By Steinke G. F.

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5. 2. Theorem, There exist polynomials b,, and D,,A with values in C and U(gIK, respectively, with b,, A # 0 and such that forf E I," and R e b , a ) > c , for a E @ ( P , A) . 5. We now begin the proof. 3. 2 (1) are continuous in f in the indicated range of the u parameter. Thus, if we prove (1) for f right K-finite then the result will follow by continuity. Let Z, be the space of all right K-finite elements of I:. , IF,,,^) be the (g, K)-module I, with the action corresponding to T ~ , , ,(resp.

5) the Lebesgue dominated convergence theorem gives the justification for the interchange. 2. Corollary. Assume that in addition to the preceding condition (a,H,) is irreducible. Zf Re(v, a) 2 ( p , a) for a E W', A ) and i f f E F,, k such that JpIp(v) f # 0, then f is a cyclic uector for rP, ,, , . Let V denote the closure of span ( ~ ~ , , , ~ ( g ) fEl gGI. " under the conjugate dual representation T* of T ~ , , , ~Now . T* = =p,u*. 4). Hence, if W # 0 then W n IF,,,, = W" # 0. Let g E W". (amk)f,g) = ( u ( m ) J p , p ( v ) f ( k )g,( 1 ) ) .

Element of Let S,,, E HomK$V,,W,) be such that Qj,,Sj,, = aj,j6r,aPj,,. Sj,, exists and is uniquely determined. If T E HomK$V,,V ) ,then T= C TPj,p = C (TQj,p)Sj,p i. 2. We now assume that V is a K,-module, and that it is a direct sum of irreducible continuous K,-modules with finite multiplicities. 1 ( * ) in this more general situation since the sum in ( * ) is finite. As before, we have: Lemma. 6 is a linear bijection. Indeed, if T is fixed then there exists a finite dimensional K,-submodule V f of V such that HomK2(V,,V’>= HomK2(V,,V ) and such that Hom,$W,, V ) 0 HomKl(V,, W,) = HomK$Wp, V’) €3 HomKl(VT,W,) (take the sum of the K,-isotypic components of V that correspond to K,-types that occur in V,).

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