TITLE: Difference between automorphic forms for SL(2) and GL(2)?
QUESTION [8 upvotes]: 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.

REPLY [4 votes]: 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.