# Is there an analogue of spin/oscillator representation for the general linear Lie algebra?

(Work over complex numbers)

Let $V$ be an orthogonal space. Let $Pin(V)$ be the double cover of the orthogonal group $O(V)$. Then $Pin(V)$ has a basic spin representation which we can think of as the exterior algebra on a maximal isotropic subspace $V'$ in $V$.

Analogously, let $W$ be a symplectic vector space. Let $sp(W)$ be the symplectic Lie algebra. Then this has a basic "oscillator" representation which we can think of as the symmetric algebra on a maximal isotropic subspace $W'$ in $W$.

Both constructions are completely analogous and proceed by embedding the Pin group in a Clifford algebra for $V$ in the first case and by embedding $sp(W)$ in the Weyl algebra for $W$ in the second case. These algebras are fundamental objects associated to an orthogonal/symplectic form.

My question is if there is an analogue of the above picture for the general linear group?

A possibly related point: over the real numbers, the symplectic group has a nontrivial double cover, the metaplectic group, and the oscillator representation can be integrated to this group. Again over the real numbers, the general linear group $GL(n, {\bf R})$ has a nontrivial double cover $\widetilde{GL(n, {\bf R})}$ (what is its name and where can I find basic information about it?). Specific questions:

1. What is an analogue of the spin/oscillator representation for $gl(n)$?
2. What is the analogue of the Clifford/Weyl algebra for $gl(n)$? Is it some algebra we build from ${\bf C}^n \oplus ({\bf C}^n)^*$?
3. Is there a natural "smallest" faithful representation of $\widetilde{GL(n, {\bf R})}$? Does it coincide with the answer to 1.?
• Steven, what properties of the spin/oscillator representations are you trying to generalize? And what properties of the Clifford/Weyl algebras? – MTS May 17 '13 at 13:36
• It's a bit of a loose question in that I don't have a specific goal in mind. Perhaps question 3 is easier to answer in a precise way. When I asked Joseph Wolf about this, he mentioned Joseph ideals and the "most nonsingular" representation in relation to characterizing the oscillator representation (so these may be clues), but I haven't yet attempted to go through the relevant literature. – Steven Sam May 17 '13 at 15:27
• Let $V$ have a positive-definite inner product. Possibly one source of confusion is that Clifford algebras give two double covers of $O(V)$, depending on whether your Clifford algebra has a positive definite or a negative definite quadratic form, denoted $\mathit{Pin}^+(V)$ and $\mathit{Pin}^-(V)$. In general, these groups are nonisomorphic (see, e.g., here). As the inclusion $O(V)\hookrightarrow\mathit{GL}(V)$ is a homotopy equivalence, there are also two double covers of $\mathit{GL}(V)$, which are generally nonisomorphic. – Arun Debray Apr 20 '18 at 22:01
• I assume that you know the classic result that there is no finite dimensional faithful representation of the nontrivial double cover of $\mathrm{GL}(n,\mathbb{R})$. – Robert Bryant Apr 27 '18 at 19:38

The Clifford / Weyl algebra duality stems from the symmetric / antisymmetric duality. For me the most natural way to generalize this to $GL(V)$ is kind of vacuous, i.e. the full tensor algebra $\bigotimes V$. Perhaps for $SL(V)$ we could take some $n$-ary operation and quotient out some relation cooked up from the volume form...
There is a defined and studied notion of oscillatory representation for $A$-series algebras but the real form is different, namely $\mathfrak{su}(p,q)$. See "On the Segal-Shale-Weil representations and harmonic polynomials" by Kashiwara and Vergne. This generalization considers the oscillatory representation of $Sp$ as a prominent example of a highest weight unitarizable module. In this framework the spin representation disappears, however the kernel of the Dirac operator in Lorentzian signature is an example of such a representation. All these representations are quite special and much is known about them. For example they all can be realized as kernels of systems of invariant differential operators (see "Differential Operators and Highest Weight Representations" by Davidson, Enright and Stanke) and if I remember correctly, they are in some sense attached to minimal nilpotent orbits. I guess the reference for this last fact could be "Annihilators and associated varieties of unitary highest weight modules" by Joseph. Finally, unitarizable highest weight modules are examples of so called minimal representations which also exists for other real forms and perhaps some of these minimal representations could serve as generalization of the oscillatory representation to the $GL$ case.
• These unitary representations are for $SU(p,q)$. I don't know whether they are faithful but since they can be realized as (kernels of differential operators acting on) holomorphic functions on bounded symmetric domains I am inclined to believe that they are. – Vít Tuček Apr 29 '18 at 7:52