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.



Is there a definition of supercupidal parameter in the Local Langland correspondence?
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!


