An Algebraic Approach to Duflo's Polynomial Conjecture in the Nilpotent Case

We introduce a new algebraic approach to Duflo's polynomial conjecture (Problem 1.1) in the nilpotent case.Duflo's polynomial conjecture is an algebraic abstraction of the problem about the center of the algebra of all invariant differential operators on a homogeneous linear bundle.In previous research on Duflo's polynomial conjecture in the nilpotent case, one already used an analytic approach to Corwin-Greenleaf's polynomial conjecture.Corwin-Greenleaf's polynomial conjecture is a restriction of Duflo's polynomial conjecture to the case where all differential operators are commutative each other.Especially, in the nilpotent case, there were no approach to Duflo's polynomial conjecture in the case where there exist two non-commutative invariant differential operators, in the knowledge of the author.In this paper, we introduce a new approach to Duflo's polynomial conjecture in the case where invariant differential operators are not necessarily commutative.This approach is based on the split symmetrization map (Definition 1.3), which is a rustic and algebraic map introduced in this paper.Furthermore, by our new approach, we solve Duflo's polynomial conjecture completely in the 2-step nilpotent case and the special nilpotent Lie algebra case (Theorem 1.7).