[Paper Review] New family of simple $\mathfrak{gl}_{2n}(\mathbb{C})$-modules

Jonathan Nilsson|arXiv (Cornell University)|Sep 7, 2015

Algebraic structures and combinatorial models26 references3 citations

TL;DR

This paper constructs a new family of simple $χ$-generalized Whittaker modules for $χτ_{2n}(ℂ)$, parameterized by $n \times n$ invertible complex matrices. The modules are realized as twisted regular $χτ_n$-modules, with explicit actions defined via trace formulas, and are shown to have Gelfand-Kirillov dimension $n^2$, locally finite $χτ_n$-action, and non-isomorphic modules for distinct parameters.

ABSTRACT

We construct a new family of simple $\mathfrak{gl}_{2n}$-modules which depends on $n^2$ generic parameters. Each such module is isomorphic to the regular $U(\mathfrak{gl}_{n})$-module when restricted the $\mathfrak{gl}_{n}$-subalgebra naturally embedded into the top-left corner.

Motivation & Objective

• To construct a new family of simple $χτ_{2n}(ℂ)$-modules with large parameter space.
• To realize these modules as generalized Whittaker modules for the Whittaker pair $(χτ_n, χτ_{2n})$.
• To provide explicit formulas for the $χτ_{2n}$-action on a basis isomorphic to the regular $χτ_n$-module.
• To prove that different parameters yield non-isomorphic modules, establishing a large family of simple modules.

Proposed method

• Construct an $χτ_n + χτ_n$-module structure on the universal enveloping algebra $χτ(χτ_n)$ using trace functions and matrix parameters.
• Define a Lie algebra automorphism $φ_S$ on $χτ_{2n}$ to twist the module structure, enabling parameterization by invertible matrices.
• Explicitly define the $χτ_{2n}$-action on monomials in $χτ(χτ_n)$ via a formula involving traces of matrix products and the parameter matrix $Q$.
• Use the twisting functor to transfer the module structure from the identity parameter case to arbitrary invertible $Q$, ensuring simplicity and parameter dependence.
• Verify that the restricted action on $χτ_n$ is isomorphic to the regular $χτ(χτ_n)$-module and that the $χτ_n$-action is locally finite.
• Prove non-isomorphism of modules for different $Q$ by showing that any isomorphism would force $Q = Q'$.

Experimental results

Research questions

• RQ1 Can a new family of simple $χτ_{2n}(ℂ)$-modules be constructed that are not isomorphic to known families like Gelfand-Zeitlin or highest weight modules?
• RQ2 How can generalized Whittaker modules be systematically constructed for the Whittaker pair $(χτ_n, χτ_{2n})$ with non-trivial parameter dependence?
• RQ3 What is the precise form of the $χτ_{2n}$-action on a module whose $χτ_n$-restriction is the regular module?
• RQ4 Can such modules be parameterized by invertible $n \times n$ matrices, and are these parameters sufficient to distinguish non-isomorphic modules?
• RQ5 What is the Gelfand-Kirillov dimension of such modules, and how does it relate to the structure of the parameter space?

Key findings

• The constructed $χτ_{2n}$-modules have Gelfand-Kirillov dimension exactly $n^2$, matching the dimension of the parameter space.
• For each invertible $n \times n$ complex matrix $Q$, a simple $χτ_{2n}$-module $V_Q$ exists such that its restriction to $χτ_n$ is isomorphic to the regular $χτ(χτ_n)$-module.
• The restriction of $V_Q$ to $χτ_n$ is locally finite, confirming that $V_Q$ is a generalized Whittaker module for the Whittaker pair $(χτ_n, χτ_{2n})$.
• The action of $χτ_{2n}$ on monomials in $χτ(χτ_n)$ is explicitly given by a trace formula involving $Q$, $Q^{-T}$, and matrix products.
• Modules corresponding to different invertible matrices $Q$ and $Q'$ are non-isomorphic, as any isomorphism would force $Q = Q'$.
• The action of each generator from $χτ_n$, $χτ_n$, $χτ_n$, and $χτ_n$ on $χτ(χτ_n)$ has degree $1$, $0$, $2$, and $1$, respectively, under the PBW basis.