BERNSTEIN-GELFAND-GELFAND SEQUENCES (Lie Groups, Geometric Structures and Differential Equations : One Hundred Years after Sophus Lie)

This survey follows the lecture presented at the conference “100 years after Sophus Lie”, RIMS Kyoto, December 12, 1999. The aim is to describe the recent geometric treatment of the distinguisilcd complexes of invariant differential operators between homogeneous vector bundles, known under the name Bernstein-Gelfand-Gelfand resolutions, in the realm of the so called parabolic geometries. The basic reference for this paper is [12], the exposition has been influenced essentially by $[14, 10]$ . The talk presents some results of a long time joint effort with Andreas \v{C}ap and Vladim\’ir Sou\v{c}ek. Further essential influence comes from the recent extensive cooperation with Michael Eastwood, Rod Gover, and Gerd Schmalz. 1. GENERAL BACKGROUND 1.1. Klein’s geometries. We shall deal with invariant operators for certain geometries. First we discuss such operators in the cases where the underlying geometry is that of a homogeneous space $G/P$ for some Lie subgroup $P$ in a Lie group $G$ . This leads to problems studied for several decades in representation theory in terms of Verma module homomorphisms. Later on, we pass to the so called parabolic geometries and the homogeneous cases play then the r\^oles of the flat models. Our considerations apply to both smooth and holomorphic categories and we shall not distinguish these two cases explicitly. (The main difference is the local existence of the holomorphic sections.) On the other hand, we shall deal with complex representations only in order not to have to distinguish between many real forms of the complex groups. In order to enjoy the general features in terms of explicit examples, we shall pay special attention to several flat models: four different geometries on the three-sphere (projective, conformal Riemannian, projective contact, and $CR$-contact), accomplished with the conformal Riemannian four-sphere. In the two projective cases, the sphere $i\dot{s}$ considered as the space of the rays emanating from tlle origin, but with different group actions: $SL(4, \mathbb{R})$ and $Sp(4, \mathbb{R})$ , respectively. The conformal spheres are regarded as projective quadrics in $\mathbb{R}^{n+2},$ $n=3,4$ , and the corresponding symmetry groups are $O(n+1,1)$ . The $CR$-sphere is understood as the non-degenerate real quadric in $\mathbb{C}^{2}$ , and the symmetry group is $SU(2,1)$ . The isotropy groups of distinguished fixed points form the subgroups $P$ in all cases. For each Kleinian geometry $G/P$ , there are the homogeneous vector bundles $\mathcal{E}(G/P)$ corresponding to $P$-modules E. More explicitly, we consider $Garrow G/P$ as the principal $P$-bundles and $\mathcal{E}(G/P)$ is the associated vector bundle $G\cross_{P}$ E. This is a functorial construction and, in particular, the left action of $G$ on the homogeneous space induces the action on the (sheaf of local) sections of $\mathcal{E}(G/P)$ . Moreover, each (local) section $s:G/Parrow \mathcal{E}(G/P)$ is expressed (in its frame form) as a $P$-equivariant function $Garrow E$ and, in this picture, the action of $G$ on sections is given by the left shifts: $g\cdot s=s\circ\ell_{g^{-1}}$ . The invariant differential operators are those operators between sections of homogeneous bundles which intertwine these actions. 数理解析研究所講究録 1150巻 2000年 14-23 14