Thanks,

I solved this problem. But without knowing why.

Originally, I did like this:
export ORACLE_SID=ODLILY( the aux database)
and then log into rman.
rman> CONNECT AUXILIARY sys/lily@ODLILY
got the error.

Then, I changed to:
rman> CONNECT AUXILIARY sys/lily
(without the @ODLILY)
It worked.