has the
following expression (see \cite[p. 125]{GriffithsHarris},
\cite[p. 208]{Wells}) in terms of its Hodge numbers $h^{i,j}(G/P)$:
$$
\sigma(G/P) = \sum_{i,j} (-1)^i h^{i,j}(G/P).
$$
Since the homology of $G/P$ is additively generated by the
fundamental classes of the Schubert subvarieties,
$h^{i,j}(G/P)=0$ for $i \neq j$.  Using this and the equation 
$\beta_k(G/P) = \sum_{i+j=k} h^{i,j}(G/P)$, we conclude that
$$
I(G/P) = \sum_{i,j} (-1)^{i} h^{i,j}(G/P)
       = \sum_{i} (-1)^{i} \beta_{2i}(G/P)
       = \left[ \sum_{w \in W^J} q^{l(w)} \right]_{q=-1}
$$
On the other hand, Connolly and Nagano \cite[p.38]{ConnollyNagano}
calculated this index $I(G/P)$, using the fact that the Schubert class
corresponding to the coset $wW_J$ is dual under the intersection pairing
to the Schubert class corresponding to $w_0wW_J$.  Hence the intersection
form when written with respect to the Schubert cycle basis looks like the
permutation matrix for the involution $w_0$ permuting the cosets
$\{wW_J\}_{w \in W^J}$.  Consequently, its signature $I(G/P)$ is the
number of cosets fixed by this involution, which is the right-hand
side of Eng's Theorem.

\begin{thebibliography}{9}

\bibitem{ConnollyNagano}
Connolly and Nagano,
The intersection pairing of a homogeneous K\"ahler manifold,
Mich. Math. J. {\bf 24} (1977), 33--39.

\bibitem{Eng}
O. Eng,
Quotients of Poincar\'e polynomials evaluated at $-1$.
J. Algebraic Combin. {\bf ??} (2000), ??--??.

\bibitem{GriffithsHarris}
P. Griffiths and J. Harris,
{\it Principles of algebraic geometry}.
Wiley Classics Library. John Wiley \& Sons, Inc., New York, 1994.

\bibitem{Hiller}
H. Hiller, 
{\it Geometry of Coxeter groups}. Research Notes in Mathematics, {\bf 54}.
Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982.

\bibitem{Wells}
R.O. Wells,
Differential analysis on complex manifolds. Second edition.
Graduate Texts in Mathematics, {\bf 65}. Springer-Verlag,
New York-Berlin, 1980. 


\end{thebibliography}

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