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Unitary and Uniformly Bounded Representations Of Some Simple Lie Groups

  • Chapter
Harmonic Analysis and Group Representation

Part of the book series: C.I.M.E. Summer Schools ((CIME,volume 82))

Abstract

We denote by F the real or complex numbers (R or C) or the quaternions (Q). We consider R as a subfield of C, and C as a subfield of Q. If z ∈ F, then we may write

$$ {\text{z}}\,{\text{ = }}\,{\text{s}}\,{\text{ + t}}\underline {\text{i}} + {\text{u}}\underline {\text{j}} + {\text{v}}\underline {\text{k}} , $$

with s, t, u, and v in R. The conjugate z̄ is now described thus:

$$ \overline {\text{z}} \,{\text{ = }}\,{\text{s}}\,{\text{ - t}}\underline {\text{i}} + {\text{u}}\underline {\text{j}} + {\text{v}}\underline {\text{k}} . $$

Note that zz̄ = z̄z = ǀzǀ2 . The real and imaginary parts of z in F are given by the formulae:

$$ 2\operatorname{Re} \left( {\text{z}} \right) = z + \overline z ,\,\,\,\,\,\,2\operatorname{Im} \left( {\text{z}} \right) = {\text{z - }}\overline {\text{z}} . $$

This is not the usual imaginary part in the complex case.

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Authors
  1. Michael Cowling
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A. Figà Talamanca

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Cowling, M. (2010). Unitary and Uniformly Bounded Representations Of Some Simple Lie Groups. In: Talamanca, A.F. (eds) Harmonic Analysis and Group Representation. C.I.M.E. Summer Schools, vol 82. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11117-4_2

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