Tempered Representation - Examples

Examples

  • The Plancherel theorem for a semisimple Lie group involves representations that are not the discrete series. This becomes clear already in the case of the group SL2(R). The principal series representations of SL2(R) are tempered and account for the spectral decomposition of functions supported on the hyperbolic elements of the group. However, they do not occur discretely in the regular representation of SL2(R).
  • The two limit of discrete series representations of SL2(R) are tempered but not discrete series (even though they occur "discretely" in the list of irreducible unitary representations).
  • For non-semisimple Lie groups, representations with matrix coefficients in L2+ε do not always suffice for the Plancherel theorem, as shown by the example of the additive group R of real numbers and the Fourier integral; in fact, all irreducible unitary representations of R contribute to the Plancherel measure, but none of them have matrix coefficients in L2+ε.
  • The complementary series representations of SL2(R) are irreducible unitary representations that are not tempered.
  • The trivial representation of a group G is an irreducible unitary representation that is not tempered unless G is compact.

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