Geometry of coadjoint orbits and noncommutativity of invariant differential operators on nilpotent homogeneous spaces

A monomial representation τ = Ind(H ↑ G, _χ) induced from a character on a connected subgroup H of a nilpotent Lie group G has a primary decomposition whose multiplicities are either purely infinite (m (τ) = ∞) or uniformly bounded (m(τ) < ∞). The multiplicities are completely determined by the geometry of coadjoint orbits in g*, and there are strong indications that orbit geometry also determines the structure of the algebra D_τ of τ-invariant differential operators on smooth sections. One unresolved conjecture says that D_τ is commutative ⇔ m (τ) < ∞; (⇐) is well known, and in this note we report significant progress toward the converse by proving that (⇒) holds when CASE I: m(τ₀) < ∞, m(τ) = ∞, and CASE II: D_τ0 ≠ Dτ, where τ₀ = Ind(H ↑ G₀, χ) and G₀ ⊇ H is a codimension-1 subgroup. When m(τ) = ∞, one can always reduce to Case I; all evidence so far suggests that II is always valid when I holds (which would resolve the conjecture), but no general proof is known. Similar results have been reported recently by H. Fujiwara, G. Lion, and S. Medhi [5] using traditional methods of induction on dimension. Our methods are completely noninductive and rest entirely on analysis of coadjoint orbit geometry. The same methods may prove useful in an ultimate orbital description of D_τ, along the lines of the structure theorems known to hold when m(τ) < ∞.