Title:

Height basis of Weyl arrangements

Abstract:

Let $A$ be the Weyl arrangement associated with an ireducible root system.   
$\Phi$.  In this talk we extenstively discuss the construction of
derivation bases of various "deformations" of $A$ including the extended Shi and
Catalan arrangements.
In particular,  we pose a conjecture, called the height conjecture,
which claims that any "height-increasing order" of the hyperplanes of
$A$ gives an inductive-free path from scratch to the entire $A$.
The celebrated classical theorem concerning the numerical relation
between the exponents and the height distribution of roots, first
observed by A. Shapiro and R. Steinberg and later proved by B. Kostant
and I. G. Macdonald without using the classification, is a consequence
of the height conjecture.
We explicitly prove the conjecture in the case of the root system of type $A$.
(The work is with Takuro Abe and Daisuke Suyama.)