symplectic spinor bundle について

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

\,

is called a symplectic spinor field.
==Formal definition==
Let

(,F_)

be a metaplectic structure on a symplectic manifold

(M, \omega),\,

that is, an equivariant lift of the symplectic frame bundle

\pi_\colon\to M\,

with respect to the double covering

\rho\colon )\to ).\,

The symplectic spinor bundle

\,

is defined 〔 page 37
〕 to be the Hilbert space bundle
:

=\times_L^2(^n)\,

associated to the metaplectic structure

via the metaplectic representation

\colon )\to (L^2(^n)),\,

also called the Segal-Shale-Weil 〔
〕 representation of

).\,

Here, the notation

()\,

denotes the group of unitary operators acting on a Hilbert space

.\,

The Segal-Shale-Weil representation 〔
〕 is an infinite dimensional unitary representation
of the metaplectic group

)

on the space of all complex
valued square Lebesgue integrable functions

L^2(^n).\,

Because of the infinite dimension,
the Segal-Shale-Weil representation is not so easy to handle.

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