By Anthony W. Knapp
This booklet deals a scientific treatment--the first in booklet 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 entire team. Later a parallel building utilizing complicated research and its linked co-homology theories grew up due to paintings through Borel, Weil, Harish-Chandra, Bott, Langlands, Kostant, and Schmid. Cohomological induction, brought by way of Zuckerman, is an algebraic analog that's technically extra workable than the complex-analysis development and results in a wide repertory of irreducible unitary representations of reductive Lie groups.
The booklet, that is obtainable to scholars past the 1st yr of graduate institution, will curiosity mathematicians and physicists who are looking to know about and benefit from the algebraic facet of the illustration conception of Lie teams. Cohomological Induction and Unitary Representations develops the required historical past in illustration concept and comprises an introductory bankruptcy of motivation, a radical therapy of the "translation principle," and 4 appendices on algebra and analysis.
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Extra resources for Cohomological Induction and Unitary Representations
The only consolation is that the Dolbeault cohomology has a parallel problem: For Z finite-dimensional, it follows from the work of Wong that H 0J ( G / L , Z#) can carry an invariant Hermi tian form only when the cohomology is finite dimensional. The forms arising in Schmid’s construction of the discrete series, for example, are defined only on certain dense subspaces of cohomology. So we start over. Suppose again that Z is an ([, L n K) module. The first step is to regard Z# as a (q, L n K) module on which u acts by 0.
Moreover, xx = Xxr if and only if Weyl group. ) X and Xr are in the same orbit of the 33 7. 87 below) says that V has an infinitesimal character. For our situation with L and G, let p be a Cartan subalgebra of 1. Then p is also a Cartan subalgebra of g, and infinitesimal characters for [ and g can both be given as members of p*. We shall assume from now on that our (I, L (1 K) module Z has an infinitesimal character, as well as finite length. Let us pause for some examples. Let A(g, p) be the set of roots of g, and let A(u) and A([, p) denote the subsets of roots whose root vectors lie in u and [, respectively.
Part (a) is most of Hard Duality, and part (b) is an instance of Easy Duality. Part (c) is then a formal consequence. If, as an interim measure as suggested above, (a) is taken as a definition of n and its derived functors, then (b) is substantially the Duality Theorem in its original form as stated by Zuckerman and Enright-Wallach . 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.
Cohomological Induction and Unitary Representations by Anthony W. Knapp