For each connected real semisimple matrix group, one obtains a constructive list of the irreducible tempered unitary representations and their characters. These irreducible representations all turn out to be instances of a more general kind of representation, here called basic. The result completes Langland's classification of all irreducible admissible representations for such groups. Since not all basic representations are irreducible, a study is made of character identities relating different basic representations and of the commuting algebra for each basic representation.
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