Let G be a simple algebraic group over the complex numbers containing a Borel subgroup B. Given a B-stable ideal I in the nilradical of the Lie algebra of B, we define natural numbers m 1 , m 2 , . . . , m k which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types A n , B n , C n and some other types. When I = 0, we recover the usual exponents of G by Kostant (1959) , and one of our conjectures reduces to a well-known

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... ation of the Poincaré polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.