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(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.?
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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

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