
Journal of Lie Theory 29 (2019), No. 3, 839879 Copyright Heldermann Verlag 2019 An Algebraic Approach to Duflo's Polynomial Conjecture in the Nilpotent Case Yoshinori Tanimura Grad. School of Math. Sciences, The University of Tokyo, Tokyo 1538914, Japan yotabaito@gmail.com We introduce a new algebraic approach to Duflo's polynomial conjecture 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 CorwinGreenleaf's polynomial conjecture. CorwinGreenleaf'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 noncommutative 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, 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 2step nilpotent case and the special nilpotent Lie algebra case. Keywords: Polynomial conjecture, Poisson algebra, Fmethod, harmonic analysis, Lie group, representation theory. MSC: 22E30, 22E25 [ Fulltextpdf (283 KB)] for subscribers only. 