2024/06/11 by Masatoshi Kitagawa, Kitagawa, Masatoshi
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Representation Theory (math.RT) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2406.07279
openalex publication_date 2024/06/11 · openalex created_date 2024/06/13 · openalex updated_date 2026/07/28
In this paper, we treat \mathscrD-modules on the basic affine space G/U and their global sections for a semisimple complex algebraic group G. Our aim is to prepare basic results about large non-irreducible modules for the branching problem and harmonic analysis of reductive Lie groups. A main tool is a formula given by Bezrukavnikov--Braverman--Positselskii. The formula is about a product of functions and their Fourier transforms on G/U like Capelli's identity. Using the formula, we give a generalization of the Beilinson--Bernstein correspondence. We show that the global sections of holonomic \mathscrD-modules are also holonomic using the formula. As a consequence, we give a large algebra action on the \mathfraku-cohomologies Hi(\mathfraku; V) of a \mathfrakg-module V when V is realized as a holonomic \mathscrD-module. We consider affinity of the supports of the \mathfrakt-modules Hi(\mathfraku; V).