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By the recent works of Mok, and Kaletha, Shin, White, James, I know that there is a notion of tempered $L$-parameter, square integrable $L$-parameter and generic $L$-parameter of unitary groups.

However, it seems that there is no notion of supercuspidal $L$-parameter corresponding the packet of supercuspidal represenations of unitary group.

Is there a notion of supercuspidal $L$-parameter?

If it exists, does it correspond to some packet of supercuspidal representation?

If you would know the reference of it, please let me know the reference.

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Not really. This is because you get two kinds of L-packets which see supercuspidal representations: there are packets consisting purely of supercuspidals (which correspond to what should probably be called "regular" discrete parameters, at least when your group has connected centre), but when your group is non-split there can also exist packets which contain both supercuspidals and non-cuspidal discrete series representations.

Of course, in GL(n) a representation is supercuspidal if and only if it corresponds to a parameter whose underlying Weil group representation is irreducible. The notion of regularity (where it had been defined; as far as I know this has only been done under tameness of ramification assumptions, see DeBacket--Reeder and Kaletha's papers) is essentially trying to capture the analogue of irreducibility ("large" image in a rather specific sense), but this doesn't work anywhere near as nicely as you might initially hope for the above reasons!

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  • $\begingroup$ I really thank you for your answer. It helped me very much. May I ask one more question? I am wondering whether the tame regular discrete parameters they [DeBAcker-Reeder] defined is tempered. I know it is generic but I am not certain it is tempered. If it is not tempered in general, can we know that there exists at least one $L$-paramater which is not only tempered but also tame regular discrete. Do you have any hint of it? $\endgroup$
    – Monty
    Commented Jun 28, 2017 at 5:48
  • $\begingroup$ Off the top of my head, no. But this is the same as asking whether the corresponding L-packet contains a tempered supercuspidal, right? Is it not the case that a supercuspidal (more generally: a discrete series) is tempered if and only if it is unitary, which is true if and only if it has unitary central character? I don't really think about tempered-ness very often, but I think that's true? $\endgroup$
    – PL.
    Commented Jun 28, 2017 at 6:35
  • $\begingroup$ I see. I learned much from your answer. I think we can produce a tempered tame regular discrete parameters from a tame regular discrete parameters by twisting it with inverse of the central character of given parameter. Your answer helped me a lot. Thank you again very much! $\endgroup$
    – Monty
    Commented Jun 28, 2017 at 10:45
  • $\begingroup$ You're welcome! Yes, that should be the correct interpretation on the Galois side of my comment about central characters. $\endgroup$
    – PL.
    Commented Jun 30, 2017 at 16:36

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