By Frédéric V. Bien
The idea of D-modules offers with the algebraic facets of differential equations. those are fairly fascinating on homogeneous manifolds, because the infinitesimal motion of a Lie algebra contains differential operators. consequently, it's attainable to connect geometric invariants, just like the help and the attribute sort, to representations of Lie teams. by means of contemplating D-modules on flag types, one obtains an easy category of all irreducible admissible representations of reductive Lie teams. nonetheless, it truly is ordinary to check the representations discovered by means of features on pseudo-Riemannian symmetric areas, i.e., round representations. the matter is then to explain the round representations between all irreducible ones, and to compute their multiplicities. this can be the objective of this paintings, completed relatively thoroughly not less than for the discrete sequence representations of reductive symmetric areas. The publication presents a basic creation to the speculation of D-modules on flag kinds, and it describes round D-modules by way of a cohomological formulation. utilizing microlocalization of representations, the writer derives a criterion for irreducibility. The relation among multiplicities and singularities can also be mentioned at size.
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E. 5, one replaces K ("1 fT-invariants by the e-isotropic subspace of for the action oi K H. 9 One more example: Let us remaxkthat if . is a standard fT-spherical -module, may not be ff-spherical. 52 II SPHERICAL T>-MODULES This happens for example if diag . H has two orbits and in X and K has nine orbits. Let F be a if-orbit consisting of a line minus a point, then itOy is if-spherical but ii» is not. 8). 6 D-modules and hyperfunctions. The goal of this section is to show that astandard {V, X)-modules on the flag space X can be mapped into the sheaf of hyperfunctions on X supported on a real analytic subvariety which is the largest flag variety of the real form of G.
Bott and Tu p. 88]. The suffice cv denotes cohomology with compact support in the vertical directions. In the derived category we have: and there is a unique isomorphism such that is the identity on If / is a closed imbedding, then S is quasi-isomorphic to the cohomology of S restricted to Y, while is quasi-isomorphic to the local cohomology of S along Y. Let us first take S C . Then This is seen by working in a slice of X transversal to Y and by using the fact that, for a point y X , where bundle is a small ball around y .
Hence contains a closed if-orbit say K-Bin B. By Matsuki-Springer's characterization, B is stable and since there is a if-conjugate of P which contains B. Hence P contains a 6 stable Borel subgroup. Conversely, if P contains a 0-stable Borel subgroup B, then Y = [K • B). Again by Matsuki-Springer, if • B is closed, hence compact. Therefore Y is compact. • Note that if if • P is closed, then if P is parabolic in if and if • P is isomorphic to the flag space of if of type P if. The map h-B^BxB. B*-* {B, B) can easily be related to Springer's parametrization of if-orbits on B .
