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In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on , functions on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle. In an invariant differential operator , the term ''differential operator'' indicates that the value of the map depends only on and the derivatives of in . The word ''invariant'' indicates that the operator contains some symmetry. This means that there is a group with a group action on the functions (or other objects in question) and this action is preserved by the operator: : Usually, the action of the group has the meaning of a change of coordinates (change of observer) and the invariance means that the operator has the same expression in all admissible coordinates. ==Invariance on homogeneous spaces== Let ''M'' = ''G''/''H'' be a homogeneous space for a Lie group G and a Lie subgroup H. Every representation gives rise to a vector bundle : Sections can be identified with : In this form the group ''G'' acts on sections via : Now let ''V'' and ''W'' be two vector bundles over ''M''. Then a differential operator : that maps sections of ''V'' to sections of ''W'' is called invariant if : for all sections in and elements ''g'' in ''G''. All linear invariant differential operators on homogeneous parabolic geometries, i.e. when ''G'' is semi-simple and ''H'' is a parabolic subgroup, are given dually by homomorphisms of generalized Verma modules. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「invariant differential operator」の詳細全文を読む スポンサード リンク
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