Invariant Differential Operators
W. Smoke · Transactions of the American Mathematical Society · 1967
Introduction.Let B be a differentiable manifold and let "V and iV be the sheaves of germs of differentiable sections of a pair V and W of differentiable vector bundles on B. A problem that has had recent attention, notably from Spencer [5], is that of constructing a resolution of the kernel, or solution sheaf, of a differential operator 3>: y -*■ IV.We consider this problem in the case where B is the homogeneous space G/77 of a Lie group G modulo a closed subgroup 77, with V and W homogeneous vector bundles, and under the assumption that 3> is invariant under the natural action of G on the sheaves.Invariant differential operators have been studied by Bott [1] in connection with the Atiyah-Singer index formula.In the special case where Y~=iV is a sheaf of germs of differentiable functions, they generalize the constant-coefficient differential operators of analysis, and have been studied by many authors.See, e.g., Helgason [3], where further references may be found.In the case of an invariant differential operator, one would like a resolution of its solution sheaf which leads to Lie algebra cohomology.In keeping with this aim, and as a matter of general interest, we find it useful to transfer the basic formal apparatus of jet bundles, etc., to the Lie algebra setting.Thus, if F is a homogeneous vector bundle associated to an 77-module V, one would expect the ^ F, defined in terms of the Lie algebras g and 6, to which Vq is associated.The 77-modules Jq($f) (gfi V have an inverse limit J(q*) (g>* V, which is a g-module.Modifying a definition of Hochschild [4], we call an element of J(&*) (8)* F representative if it generates ä finite-dimensional semisimple submodule of J(8*) * F under the action of g.The representative elements form a submodule P(q*) * F which decomposes R(q*) ®*> V ~ 2 Mi ® Hom^Mi, V) i if the field is algebraically closed.Here, 7 indexes a complete family of inequivalent finite-dimensional simple g-modules.With 77 connected, it turns out that the G-invariant differential operators 3i : 'f ->■ "W are in 1-1 correspondence with certain g-invariant homomorphisms D:J(q*) ®« F->/(g*) & W.