Solvability of Non-Invariant Differential Operators on Homogeneous Spaces

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We consider the solvability of non-invariant differential operators on homogeneous spaces. Such operators cannot be expected to have solutions in smooth functions (an illustrative example is provided). However, Lion has shown that, under suitable growth conditions on the infinitesimal components of the operators in a representation-theoretic decomposition, one can deduce solvability in a space of distributions. In this paper we prove that Lion's result can be improved to yield solvability in square-integrable functions. AN IMPROVEMENT OF LION'S RESULT Let G be a connected Lie group, H a closed subgroup, v a unitary representation of H and T = IndH Gv. Denote by it(g) the (complexified) universal enveloping algebra of the Lie algebra g of G. The elements of it(g) act on smooth sections in the space A of T by differential operators. The action is as follows. Let JH ,G denote the quotient of the modular functions of H and G. Let 'G ,H denote a relatively invariant measure on G/H. Consider C (T) = {f E C (G, ): f(gh) = 2 (h)v(h)lf(g), hEH,gEG}. The space t?? of C?-vectors of T lies inside C??(T). The action of G on L2 (G, , dl' ,H) by left translation is unitary, differentiates to g and extends canonically to it(g). The resulting action of it(g) on C??(T) gives the desired differential operators. For L E 11(g), we denote the corresponding differential operator on C??(T) by LT or T(L); and we denote the totality of them by iTA. We write itA for those that commute with the action of G, and refer to them as invariant. Typically, one has nice solvability results on the operators in tA.0 For example, if G/H is nilpotent and symmetric, and v = 1, we have global solvability [1]; or if G = G, x G,, H = AG the diagonal (v =1 again), then itA equals 3(go), the center of it(gl), and one has local solvability [3]. The main thrust of the subject (see [2], [7]) is to exploit the solvability of the operators Received by the editors November 17, 1988 and, in revised form, January 30, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 22E99; Secondary 58G99. This research was supported by NSF under MCS 87-00551AO1. ? 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page