16 TOSHIYUKI KOBAYASHI (notation §1.1). W (or 7) is called weakly good (respectively weakly fair) if the weak inequalities ( ) hold. One should notice that good implies fair if [[, [] acts by zero on W. Conversely, there exists in general a strip of the fair range that is not in the good range. Roughly, the size of this strip corresponds to that of a Levi part of q and it is empty when q is a Borel subalgebra. 1.3. cohomological parabolic induction As an algebraic analogue of Dolbeault cohomology on a homogeneous complex man- ifold G/L, Zuckerman introduced the cohomological parabolic induction (we follow [32] Definition 6.20) Ki = {K*Y (jeN), which is a covariant functor from the category of metapletic (I, (L fl if )~)-modules to the category of (g, if)-modules. With notation as before, fix a Cartan subalgebra r) C L The definition here dif- fers from [28] Definition 6.3.1 only by a p-shift. In our normalization, if a metapletic (I, (L fl # )~)-module W has 3 ( 0 - m f i n i t e s i m a l character 7 e f)*, then ft{{W) has 3(g)- infinitesimal character 7 in the Harish-Chandra parametrization. 1.4. results from Zuckerman and Vogan Retain notations as in §1.1-2. The following theorem is due to Zuckerman and Vogan (see [28],[32],[35]). Fact 1.4.1. In the setting of §1.1, suppose that W is a metapletic ([, (LC\K)~)-module. 1) Assume that W is weakly good. a) n3q{W) = 0 for all j ^ S.
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