Topics in Representation Theory: The Heisenberg Algebra

We’ll now turn to a topic which is a precise analog of the previous discussion of the Clifford algebra and spinor representations. By replacing the symmetric two-form (the inner product) in the earlier discussion by an antisymmetric two-form, we get a new algebra, the Heisenberg algebra. The group of automorphism of this algebra is now a symplectic group, and we again get a projective representation of this group, called the metaplectic representation. A similar discussion to ours of these topics can be found in [2] Chapter 17, a much more detailed one in [1].