JournalsprimsVol. 49 , No. 2pp. 361–391

Localization of Cohomological Induction

  • Yoshiki Oshima

    The University of Tokyo, Chiba, Japan
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Abstract

We give a geometric realization of cohomologically induced (g,K)(\mathfrak{g},K)-modules. Let (h,L)(\mathfrak{h}, L) be a subpair of (g,K)(\mathfrak{g},K). The cohomological induction is an algebraic construction of (g,K)(\mathfrak{g},K)-modules from a (h,L)(\mathfrak{h},L)-module VV. For a real semisimple Lie group, the duality theorem by Hecht, Mili{\v{c}}i{\'c}, Schmid, and Wolf relates (g,K)(\mathfrak{g},K)-modules cohomologically induced from a Borel subalgebra with D{\mathcal D}-modules on the flag variety of g\frak{g}. In this article we extend the theorem for more general pairs (g,K)(\mathfrak{g},K) and (h,L)(\mathfrak{h},L). We consider the tensor product of a D{\mathcal D}-module and a certain module associated with VV, and prove that its sheaf cohomology groups are isomorphic to cohomologically induced modules.

Cite this article

Yoshiki Oshima, Localization of Cohomological Induction. Publ. Res. Inst. Math. Sci. 49 (2013), no. 2 pp. 361–391

DOI 10.4171/PRIMS/108