2025/07/23 by Nang, Philibert
#15B57 #32C38 #32S25 #32S60 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2507.17970
Consider (G, V) a finite-dimensional representation of a connected reductive complex Lie group G and ℙ( V) the projective space of V. Denote by G' the derived subgroup of G and assume that the categorical quotient is one dimensional. In the case where the representation (G, V) is also multiplicity-free, it is known from Howe-Umeda [4] that the algebra of G-invariant differential operators Γ(V, DV)G is a commutative polynomial ring. Suppose that the representation (G, V) satisfies the abstract Capelli condition: (G, V) is an irreducible multiplicity-free representation such that the Weyl algebra Γ(V, DV)G is equal to the image of the center of the universal enveloping algebra of Lie(G) under the differential τ: Lie(G) \longrightarrow Γ(V, DV) of the G-action. Let A be the quotient algebra of all G'-invariant differential operators by those vanishing on G'-invariant polynomials. The main aim of this paper is to prove that there is an equivalence of categories between the category of regular holonomic Dℙ( V)-modules on the complex projective space ℙ( V) and the quotient category of finitely generated graded A-modules modulo those supported by \ 0\ . This result is a generalization of [12, Theorem 3.4] and of [13, Theorem 8]. As an application we give an algebraic/combinatorial classification of regular holonomic Dℙ( V) -modules on the projective space of skew-symmetric matrices.