Endo-parameters for p-adic classical groups

Abstract

For a classical group over a non-archimedean local field of odd residual characteristic p, we prove that two cuspidal types, defined over an algebraically closed field $${\mathbf {C}}$$ C of characteristic different from p, intertwine if and only if they are conjugate. This completes work of the first and third authors who showed that every irreducible cuspidal $${\mathbf {C}}$$ C -representation of a classical group is compactly induced from a cuspidal type. We generalize Bushnell and Henniart’s notion of endo-equivalence to semisimple characters of general linear groups and to self-dual semisimple characters of classical groups, and introduce (self-dual) endo-parameters. We prove that these parametrize intertwining classes of (self-dual) semisimple characters and conjecture that they are in bijection with wild Langlands parameters, compatibly with the local Langlands correspondence.

Intertwining Semisimple Characters for -Adic Classical Groups

“…This is the first step in an “intertwining implies conjugacy” result for cuspidal types proved in the sequel [KSS16], which then completes the classification of cuspidal representations of .…”

“…In this paper, we prove many of these rigidity results for semisimple characters, which are new even in the case of general linear groups—in particular, we prove “intertwining implies conjugacy” and Skolem–Noether results (see below for details). In a sequel [KSS16], jointly with Kurinczuk, we are then able to put this together with other work of Kurinczuk and the second author [KS15], to turn the construction of cuspidal representations into a classification, for both complex and -modular representations, with prime. More precisely, we establish the following conjugacy result for cuspidal types in -adic classical groups: if and are two types from the construction in [Ste08] which induce to give equivalent irreducible cuspidal representations, then they are conjugate.…”

“…Suppose that our underlying strata are skew —that is, is in the Lie algebra of , the associated decomposition of is orthogonal with respect to the hermitian form, and is in the building of the centralizer in of (see [BS09]). Our first main result here is a Skolem–Noether theorem for semisimple strata, which is crucial in the sequel [KSS16].…”

Semisimple characters for inner forms II: Quaternionic forms of 𝑝-adic classical groups (𝑝 odd)

“…On the other hand, positive depth representations, where the arithmetic information of F reside, are more challenging. Among many others let me mention the work of Adler [1], Yu and Kim [32], [16], Kaletha [15] for tame cases and Bushnell-Kutzko [10], Sécherre-Stevens [23], and Kurinczuk-Skodlerack-Stevens [17] for general linear and classical groups, the latter if F has odd residue characteristic.…”

“…We come 324 DANIEL SKODLERACK to that later. For endo-parameters in the non-quaternionic case (for example for G(L) where L|F is a quadratic unramified extension of F ) see [17].…”