By Anthony W. Knapp
This e-book bargains a scientific treatment--the first in publication form--of the advance and use of cohomological induction to build unitary representations. George Mackey brought induction in 1950 as a true research building for passing from a unitary illustration of a closed subgroup of a in the neighborhood compact team to a unitary illustration of the entire workforce. Later a parallel building utilizing complicated research and its linked co-homology theories grew up due to paintings by means of Borel, Weil, Harish-Chandra, Bott, Langlands, Kostant, and Schmid. Cohomological induction, brought via Zuckerman, is an algebraic analog that's technically extra plausible than the complex-analysis development and ends up in a wide repertory of irreducible unitary representations of reductive Lie groups.
The booklet, that's available to scholars past the 1st yr of graduate tuition, will curiosity mathematicians and physicists who are looking to find out about and reap the benefits of the algebraic part of the illustration concept of Lie teams. Cohomological Induction and Unitary Representations develops the mandatory history in illustration conception 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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Additional resources for Cohomological Induction and Unitary Representations
D e f i n i t i o n 0 . 5 2 . With (•, •) as above, suppose that the (I, L n K) module Z has an infinitesimal character A,. We say that Z or A isinthe fair range or that Z is fair (relative to q and g) if Re (A. + 5(u), a |3) > 0 for all a e A(u). We say that Z or A is weakly fair if Re(A + <5(u), a | 3) > 0 When Z = for all a e A(u). and k = k f + SL, we have (A. ' + 6 (u),a>. ' + <5(u), a) is > 0 or > 0, respectively. Whether or not Z is one-dimensional, it is not hard to see that if Z is in the good range, then Z is in the fair range.
39) Impose 3 between spaces r(H om r(t/(fl), (Aj u ) * ®V ) LnK), and take the kernel/image as a (g, K) module analog of H 0J ( G/ L, V). 25 4. ZUCKERMAN FUNCTORS Let us bring in homological algebra, temporarily assuming that L c K . 22b). 40) r(H om q(t/( 0) ,U W ) . 41) 0 —►Homq(t/(g), V)LnK —►Hom((£/(g), (a°u)* ® V)LnK —►Homi([/(g), (a ‘u)* ® V W —> • • • in the category C(g, L n K). 41) is an injective resolution of Homq(U(g), V) l kk in the category C(g, L n K). 3) The category C(q, L n K) has enough injectives.
His idea was to adapt A + (and hence the complex structure) to the parameter, making the parameter dominant. Then the degree of interest for cohomology is S = d im c (^ / T) = dimc(u n t) . Under some hypotheses Schmid proved that the natural map from L 2 cohomology in degree S into Dolbeault cohomology is one-one. 11). 31), and suppose that A0 - S0 is analytically integral for the half sum 50 of positive roots in some (or equivalently each) positive system. 33 (Bott-Borel-Weil Theorem, third form).
Cohomological Induction and Unitary Representations by Anthony W. Knapp