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E. e. ) The significance of (6) and (7) ~s that the space of square-integrable The significance ~f will belong to cusp forms for the metaplectic of (5) is that -k/~( ~k - i) group. 1 we use a few Lemmas. 2. Suppose = ~k with y = -~2(~) ~y~S~(y)] in ~ = (g,~) belongs to S-~2(~). ~~ 9 Proof. By "strong approximation" SL2(Q)SL2(~)~ ~. More precisely, for SL(2), g = yg k 0 , with SL2(~) = g~ in SL2(~) 54 determined up to left multiplication by elements of [g,C] = {y,S~(~)} [g~,r r with r : ~&(Y,g~)~&(Yg~,k0)S&(Y)" FI(N ).

42) the Fourier and Haar measure A transform dtY = Itln/2dy. 22. Without loss of generality we assume n t :I. ,n. ika is generated V:F, [35]. 22 by checking are preserved In any case, relations. the resulting m u l t i p l i e r will be denoted rq to the quadratic form representation r and called q. 24. 21). is ordinary q 0 -I r([l 0 ])" and representation "the" Well (This of Example q i b r([0 1 ]) by the operators that these if and only if V is even dimensional. We conclude of operators this Section with some remarks.

As results have form in an odd number of This is because until recently no one seriously attacked the r e p r e s e n t a t i o n automorphic T(G)), the group representations. in the Section I, yet been obtained when variables. irreducible. 4. group. decomposes GL(1) and covering groups of SL2(F). 4. A p h i l o s o p h y for Well's The purpose which of this Subsection (though unproven) Roughly representation. speaking, the idea is that quadratic b e t w e e n automorphic and automorphic forms F representation q.

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