Depth-zero supercuspidal L-packets and their stability

Abstract

In this paper we verify the local Langlands correspondence for pure inner forms of unramified p-adic groups and tame Langlands parameters in "general position". For each such parameter, we explicitly construct, in a natural way, a finite set ("L-packet") of depth-zero supercuspidal representations of the appropriate p-adic group, and we verify some expected properties of this L-packet. In particular, we prove, with some conditions on the base field, that the appropriate sum of characters of the representations in our L-packet is stable; no proper subset of our L-packets can form a stable combination. Our L-packets are also consistent with the conjectures of B. Gross and D. Prasad on restriction from SO 2n+1 to SO 2n [24].These L-packets are, in general, quite large. For example, Sp 2n has an L-packet containing 2 n representations, of which exactly two are generic. In fact, on a quasi-split form, each L-packet contains exactly one generic representation for every rational orbit of hyperspecial vertices in the reduced BruhatTits building. When the group has connected center, every depth-zero generic supercuspidal representation appears in one of these L-packets.We emphasize that there is nothing new about the representations we construct. They are induced from Deligne-Lusztig representations on subgroups of finite index in maximal compact mod-center subgroups, see [42], [44], [61]. The point here is to assemble these representations into L-packets in a natural and explicit way and to verify that these L-packets have the required properties.

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“…In fact, inspired by the work of Adams and Vogan, we provide a refinement of their conjecture to the level of representations, rather than packets, for the tempered local Langlands correspondence. We prove this refinement when G is either a quasisplit connected real reductive group (more generally, quasisplit real K -group), a quasisplit symplectic or special orthogonal p-adic group, and in the context of the constructions of [DeBacker and Reeder 2009] and [Kaletha 2012]. In the real case, the results of Adams and Vogan are a central ingredient in our argument.…”

“…Fix a Langlands parameter ϕ : W → L G of the type considered in [DeBacker and Reeder 2009] or [Kaletha 2012]. Fix a -invariant splitting (T ,B, {Xα}) of G and arrange thatT is the unique torus normalized by ϕ.…”

“…The first construction is that of [DeBacker and Reeder 2009], in which L-packets consisting of depth-zero supercuspidal representations are constructed for each pure inner form of an unramified p-adic group. This construction was then extended to inner forms of p-adic groups arising from isocrystals with additional structure [Kaletha 2011].…”

“…After that, the argument is the same as for the real case. The constructions of [DeBacker and Reeder 2009] and [Kaletha 2012] are inspected directly.…”

“…Other cases include the group U 3 by work of Rogawski, Sp 4 and GSp 4 by work of Gan-Takeda. For general connected reductive groups, there are constructions of the correspondence for specific classes of parameters, including the classical case of unramified representations, the case of representations with Iwahori-fixed vector by work of Kazhdan-Lusztig, unipotent representations by work of Lusztig, and more recently regular depthzero supercuspidal representations by [DeBacker and Reeder 2009], very cuspidal representations [Reeder 2008], and epipelagic representations [Kaletha 2012]. …”