  For any finite Coxeter system $(W,S)$ we construct a certain noncommutative
algebra, so-called {\it bracket algebra}, together with a familiy of commuting
elements, so-called {\it Dunkl elements.} Dunkl elements conjecturally generate
an algebra which is canonically isomorphic to the coinvariant algebra of the
group $W.$ We prove this conjecture for classical Coxeter groups and $I_2(m)$.
We define a ``quantization'' and a multiparameter deformation of our
construction and show that for Lie groups of classical type and $G_2,$ the
algebra generated by Dunkl elements in the quantized bracket algebra is
canonically isomorphic to the small quantum cohomology ring of the
corresponding flag variety, as described by B. Kim. For crystallographic
Coxeter systems we define {\it quantum Bruhat representation} of the
corresponding bracket algebra. We study in more detail relations and structure
of $B_n$-, $D_n$- and $G_2$-bracket algebras, and as an application, discover
{\it Pieri type formula} in the $B_n$-bracket algebra. As a corollary, we
obtain Pieri type formula for multiplication of arbitrary $B_n$-Schubert
classes by some special ones. Our Pieri type formula is a generalization of
Pieri's formulas obtained by A. Lascoux and M.-P. Sch\"utzenberger for flag
varieties of type $A.$ We also introduce a super-version of the bracket algebra
together with a family of pairwise anticommutative elements which describes
``noncommutative differential geometry on a finite Coxeter group'' in a sense
of S. Majid.
