Today I want to talk about an amazing theorem of Moeglin, which I learned from WTG.

Let be a finite extension, the split odd special orthogonal group over , its unique inner form. By work of Arthur and Moeglin, there is a natural bijection between discrete series representations of or , and pairs where is a discrete L-parameter and is a character of the centralizer group . In this setting, is discrete if it is the sum of pairwise-distinct irreducible representations with . Here is the usual d-1st symmetric power representation. The associated centralizer group is of the form , and has a canonical basis indexed by the irreducible summands . Given , let be the associated discrete series representation. Note that is a representation of if is trivial on the evident subgroup , and is a representation of otherwise. This splits the representations up evenly: for fixed, there are possible ‘s, and we get a Vogan L-packet where and each contain elements.

Question. When is a supercuspidal representation?

For I had previously memorized the (already complicated) answer to this question, so you can imagine my pleasure when I learned that Moeglin found a simple conceptual criterion which works in general! To state her theorem, we need a little vocabulary.

Definition. A discrete parameter is **without gaps** if for every occurring among the ‘s with , then also occurs among the ‘s.

Definition. Suppose is without gaps. A character of the component group is **alternating** if for every pair and (with ) occurring among the ‘s, . Moreover we require that on every summand of the form , we have

**Theorem (Moeglin)**. The representation is supercuspidal iff is without gaps and is alternating.

Example 0. By definition, is supercuspidal if for all summands. In this case, is (vacuously) without gaps and every is (vacuously) alernating, so consists entirely of supercuspidal representations. The converse – if consists only of supercuspidals then necessarily is supercuspidal – is also immediate!

Example 1. Let be distinct supercuspidal parameters. Then is a discrete parameter without gaps, with component group of size 8. It is easy to see that four of the possible ‘s are alternating, and two of these are trivial on the center of . Thus, the packets and each contain four elements, two of which are supercuspidal and two of which are non-supercuspidal.

Example 2. Let and be supercuspidal parameters. Then is a discrete parameter without gaps, with component group again of size 8. Now only two of the possible ‘s are alternating, and one of these is trivial on the center of . Thus, the packets and each contain four elements, one of which is supercuspidal and three of which are non-supercuspidal.

Example 3. Let be a nontrivial character. Then is a discrete parameter (vacuously) without gaps, with component group of size 4. Now only one of the possible ‘s is alternating, and it is trivial on the center of . Thus, the packets and each contain two elements, with containing one supercuspidal and containing no supercuspidals.

More generally, if is without gaps and all ‘s are even, then only one is alternating, so the packet contains a single supercuspidal representation (which may be a representation of or – both possibilities occur) swimming in a sea of discrete series representations.

References:

Colette Moeglin, Classification des series discretes pour certains groupes classiques p-adiques, 2006

Bin Xu, On the cuspidal support of discrete series for p-adic quasisplit Sp(N) and SO(N), 2015

I feel very tempted to mention the work of Aubert, Moussaoui etc. (Conjecture 6.10 of their Manuscripta 2018 paper, I think), interpreting this in terms of “cuspidal pairs”.

(On a different note, the supercuspidality condition for G’ could be milder than for G?)

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Yes, that is a deep and beautiful conjecture, but I really like the explicit nature of Moeglin’s result for these particular groups.

I don’t understand your final remark.

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I had heard of Moeglin’s criterion only for quasi-split groups G, and always assumed that it should be easier for a representation of the non-quasi-split group G’ to be supercuspidal, just as the Steinberg parameter is supercuspidal for a division algebra but not for GL. So I am a bit surprised but happy to learn that exactly the same rule as for G applies to G’ as well.

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