By David A. Vogan

This publication is an improved model of the Hermann Weyl Lectures given on the Institute for complex research in January 1986. It outlines a few of what's referred to now approximately irreducible unitary representations of actual reductive teams, delivering particularly whole definitions and references, and sketches (at least) of such a lot proofs.

The first 1/2 the publication is dedicated to the 3 kind of understood buildings of such representations: parabolic induction, complementary sequence, and cohomological parabolic induction. This culminates within the description of all irreducible unitary illustration of the overall linear teams. For different teams, one expects to wish a brand new building, giving "unipotent representations." The latter 1/2 the booklet explains the proof for that expectation and indicates a partial definition of unipotent representations.

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**Sample text**

This suggests that all the essential obstructions to passing from the Lie algebra to the Lie group should involve only 50 HARISH-OIANDRA MODULES K. 51 Harish-cbandra's results make this precise for represen- tation theory. 0 , positive definite on s 0 and negative defi- and making these two subspaces orthogonal. , We will use the same notation for various restrictions and complexifications of the form. 2) It turns out that involution. 8 from G to G, by = k•exp(-X). 3. Lie group. of those 8 Suppose g0 .

23) The inner product < , =X - > on iY (X,Y in f). ) Since u(n-) is n, an easy formal argument now shows that the orthogonal complement of T(n-)V is V+. The assertion of the lemma follows. 24 (Borel-Weil). group, T Suppose K is a compact Lie is a Cartan subgroup, and algebra normalized by T (Definition irreducible representation act trivially. 1~). 21). Let V = I'(KIT ,W), COMPACT GROUPS AND BOREL-WEIL THEOREM K act on V by left translation of sections; the resulting representation is denoted and only if that case, (T,W) (v,V) attached to v.

L representations of H. Then passage to differentials Aq; with defines an identification of H. Y H u ts exhibited as a real manifold with H as a cornplexiftcaHon. Part (a) of this lenuna. is a consequence of spectral theory in Hilbert spaces (cf. [Weil, 1940]). The rest follows inunediately from (for example) the identification of H with given by the exponential map. ~ 0 /L We turn now to the specific structure theory for compact Lie groups that we will need. 3. l torus in K of K is a compact Lie group.