[Paper Review] Projective Oscillator Representations of sl(n+1) and sp(2m+2)

Xiaoping Xu|arXiv (Cornell University)|Mar 28, 2014

Coding theory and cryptography3 references3 citations

TL;DR

This paper constructs new multi-parameter families of infinite-dimensional irreducible representations for $\mathfrak{sl}(n+1)$ and $\mathfrak{sp}(2m+2)$ by generalizing projective oscillator representations through partial swapping of differential and multiplication operators. The key contribution is the explicit realization of irreducible weight modules on polynomial and exponential-polynomial algebras, extending Howe’s oscillator construction and providing new realizations for cuspidal modules.

ABSTRACT

The n-dimensional projective group gives rise to a one-parameter family of inhomogeneous first-order differential operator representations of sl(n+1). By partially swapping differential operators and multiplication operators, we obtain more general differential operator representations of sl(n+1). Letting these differential operators act on the corresponding polynomial algebra and the space of exponential-polynomial functions, we construct new multi-parameter families of explicit infinite-dimensional irreducible representations for s(n+1) and sp(2m+2) when n=2m+1. Our results can be viewed as extensions of Howe's oscillator construction of infinite-dimensional multiplicity-free irreducible representations for sl(n).

Motivation & Objective

• To construct explicit infinite-dimensional irreducible representations of $\mathfrak{sl}(n+1)$ and $\mathfrak{sp}(2m+2)$ when $n = 2m+1$.
• To extend Howe’s oscillator construction for $\mathfrak{sl}(n)$ to higher-rank Lie algebras using differential operator representations.
• To provide natural realizations of cuspidal modules as irreducible weight modules with finite-dimensional weight subspaces.
• To generalize the one-parameter projective oscillator representation of $\mathfrak{sl}(n+1)$ by partially swapping differential and multiplication operators.
• To establish irreducibility of representations on polynomial and exponential-polynomial function spaces via induction and commutator analysis.

Proposed method

• Generalize the projective representation $\pi_c$ of $\mathfrak{sl}(n+1)$ to $\pi_{c,S}$ by swapping $x_r \leftrightarrow \partial_{x_r}$ for $r \in S$, yielding projective oscillator representations.
• Define representations $\pi_{c,S}^{\vec{a}}$ on the space $\mathscr{A}_{\vec{a}} = \{ f e^{\vec{a} \cdot \vec{x}} \mid f \in \mathbb{F}[x_1,\dots,x_n] \}$, where $\mathscr{A}$ is the polynomial algebra.
• Use induction on the degree of polynomials to prove irreducibility of $\mathscr{A}_{\vec{a}}$ as a $\mathfrak{sp}(2m+2)$-module under $\pi_{c,S}^{\vec{a}}$.
• Apply commutator identities and differential operator relations to show that generators of the algebra are preserved under the action of $\mathfrak{sp}(2m+2)$.
• Leverage Howe’s oscillator theory and theta correspondence to relate the results to multiplicity-free representations and symplectic Lie algebras.
• Verify that the weight subspaces are finite-dimensional using the degree filtration and the action of $D = \sum x_s \partial_{x_s}$.

Experimental results

Research questions

• RQ1 Can one construct new infinite-dimensional irreducible representations of $\mathfrak{sl}(n+1)$ and $\mathfrak{sp}(2m+2)$ using generalized differential operator realizations?
• RQ2 How do partial swaps of differential and multiplication operators affect the structure of oscillator representations of $\mathfrak{sl}(n+1)$?
• RQ3 Under what conditions is the space $\mathscr{A}_{\vec{a}}$ of exponential-polynomial functions irreducible under the action of $\mathfrak{sp}(2m+2)$?
• RQ4 Can these representations be viewed as extensions of Howe’s oscillator construction for $\mathfrak{sl}(n)$?
• RQ5 What is the role of the parameter $c$ and the subset $S \subset \overline{1,n}$ in determining irreducibility and weight space structure?

Key findings

• The representation $\pi_{c,S}^{\vec{a}}$ on $\mathscr{A}_{\vec{a}}$ is irreducible as a $\mathfrak{sp}(2m+2,\mathbb{F})$-module when $n = 2m+1$.
• The space $\mathscr{A}_{\vec{a}}$ is an infinite-dimensional weight module with finite-dimensional weight subspaces under $\pi_{c,S}^{\vec{0}}$.
• Irreducibility is proven via induction on polynomial degree, showing that all monomials are generated from a single nonzero vector.
• The action of $\mathfrak{sp}(2m+2)$ on $\mathscr{A}_{\vec{a}}$ preserves the filtration by total degree and generates all components through differential operators.
• The construction generalizes the one-parameter projective oscillator representation $\pi_c$ to a multi-parameter family $\pi_{c,S}^{\vec{a}}$.
• The results extend Howe’s oscillator construction by providing explicit realizations of cuspidal modules for $\mathfrak{sl}(n+1)$ and $\mathfrak{sp}(2m+2)$.