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Hi, Let $A$ denote the adeles of $Q$. I know how to decompose $L^2(SL(2,A)/SL(2,Q))$ into irreducible $SL(2,A)$-representations. What is the difference between this decomposition and the corresponding decomposition for $GL(2)$? Can I deduce the $GL(2)$-case from the $SL(2)$-case? Thanks for answering this basic question. |
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I think GH's link and K.Buzzard summary of Labesse-Langlands explains that there is no easy comparison possible. But I think at the heart of your question is something else. Are you asking about a decomposition into cuspidal, continuous and residual part? If yes, the decomposition is completly analogous (of course for technical convenience you should rather fix a central character - say trivial - in GL(2)). You get 1) a direct sum of cuspidal representation (all single multiplicity) 2) a sum over the one-dimensional representations $\chi \circ \det$ with $\chi$ Hecke character and $\chi^2 =1$ resp. for SL(2) only the trivial rep. 3) a direct integral over parabolic induced representation $\chi_1,\chi_2$ with $\chi_j$ Hecke quasi character and $\chi_1 \chi_2 =1$ The proof in Gelbart-Jacquet "Analytic ..." translates easily to this situation. |
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