Intertwining semisimple characters for p-adic classical groups

Endo-parameters for p-adic classical groups

“…As conjugate cuspidal types induce isomorphic representations, this completes a classification by types of the irreducible cuspidal representations of G. This result was expected by analogy with other classifications of cuspidal representations via types for other connected reductive groups (such as Bushnell-Kutzko for GL n [9], the third author and Sécherre for inner forms of GL n [16], and Hakim-Murnaghan for all connected reductive groups under tame conditions [11]), but our proof has required a substantial amount of work and relies on the main results of a number of papers (recently, [12] and [18]). We expect this theorem to find many applications in arithmetic -and will be useful whenever detailed analysis of cuspidal representations of G is required.…”

“…it is explained that the statement also holds for inseparable extensions (cf. [13,Remark 1.4]), while a different proof for inseparable extensions can be found in the proof of [18,Theorem 4.4] where particular linear forms are chosen. However, the following simple observation shows that if the result holds for one (σ E , σ)-equivariant linear form then it holds for all such forms: Lemma 2.2.…”

“…In previous works, the third author has developed an approach to the smooth dual of G via Bushnell-Kutzko types. In this article, we accomplish an important part of this programme and complete recent work ( [12] and [18]) in refining the third author's construction [22] of all cuspidal representations of G to a classification. Moreover, we develop a natural framework for our results on semisimple characters, the theory of endo-parameters for self dual semisimple characters of G + (and of G), which contains in its core a generalisation of Bushnell-Henniart's theory of endo-class ( [5]).…”

“…In Section 8, we recall definitions of semisimple strata and characters, and define potential semisimple characters. In Section 9, making significant use of recent work of the second and third authors [18], we investigate intertwining semisimple characters. In Section 10, we can finally introduce semisimple endoequivalence and prove Theorem B.…”

“…In Section 10, we can finally introduce semisimple endoequivalence and prove Theorem B. Section 11 generalises results on intertwining and conjugacy of skew characters of the second and third authors [18] to self dual characters, and Section 12 investigates the difference between intertwining under orthogonal and special orthogonal groups. We then deduce Theorem A (intertwining implies conjugacy for cuspidal types) in Section 13, using the work of the previous sections to reduce to the level zero part where it is known thanks to recent results of the first and third authors [12].…”