## When is it supercuspidal?

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

Let $F/\mathbf{Q}_p$ be a finite extension, $G=\mathrm{SO}_{2n+1}$ the split odd special orthogonal group over $F$, $G'$ its unique inner form. By work of Arthur and Moeglin, there is a natural bijection between discrete series representations of $G$ or $G'$, and pairs $(\phi,\chi)$ where $\phi: W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{2n}$ is a discrete L-parameter and $\chi$ is a character of the centralizer group $A_\phi$. In this setting, $\phi$ is discrete if it is the sum of $m$ pairwise-distinct irreducible representations $\phi_i = \sigma_i \boxtimes [d_i] : W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{2n_i}$ with $\sum_{1 \leq i \leq m} n_i = n$. Here $[d]:\mathrm{SL}_2 \to \mathrm{SL}_d$ is the usual d-1st symmetric power representation. The associated centralizer group $A_\phi$ is of the form $\{ \pm 1\}^m$, and has a canonical basis indexed by the irreducible summands $\phi_i$. Given $(\phi,\chi)$, let $\pi(\phi,\chi)$ be the associated discrete series representation. Note that $\pi(\phi,\chi)$ is a representation of $G$ if $\chi$ is trivial on the evident subgroup $\{ \pm 1 \} = Z(\mathrm{Sp}_{2n}) \subset A_\phi$, and is a representation of $G'$ otherwise. This splits the representations up evenly: for $\phi$ fixed, there are $2^m$ possible $\chi$‘s, and we get a Vogan L-packet $\Pi_\phi = \Pi_\phi(G) \cup \Pi_\phi(G')$ where $\Pi_\phi(G)$ and $\Pi_\phi(G')$ each contain $2^{m-1}$ elements.

Question. When is $\pi(\phi,\chi)$ a supercuspidal representation?

For $n=2$ 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 $\phi=\oplus_i \phi_i$ is without gaps if for every $\sigma \boxtimes [d]$ occurring among the $\phi_i$‘s with $d \geq 3$, then also $\sigma \boxtimes [d-2]$ occurs among the $\phi_i$‘s.

Definition. Suppose $\phi$ is without gaps. A character $\chi$ of the component group is alternating if for every pair $\sigma \boxtimes [d]$ and $\sigma \boxtimes [d-2]$ (with $d \geq 3$) occurring among the $\phi_i$‘s, $\chi(\sigma \boxtimes [d]) = - \chi (\sigma \boxtimes [d-2])$. Moreover we require that on every summand of the form $\sigma \boxtimes [2]$, we have $\chi(\sigma \boxtimes [2])=-1.$

Theorem (Moeglin). The representation $\pi(\phi,\chi)$ is supercuspidal iff $\phi$ is without gaps and $\chi$ is alternating.

Example 0. By definition, $\phi$ is supercuspidal if $d_i =1$ for all summands. In this case, $\phi$ is (vacuously) without gaps and every $\chi$ is (vacuously) alernating, so $\Pi_\phi$ consists entirely of supercuspidal representations. The converse – if $\Pi_\phi$ consists only of supercuspidals then necessarily $\phi$ is supercuspidal – is also immediate!

Example 1. Let $\sigma_2, \sigma_2':W_F \to \mathrm{SL}_2$ be distinct supercuspidal parameters. Then $\phi = \sigma_2 \oplus \sigma_2' \oplus \sigma_2' \boxtimes [3]:W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{10}$ is a discrete parameter without gaps, with component group of size 8. It is easy to see that four of the possible $\chi$‘s are alternating, and two of these are trivial on the center of $\mathrm{Sp}_{10}$. Thus, the packets $\Pi_\phi(\mathrm{SO}_{11})$ and $\Pi_\phi(\mathrm{SO}_{11}')$ each contain four elements, two of which are supercuspidal and two of which are non-supercuspidal.

Example 2. Let $\sigma_2:W_F \to \mathrm{SL}_2$ and $\sigma_3:W_F \to \mathrm{O}_3$ be supercuspidal parameters. Then $\phi = \sigma_2 \oplus \sigma_3 \boxtimes [2] \oplus \sigma_3 \boxtimes [4]:W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{20}$ is a discrete parameter without gaps, with component group again of size 8. Now only two of the possible $\chi$‘s are alternating, and one of these is trivial on the center of $\mathrm{Sp}_{20}$. Thus, the packets $\Pi_\phi(\mathrm{SO}_{21})$ and $\Pi_\phi(\mathrm{SO}_{21}')$ each contain four elements, one of which is supercuspidal and three of which are non-supercuspidal.

Example 3. Let $\tau:W_F \to \{ \pm 1 \}$ be a nontrivial character. Then $\phi = 1 \boxtimes [2] \oplus \tau \boxtimes [2]:W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{4}$ is a discrete parameter (vacuously) without gaps, with component group of size 4. Now only one of the possible $\chi$‘s is alternating, and it is trivial on the center of $\mathrm{Sp}_{4}$. Thus, the packets $\Pi_\phi(\mathrm{SO}_{5})$ and $\Pi_\phi(\mathrm{SO}_{5}')$ each contain two elements, with $\Pi_\phi(\mathrm{SO}_{5})$ containing one supercuspidal and $\Pi_\phi(\mathrm{SO}_{5}')$ containing no supercuspidals.

More generally, if $\phi$ is without gaps and all $d_i$‘s are even, then only one $\chi$ is alternating, so the packet $\Pi_\phi$ contains a single supercuspidal representation (which may be a representation of $G$ or $G'$ – 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