UNITARY REPRESENTATIONS IN L2(U(p,q F)/U(p - m,q ¥)) 15 We denote by CA the one-dimensional metapletic representation of L~ whose differ- ential is given by A G I*. (Then automatically A G t* in the sense of (1.1.1) under the assumption (1.1.2).) We use the same notation CA when it is a character of L or L~. 1.2. good range, fair range Take a Cartan subalgebra rj of g and let W(g, f)) be the Weyl group of the root system A(g, f)). We usually assume J)o is a fundamental Cartan subalgebra. This means f)o Qfyo a nd f)o = tg + ag = (f)o H to) + (f)o ^ Po) such that tg is a Cartan subalgebra of to- A maximal ideal of 3(g) is identified with a W(g, f)) orbit in 1}*: H o m c-algebra ( 3 ( g ) , Q ~ ^ * / ~ W ( g , h ) via the Harish-Chandra isomorphism 3(fl) S^f))^ 8 '^ which involves a shift by p(A+(fl,f))). We follow the terminologies below from [33] Definition 2.5. Suppose that \) is con- tained in [ in the setting of §1.1. Definition 1.2.1. Let W be a metapletic (U{L 0 K)~)-module which has a 3(0" infinitesimal character represented by 7 G f)*. We say that W is in the good range if (1.2.2) Re(a, 7) 0 for each a G A(u, rj). In this case we also say that 7 is in the good range with respect to q C 0. Clearly, this condition is invariant under the action of the Weyl group W((, rj). Suppose that [[, I] acts by zero on W. We say that W is in the fair range if (1.2.2) Re(a, 7|t) 0 for each a G A(u, Fj)
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