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I shall continue lecturing on foundational results in the representation theory of semisimple Lie groups.
My first objective in this set of three lectures will be an important theorem of Harish-Chandra, asserting that every irreducible unitary representation of a semisimple group G is admissible. This means that if K is a maximal compact subgroup of G, then the K-isotypical components of an irreducible unitary representation of G are all finite-dimensional. Harish-Chandra's theorem answered a question of Mautner and made possible the framing of an abstract Plancherel theorem for semisimple groups.
The same infinitesimal techniques that Harish-Chandra used to prove admisibility led him to the famous subquotient theorem, which gives a uniform (but not very explicit) description of all irreducible representations. We shall examine this result too, using the later-developed concept of the Hecke algebra of a semisimple group.
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