\S {\bf 1.{Introduction}}
\label{1}

\indent Let $G$ be a finite group and let $k(G)$ be the      
number of  
its conjugacy classes or simple ${\bf C} [G]$-modules.  The 
origin of this paper lies in a remarkable formula of 
Kn\"orr and Robinson \cite{kr} which involves the numbers    
$k(\ldots)$ for parabolic subgroups of a finite group $G$   
with a split $(B,N)$ pair.  The problem posed and partially 
solved in this paper concerns an analogous formula which    
involves the numbers $k(\ldots)$ for parabolic subgroups of 
a finite Coxeter group $W$.  The work here does not depend  
on \cite{kr} and has, at least for the present, no direct   
contact with it.  Nevertheless it seems appropriate to      
state some of the results of \cite{kr} in this              
Introduction.

The main results of \cite{kr} give a reformulation, in      
homological terms, of a conjecture of Alperin \cite{alp}    
about the representations of any finite group over an       
algebraically closed field $K$ of characteristic $p >    
0$, as well as certain applications to groups where the     
conjecture is known to be true.  Alperin's conjecture       
states that the number of isomorphism classes of simple     
$K[G]$-modules is equal to the number of weights for $G$.   
A weight for $G$ is, by definition, a pair $(Q,S)$ where    
$Q$ is a $p$-subgroup and $S$ is a simple $K[N(Q)]$-module   
which is projective when regarded as a module for           
$K[N(Q)/Q]$; two weights are considered to be the same if   
the subgroups are conjugate and the modules are isomorphic.

Suppose now that $G$ has a split $(B,N)$ pair and that $p$  
is the natural characteristic for $G$.  Cabanes \cite{cab}   
has shown that Alperin's conjecture is true in this case.   
Let $W$ be the Weyl group and let $I$ be a set of Coxeter   
generators for $W$.  If $J \subseteq I$ let $W_{J}$ be the  
corresponding parabolic subgroup of $W$ and let $P_{J} =    
BW_{J}B$ be the corresponding parabolic subgroup of $G$.    
Let $l(P_{J})$ be the number of $p$-regular conjugacy       
classes or simple $K[P_{J}]$-modules.  Kn\"orr and     
Robinson show using their general results, Cabanes' theorem 
and the homology of the Tits building that
\begin{formula}
$$ \sum_{J \subseteq I} (-1)^{|I-J|} l(P_{J}) = f_{0}(G) $$
\label{101}
\end{formula}
and also 
\begin{formula}
$$ \sum_{J \subseteq I} (-1)^{|I-J|} k(P_{J}) = f_{0}(G)  $$
\label{102}
\end{formula}
where $f_{0}(G)$ is the number of $p$-blocks of $G$ of      
defect zero.
