Symplectic spinor bundle

In differential geometry, given a metaplectic structure formula_1 on a formula_2-dimensional symplectic manifold formula_3 the symplectic spinor bundle is the Hilbert space bundle formula_4 associated to the metaplectic structure via the metaplectic representation. The metaplectic representation of the metaplectic group — the two-fold covering of the symplectic group — gives rise to an infinite rank vector bundle; this is the symplectic spinor construction due to Bertram Kostant.
A section of the symplectic spinor bundle formula_5 is called a symplectic spinor field.
Formal definition.
Let formula_6 be a metaplectic structure on a symplectic manifold formula_3 that is, an equivariant lift of the symplectic frame bundle formula_8 with respect to the double covering formula_9 
The symplectic spinor bundle formula_5 is defined to be the Hilbert space bundle
associated to the metaplectic structure formula_12 via the metaplectic representation formula_13 also called the Segal–Shale–Weil representation of formula_14 Here, the notation formula_15 denotes the group of unitary operators acting on a Hilbert space formula_16
The Segal–Shale–Weil representation is an infinite dimensional unitary representation
of the metaplectic group formula_17 on the space of all complex
valued square Lebesgue integrable square-integrable functions formula_18 Because of the infinite dimension,
the Segal–Shale–Weil representation is not so easy to handle.