[Paper Review] On the Weil representation of infinite-dimensional symplectic group over a finite field

Yury A. Neretin|arXiv (Cornell University)|Mar 21, 2017

Advanced Algebra and Geometry6 references3 citations

TL;DR

This paper extends the Weil representation of the infinite-dimensional symplectic group over a finite field to a representation on a category of linear relations, establishing that the composition of perfect Lagrangian linear relations yields another perfect Lagrangian relation. The key contribution is a unitary representation of the symplectic group on $ε^2(V^{d}_{∞})$ that intertwines with the Heisenberg group action, and the construction is shown to be closed under composition via duality and orthogonality arguments in locally compact abelian groups.

ABSTRACT

We extend the Weil representation of infinite-dimensional symplectic group to a representation a certain category of linear relations.

Motivation & Objective

• To extend the Weil representation of the infinite-dimensional symplectic group $\mathrm{Sp}(\mathbb{V}_{2\infty})$ to a representation on a category of linear relations.
• To characterize the structure of perfect Lagrangian linear relations in the context of locally compact abelian groups.
• To prove that the composition of perfect Lagrangian linear relations is again perfect Lagrangian, ensuring closure under composition.
• To establish a unitary representation $W(g)$ on $\ell^2(V^{d}_{\infty})$ satisfying the intertwining relation $a(vg) = W(g)^{-1}a(v)W(g)$.

Proposed method

• Define the infinite-dimensional symplectic space $\mathbb{V}_{2\mu} = V^{d}_{\mu} \oplus V^{c}_{\mu}$ with a skew-symmetric form $\{\cdot,\cdot\}$, equipped with Pontryagin duality.
• Construct a projective unitary representation of the Heisenberg group via operators $a(v)$ on $\ell^2(V^{d}_{\mu})$, satisfying $a(v)a(w) = \mathsf{Exp}(\frac{1}{2}\{v,w\})a(v+w)$.
• Define the Weil representation $W(g)$ as the unique unitary operator up to scalar satisfying $a(vg) = W(g)^{-1}a(v)W(g)$ for $g \in \mathrm{Sp}(\mathbb{V}_{2\infty})$.
• Introduce the category of linear relations $T: \mathbb{V}_{2\mu} \rightrightarrows \mathbb{V}_{2\nu}$, with kernels, domains, images, and indefiniteness defined via projections.
• Define perfect Lagrangian linear relations as maximal isotropic subspaces $T \subset \mathbb{V}_{2\mu} \oplus \mathbb{V}_{2\nu}$ satisfying compactness and codiscreteness conditions on kernel, image, and their orthogonals.
• Use duality and orthogonality in quotient spaces to prove closure under composition: if $T$ and $S$ are perfect Lagrangian, then $ST$ is also perfect Lagrangian.

Experimental results

Research questions

• RQ1 Can the Weil representation of the infinite-dimensional symplectic group over a finite field be extended to a representation on a category of linear relations?
• RQ2 What conditions define a perfect Lagrangian linear relation in the context of locally compact abelian groups?
• RQ3 Is the composition of two perfect Lagrangian linear relations again a perfect Lagrangian linear relation?
• RQ4 How does the Weil representation $W(g)$ interact with the Heisenberg group action via the intertwining relation $a(vg) = W(g)^{-1}a(v)W(g)$?

Key findings

• The Weil representation $W(g)$ exists as a unitary operator on $\ell^2(V^{d}_{\infty})$ satisfying the intertwining condition $a(vg) = W(g)^{-1}a(v)W(g)$ for all $g \in \mathrm{Sp}(\mathbb{V}_{2\infty})$.
• The representation $W(g)$ is projective with values in $\mathbb{C}^\times$ of modulus 1, though it is not known whether it is linear or projective in the infinite-dimensional case.
• Perfect Lagrangian linear relations are closed, maximal isotropic subspaces with compact kernel and indefiniteness, and codiscrete domain and image.
• The composition $ST$ of two perfect Lagrangian linear relations $T: \mathbb{V}_{2\mu} \rightrightarrows \mathbb{V}_{2\nu}$, $S: \mathbb{V}_{2\nu} \rightrightarrows \mathbb{V}_{2\varkappa}$ is again perfect Lagrangian, as shown via orthogonality in quotient spaces.
• The pseudo-inverse $T^\square$ of a perfect Lagrangian relation $T$ satisfies $T^\square = (T^\lozenge)^\lozenge$, and the relation is closed under taking adjoints.
• The adjoint of the Weil operator $W(P)$ satisfies $W(P)^* a(-w) = a(-v) W(P)^*$, which identifies $W(P)^*$ with $W(P^\square)$, confirming compatibility with duality.