Nature of the GL_n-invariant subalgebra in U(so(m+2n))

Characterize the structure of the GL_n-invariant subalgebra U(so(m+2n))^{GL_n} that acts on the Schur modules arising from the Jacobi–Trudi structure on quadric hypersurface rings. Determine its algebraic and representation-theoretic properties, including generators, relations, and how these Schur modules decompose or behave as modules for this invariant subalgebra.

Background

The authors construct Schur modules for quadric hypersurface rings as GL_n-isotypic components of quotients of universal enveloping algebras tied to parabolic subalgebras in so(m+2n). While these Schur modules are typically reducible as O(V)-representations, invoking a form of Howe duality implies they are irreducible as modules for the GL_n-invariant subalgebra of U(so(m+2n)).

Despite this observation, the authors do not yet understand the structural nature of the invariant subalgebra U(so(m+2n))^{GL_n}. Clarifying its structure is essential for deeper understanding of the representation theory governing the Jacobi–Trudi constructions and their connections to Lie-theoretic dualities.