We know that if $G=SL(2,R)$ and $H=SO(2,R)$ as a compact subgroup of $G$, then $\{\xi \in \hat{G}; \xi|_H=1\}$is triviall. Can we conclude that each multiplicative linear functional on $\{f\in L^1(G); R_\xi f=f(a.e)\forall \xi \in H\}$ is triviall, too?
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$\begingroup$ If memory serves right, $L^1(SL(2,R),SO(2,R))$ is a commutative Banach algebra -- this is a Gelfand pair, isn't it? -- and it is moreover semisimple. I don't recall the proof but I think you can write down explicit characters on this algebra using the spherical functions associated to this Gelfand pair. $\endgroup$– Yemon ChoiCommented Jan 28, 2015 at 14:44
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$\begingroup$ Thank you. But it isn't commutative B.al. $\endgroup$– B.GillanCommented Jan 29, 2015 at 4:51
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$\begingroup$ Oh, my mistake: I was thinking of SO(3)-bi-invariant functions. Your question is asking about one-sided SO(3)-invariant functions, right? $\endgroup$– Yemon ChoiCommented Jan 29, 2015 at 12:13
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$\begingroup$ Dear Yemon Choi, Can you refer to a good reference in this regard. $\endgroup$– B.GillanCommented Jan 31, 2015 at 11:45
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$\begingroup$ For one-sided invariant functions, I guess the book of Reiterand Stegemann may have something; but it may not deal with the particular case that you are after $\endgroup$– Yemon ChoiCommented Jan 31, 2015 at 13:07
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