Semisimple types for $p$-adic classical groups

Abstract

We construct, for any symplectic, unitary or special orthogonal group over a locally compact nonarchimedean local field of odd residual characteristic, a type for each Bernstein component of the category of smooth representations, using Bushnell–Kutzko’s theory of covers. Moreover, for a component corresponding to a cuspidal representation of a maximal Levi subgroup, we prove that the Hecke algebra is either abelian, or a generic Hecke algebra on an infinite dihedral group, with parameters which are, at least in principle, computable via results of Lusztig. In an appendix, we make a correction to the proof of a result of the second author: that every irreducible cuspidal representation of a classical group as considered here is irreducibly compactly-induced from a type.

Appendix A: Correction to the proof of [29, Theorem 7.14]

In this appendix, we give the correction to the proof of the main result of [29] (Theorem 7.14), with the newly corrected definition of cuspidal type from Definition 4.3, which we repeat here, recalling that $\mathrm{G}$ has compact centre:

A cuspidal type for $\mathrm{G}$ is a pair $(\mathrm{J},\lambda )$, where $\mathrm{J}=\mathrm{J}(\beta ,\varLambda )$ for some skew semisimple stratum $[\varLambda ,n,0,\beta ]$ such that

• $\mathrm{G}_\mathrm{E}$ has compact centre and

• $\mathrm{P}^{\mathsf{{o}}}(\varLambda _{{{\mathcal{O }}_\mathrm{E}}})$ a maximal parahoric subgroup of $\mathrm{G}_\mathrm{E}$,

and $\lambda =\kappa \otimes \tau$, for $\kappa$ a $\beta$-extension and $\tau$ the inflation of an irreducible cuspidal representation of $\mathrm{J}/\mathrm{J}^1\simeq \mathrm{P}(\varLambda _{{{\mathcal{O }}_\mathrm{E}}})/\mathrm{P}_1(\varLambda _{{{\mathcal{O }}_\mathrm{E}}})$.

Remark A.1

We thank Laure Blasco, Corinne Blondel and Van-Dinh Ngo for pointing out the problem with the definition in [29, Definition 6.17]. There, the two conditions on the stratum $[\varLambda ,n,0,\beta ]$ in Definition 4.3 are replaced by the (insufficient) condition that $\mathfrak A (\varLambda _{{{\mathcal{O }}_\mathrm{E}}})$ be a maximal self-dual ${{\mathcal{O }}_\mathrm{E}}$-order in $\mathrm{B}$.

Firstly, this is not enough to guarantee that $\mathrm{P}^{\mathsf{{o}}}(\varLambda _{{{\mathcal{O }}_\mathrm{E}}})$ be a maximal parahoric subgroup of $\mathrm{G}_\mathrm{E}$: for example, if $\mathrm{G}_\mathrm{E}$ is a quasi-split ramified unitary group in $2$ variables then, for one of the two (up to conjugacy) maximal self-dual ${{\mathcal{O }}_\mathrm{E}}$-orders, the corresponding parahoric subgroup is an Iwahori subgroup, so not maximal.

Secondly, even if $\mathrm{P}^{\mathsf{{o}}}(\varLambda _{{{\mathcal{O }}_\mathrm{E}}})$ is a maximal parahoric subgroup, it can still happen that its normalizer in $\mathrm{G}_\mathrm{E}$ is not compact: this happens precisely when $\mathrm{G}_\mathrm{E}$ has a factor isomorphic to the split torus $\mathrm{SO}(1,1)(\mathrm{F})$, which can only happen when $\mathrm{G}$ is an even-dimensional orthogonal group and $\beta _i=0$, $\dim _\mathrm{F}\mathrm{V}^i=2$, for some $i\in \mathrm{I}_0$. The condition that $\mathrm{G}_\mathrm{E}$ have compact centre rules out exactly this possibility.

In particular, with the definition of cuspidal type $(\mathrm{J},\lambda )$ given here, the proof of [29, Proposition 6.18] is valid, and c-Ind$_\mathrm{J}^\mathrm{G}\lambda$ is an irreducible cuspidal representation of $\mathrm{G}$.

1.1 A.1

In this paragraph we indicate the minor changes that must be made to [29, §7.2] in order to correct the proof of the main result there [29, Theorem 7.14]: every irreducible cuspidal representation of $\mathrm{G}$ contains a cuspidal type. This paragraph should be read alongside that paper and we will make free use of notations from there.

Suppose $\pi$ is an irreducible representation of $\mathrm{G}$ and suppose that there is a pair $([\varLambda ,n,0,\beta ],\theta )$, consisting of a skew semisimple stratum $[\varLambda ,n,0,\beta ]$ and a semisimple character $\theta \in \mathcal C _-(\varLambda ,0,\beta )$, such that $\pi$ contains $\theta$. Suppose moreover that, for fixed $\beta$, we have chosen a pair for which the parahoric subgroup $\mathrm{P}^{\mathsf{{o}}}(\varLambda _{{\mathcal{O }}_\mathrm{E}})$ is minimal amongst such pairs. If $\kappa$ is a standard $\beta$-extension then $\pi$ also contains a representation $\vartheta =\kappa \otimes \rho$ of $\mathrm{J}^{\mathsf{{o}}}$, for $\rho$ an irreducible representation of $\mathrm{J}^{\mathsf{{o}}}/\mathrm{J}^1\simeq \mathrm{P}^{\mathsf{{o}}}(\varLambda _{{\mathcal{O }}_\mathrm{E}})/\mathrm{P}_1(\varLambda _{{\mathcal{O }}_\mathrm{E}})$. By [29, Lemma 7.4], the minimality of $\mathrm{P}^{\mathsf{{o}}}(\varLambda _{{\mathcal{O }}_\mathrm{E}})$ implies that the representation $\rho$ is cuspidal.

We suppose that either the parahoric subgroup $\mathrm{P}^{\mathsf{{o}}}(\varLambda _{{\mathcal{O }}_\mathrm{E}})$ is not maximal in $\mathrm{G}_\mathrm{E}$ or $\mathrm{G}_\mathrm{E}$ does not have compact centre and will find a non-zero Jacquet module. (This assumption takes the place of hypothesis (H) in [29, §7.2].) Most of [29, §7.2] now goes through essentially unchanged, with two small changes in the cases called (i) and (ii) in §7.2.2 (page 350).

In case (i), the change happens when the element $p$ cannot be chosen to normalize the representation $\rho$, interpreted as a representation of $\mathrm{P}^{\mathsf{{o}}}(\varLambda _{{{\mathcal{O }}_\mathrm{E}}})$ trivial on $\mathrm{P}_1(\varLambda _{{{\mathcal{O }}_\mathrm{E}}})$. (Note that $p\in \mathrm{P}^+(\varLambda _{{{\mathcal{O }}_\mathrm{E}}})$ so it does normalize the group $\mathrm{P}^{\mathsf{{o}}}(\varLambda _{{{\mathcal{O }}_\mathrm{E}}})$.) In this case, $N_\varLambda (\rho )\subseteq \mathrm{M}'$, where $\mathrm{M}'$ is the Levi subgroup of loc. cit. (Note, however, that this would not be the case if we were working in the non-connected group $\mathrm{G}^+$, rather than $\mathrm{G}$.) Thus, by [29, Corollary 6.16], we have $\mathrm{I}_\mathrm{G}(\vartheta _\mathrm{P})\subseteq \mathrm{J}_\mathrm{P}^{\mathsf{{o}}}\mathrm{M}'\mathrm{J}_\mathrm{P}^{\mathsf{{o}}}$ and, as in the proof of [29, Proposition 7.10], $(\mathrm{J}_\mathrm{P}^{\mathsf{{o}}},\vartheta _\mathrm{P})$ is a cover of $(\mathrm{J}_\mathrm{P}^{\mathsf{{o}}}\cap \mathrm{M}',\vartheta _\mathrm{P}|_{\mathrm{J}^{\mathsf{{o}}}_\mathrm{P}\cap \mathrm{M}'})$. (See also Lemma 5.8.)

In case (ii), the change happens when $m=1$. In this case $\mathrm{P}^{\mathsf{{o}}}(\varLambda _{{\mathcal{O }}_\mathrm{E}})$ is a maximal parahoric subgroup but $\mathrm{G}_\mathrm{E}$ does not have compact centre; indeed $\mathrm{G}_{\mathrm{E}_1}\simeq \mathrm{SO}(1,1)(\mathrm{F})$ and we have $\mathrm{G}_\mathrm{E}\subseteq \mathrm{M}'$. As in case (i) above, we get that $\mathrm{I}_\mathrm{G}(\vartheta _\mathrm{P})\subseteq \mathrm{J}_\mathrm{P}^{\mathsf{{o}}}\mathrm{M}'\mathrm{J}_\mathrm{P}^{\mathsf{{o}}}$ and $(\mathrm{J}_\mathrm{P}^{\mathsf{{o}}},\vartheta _\mathrm{P})$ is a cover of $(\mathrm{J}_\mathrm{P}^{\mathsf{{o}}}\cap \mathrm{M}',\vartheta _\mathrm{P}|_{\mathrm{J}^{\mathsf{{o}}}_\mathrm{P}\cap \mathrm{M}'})$.

In particular, in all cases, the representation $\pi$ containing $\vartheta$ cannot be cuspidal and we have proved the first two assertions of Proposition 4.4; the third assertion follows from [29, Lemma 7.4].

In particular, since by [28, Theorem 5.1] every irreducible cuspidal representation of $\mathrm{G}$ does contain a semisimple character, and hence a representation of $\mathrm{J}$ of the form $\kappa \otimes \tau$, this also proves [29, Theorem 7.14].