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By R. Goebel, E. Walker
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In particular, w of length m in W, we have to consider a gallery x w t c+,C', . . , c, =W(C+) which is contained in 0: (since 0: is convex), and repeat the above argument m-times. 2: A s s u m e that xa/xI = I . ' 1 for some a < j . If x" is a summand of VN then C+ and w ( C + ) are on the same side of the hyperplane {xa = 2 3 ) . To be specific, assume that xa/xI= I . I. Then X:/X:+~ = 1 . I. 11/2 = Xz+ll . [-I/? A more general form of the Bruhat-Tits decomposition implies that Indgt(7])N = @ W(X')W W€W% + where W, is the set of all w such that w(i) < w(i 1).
0 IJ, the Exercise. The proof of the above proposition shows that smooth irreducible cuspidal representations are admissible. Hint: The map i defines a slcewand V . 5). 4: Every irreducible smooth representation V of GL2(F) is admissible. Proof: We have just seen that cuspidal representations are admissible. 3, so it is again admissible. 0 7. Decomposing principal series In this section we shall describe the composition series of I n d $ ( X ) . Some critical calculations will be performed in Section 8.
0 For any smooth G representation V , let V * be the space of all linear functionals on V . Then G acts on V * by (w,7r*(g)v*)= (7r(g-1)w1 u*). v Let be the set of smooth vectors in V * ,and let iidenote the restriction of 7r* t o The representation is called the contragredient dual of V . A priori, it may not be clear that is non-trivial. However, we have the following simple proposition: v. v v G. 5: Let V be a smooth representation of G . I n particular, missible, i f and only i f V is so. v Proof: Let u be any vector in V.