# Cohomological Induction and Unitary Representations by Anthony W. Knapp

By Anthony W. Knapp

This ebook deals a scientific treatment--the first in e-book form--of the improvement and use of cohomological induction to build unitary representations. George Mackey brought induction in 1950 as a true research development for passing from a unitary illustration of a closed subgroup of a in the community compact team to a unitary illustration of the total workforce. Later a parallel development utilizing advanced research and its linked co-homology theories grew up due to paintings by way of Borel, Weil, Harish-Chandra, Bott, Langlands, Kostant, and Schmid. Cohomological induction, brought by way of Zuckerman, is an algebraic analog that's technically extra viable than the complex-analysis building and results in a wide repertory of irreducible unitary representations of reductive Lie groups.

The booklet, that's obtainable to scholars past the 1st 12 months of graduate university, will curiosity mathematicians and physicists who are looking to know about and reap the benefits of the algebraic facet of the illustration conception of Lie teams. *Cohomological Induction and Unitary Representations* develops the mandatory history in illustration concept and comprises an introductory bankruptcy of motivation, an intensive therapy of the "translation principle," and 4 appendices on algebra and analysis.

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45. , • )G on £ S(Z). Recall that we have been seeking a complex-analysis construction (or an algebraic analog of one) yielding irreducible unitary represen tations and complementing the real-analysis construction of parabolic induction. We intend for cohomological induction with Cs to be that construction. Before considering how close we are to the desired goal, we mention one more theorem as background. 46. ’^ ^ ( Z #) and pro®’£ ^ ( Z #) have finite length, and they have the same irreducible composition factors and multiplicities (b) all the (g, K) modules £/(Z) and TV(Z) have finite length, and £ (-i y(£;(z» = y,(-DW(z» j j in the Grothendieck group of finite-length (g, K) modules.

At the bottom right the quotient Gc / B is a complex manifold, and G / T therefore acquires an invariant complex structure. 10) may be identified with the holomorphic sections of the holomorphic line bundle over G / T 12 INTRODUCTION associated to the character , and G acts on the space of sections in the natural way. In short, the irreducible representation with highest weight k is realized as the space of global holomorphic sections of a certain holomorphic line bundle. Suppose now that A is analytically integral but no longer dominant.

38) to indicate the L H K finiteness. 38) is a (g, L n K) module. 38) loses information because (a) cp need not be analytic, and hence cp i->

C(fl, K) given by T(V) = sum of all finite-dimensional £ invariant subspaces of V for which the action of 6 globalizes to K , r(V 0 = ^lr(v) if ^ e Hom(V, W).