Unitary Representations of the Universal Cover of SL(2, ℝ)

  • Roger Howe
  • Eng Chye Tan
Part of the Universitext book series (UTX)


There are two main themes in this chapter. The first is to classify the unitary dual of SL(2, ℝ). This is done in Section 1. Our approach is algebraic and is similar to Lang [2], except that we cover the more general case of unitary representations of any cover of SL(2, ℝ) (see Sally [1] and Pukansky [1], the main idea follows from Bargmann [1]). The second theme introduces the oscillator representation of \( \mathop{{SL}}\limits^{ \sim } \)(2, ℝ) (the double cover of SL(2, ℝ)) on L2(ℝ), and analyzes the tensor products of the oscillator representation. The oscillator representation provides an important link between non-abelian harmonic analysis and classical analysis, and has numerous applications, which are discussed in the next chapter.


Universal Cover Unitary Representation Maximal Compact Subgroup Irreducible Unitary Representation Oscillator Representation 
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Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • Roger Howe
    • 1
  • Eng Chye Tan
    • 2
  1. 1.Department of MathematicsYale UniversityNew HavenUSA
  2. 2.Department of MathematicsNational University of SingaporeSingapore

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