Reference request for the representation of Metaplectic group and related topics

I've studied the spin geometry, Atiyah-Singer index theorem and I realize that the representation theory of spin groups, and spinor representation is very interesting and useful things in geometry.

After reading it, I wonder if this technic could apply in similar way for the symplectic geometry, and I found out that there exists some analogy due to the paper of Baez's http://www.math.columbia.edu/%7Ewoit/notes21.pdf

In addition, From D. Reutter's essay, "The Heat Equation and the Atiyah-Singer Index Theorem", It seems to exist a supersymmetric argument among spin and metaplectic group.

Moreover, I found out that the related topics oftenly studied under the name "Symplectic spinor" (or oscillator representation used in Is there an analogue of spin/oscillator representation for the general linear Lie algebra?). Wikipedia said that the representation of metaplectic is well studied by Segal-Shale-Weil.

Since there are so many things and applications related to the metaplectic group, unfortunately I cannot find any well-organized comprehensive lectures, or references.

In summary, I'm looking for references(book, reference) which describes metaplectic group, Lie algebra, its representation, symplectic spinor, and its applications in geometry(supersymmetry, etc.) in full detail.

• Note that the metaplectic group has the same Lie algebra as the symplectic group, hence the same finite-dimensional representations. The genuine representations of the metaplectic group are all infinite-dimensional. Commented Feb 10, 2021 at 7:49
• @LiorSilberman Thanks for comment. In fact, I'm aware of those facts, but I don't know the details. My goal is to understand them fully. Commented Feb 10, 2021 at 10:35
• I found a book, G. B. Follands, "Harmonic Analysis in Phase Space" Commented Feb 15, 2021 at 9:12