(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:

- What is an analogue of the spin/oscillator representation for $gl(n)$?
- What is the analogue of the Clifford/Weyl algebra for $gl(n)$? Is it some algebra we build from ${\bf C}^n \oplus ({\bf C}^n)^*$?
- Is there a natural "smallest" faithful representation of $\widetilde{GL(n, {\bf R})}$? Does it coincide with the answer to 1.?

finite dimensionalfaithful representation of the nontrivial double cover of $\mathrm{GL}(n,\mathbb{R})$. $\endgroup$ – Robert Bryant Apr 27 '18 at 19:38