Differential Operators and Cohomology Groups on the Basic Affine Space

Details Differential Operators and Cohomology Groups on the Basic Affine Space Differential Operators and Cohomology Groups on the Basic Affine Space

2004-02-25

arxiv.org/abs/math/0402416v2

Mathematics

Representation Theory

To appear in ``Studies in Lie Theory: A. Joseph Fest.'' (Progress in Mathematics, Birkhauser). Minor corrections made to the t

Scientific paper

We study the ring of differential operators D(X) on the basic affine space X=G/U of a complex semisimple group G with maximal unipotent subgroup U. One of the main results shows that the cohomology group H^*(X,O_X) decomposes as a finite direct sum of non-isomorphic simple X-modules, each of which is isomorphic to a twist of O(X) by an automorphism of D(X). We also use D(X) to study the properties of D(Y) for highest weight varieties Y. For example we prove under mild hypotheses that Y is D-simple in the sense that O(Y) is a simple D(Y)-module and produce an irreducible G-module of differential operators on Y of degree -1 and specified order.

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