Authorship：Lead author,　Last author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：Journal of Algebra  

In this paper, we deal with the U(g)-action on a g-module on which a larger algebra A acts irreducibly. Under a mild condition, we will show that the support of the Z(g)-action is a union of affine subspaces in the dual of a Cartan subalgebra modulo the Weyl group action. As a consequence, we propose a definition of a Cartan subalgebra for such a g-module. The support of the Z(g)-module is an algebraic counterpart of the support of the measure in the irreducible decomposition of a unitary representation. This consideration is motivated by the theory of the discrete decomposability initiated by T. Kobayashi. Defining a Cartan subalgebra for a g-module is motivated by the study of I. Losev on Poisson G-varieties. These are related each other through the associated variety and the nilpotent orbit associated to a g-module.

DOI： 10.1016/j.jalgebra.2026.04.052

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Authorship：Lead author,　Last author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：Journal of Algebra  

In this paper, we consider D-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. It is also shown that the global sections of holonomic D-modules are also holonomic using the formula. As a consequence, we give a large algebra action on the u-cohomologies Hⁱ(u;V) of a g-module V when V is realized as a holonomic D-module. We consider affinity of the supports of the t-modules Hⁱ(u;V).

DOI： 10.1016/j.jalgebra.2025.06.037

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Authorship：Lead author,　Last author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：Symmetry Integrability and Geometry Methods and Applications Sigma  

In this paper, we study the irreducibility of U(g)^G′-modules on the spaces of intertwining operators in the branching problem of reductive Lie algebras, and construct a family of finite-dimensional irreducible U(g)^G′-modules using the Zuckerman derived functors. We provide criteria for the irreducibility of U(g)^G′-modules in the cases of generalized Verma modules, cohomologically induced modules, and discrete series representations. We treat only discrete decomposable restrictions with certain dominance conditions (quasiabelian and in the good range). To describe the U(g)^G′-modules, we give branching laws of cohomologically induced modules using ones of generalized Verma modules when K^′ acts on K/L<inf>K</inf> transitively.

DOI： 10.3842/SIGMA.2025.095

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