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Depth-zero base change for ramified \(U(2,1)\). (English) Zbl 1233.22002

In [Automorphic representations of unitary groups in three variables. Annals of Mathematics Studies, 123. Princeton, NJ: Princeton University Press (1990; Zbl 0724.11031)], J. D. Rogawski undertook a comprehensive study of the representation theory of unitary groups in (at most) three variables, proving, among other things, the existence of a base-change lifting; but this existence result does not provide an explicit description. The paper under review is the second by the authors (the first is [J. Number Theory 114, 324–360 (2006; Zbl 1077.22021)]) that provides this description in the depth-\(0\) case. (One is reminded of the situation with the local Langlands correspondence, whose existence for general linear groups was proven by M. Harris and R. Taylor [The geometry and cohomology of some simple Shimura varieties. Annals of Mathematics Studies 151. Princeton, NJ: Princeton University Press. (2001; Zbl 1036.11027)], but which was not made completely explicit until the work of C. J. Bushnell and G. Henniart [J. Am. Math. Soc. 18, No. 3, 685–710 (2005; Zbl 1073.11070)].
By work of A. Moy and G. Prasad [Invent. Math.116, 393–408 (1994; Zbl 0804.22008); Comment. Math. Helv.71, No. 1, 98–121 (1996; Zbl 0860.22006)], depth-\(0\) representations of \(p\)-adic groups ‘are’ representations of associated finite groups of Lie types – the reductive quotients of parahoric subgroups. Adler and Lansky’s explicit description of the base-change lifting amounts to a description of what is happening on the level of the finite-group representations.
In the case where \(\underline G\) is a unitary group over an unramified quadratic extension \(E/F\), and \(\underline{\mathsf G}\) is the associated finite group (which could be, say, another unitary group over the residue field \(k_F\) of \(F\)), the Galois action on \(\underline G(E)\) descends to the Galois action on \(\underline{\mathsf G}(k_E)\), and we can hope that the base-change lifting on the level of local fields is compatible with the base-change lifting on the level of finite fields. The main result of Adler–Lansky [loc. cit.], reproduced as Theorem 1.2 of the paper under review, is the statement that this expected compatibility does hold.
The present paper treats the more complicated extension when the quadratic extension \(E/F\) is ramified. In that case, the residue fields \(k_E\) and \(k_F\) are equal, and it is not so clear what the proper lifting should be. One of the key observations of this paper is that the Galois action on \(\underline G(E)\) still descends to some (non-Galois) involutive action on \(\underline{\mathsf G}(k_E) = \underline{\mathsf G}(k_F)\), and that one can use this action to define an analogue of the classical base-change map. The authors call this the \(\varepsilon\)-lifting, where \(\varepsilon\) is the action of the non-trivial action of the Galois group; its theory is sketched here, in §3.2, and fleshed out in great generality in [“Lifting representations of finite reductive groups” I and II, arXiv:1106.0786 and arXiv:1109.0794]. The goal is then to show that the local-field base-change lifting is compatible with the \(\varepsilon\)-lifting.
The authors’ approach to this is intriguing. After a thorough review of the Bruhat–Tits structure theory of the unitary groups in at most three variables – which is worth reading in its own right, as an explicit example of the theory that is not often discussed – the construction of these representations is outlined, and then the \(L\)-packets and the effect of the base-change map on them are described explicitly, more or less by a process of elimination. The elimination process relies on a bit more explicit knowledge than is currently available, so the authors must impose an additional, technical hypothesis (Hypothesis 1.5) on endoscopic transfers of depth-\(0\) \(L\)-packets; but this is no serious obstacle, since considerable progress, which they describe on p. 5571, has already been made towards proving that the hypothesis is satisfied.
Since the base-change lifting is defined in terms of character identities, it will be no surprise that the matching requires character formulæfor supercuspidal representations; but these are, in general, quite hard to come by. Fortunately, as in early work of G. Henniart [Ann. Math. (2) 123, 145–203 (1986; Zbl 0588.12010)] on the local Langlands correspondence, it turns out to be sufficient to understand the character values on very regular elements (see Lemma 6.2), and to combine it with the combinatorial Lemma 4.2 on character sums.
In the process of analysing the various unitary groups that arise, the authors discover a conjectural Jacquet–Langlands-type map see [H. Jacquet and R. P. Langlands, Automorphic forms on GL (2). Lecture Notes in Mathematics. 114. Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0236.12010)] for the anisotropic unitary group in two variables. In fact, under the assumption that some such map exists, §5.4 completely determines it. Conjecturally, the \(\Theta\) correspondence (on the level of representations) refines this Jacquet–Langlands lift, in the sense that every \(L\)-packet contains a representation \(\pi\) appearing in the \(\Theta\) correspondence, and \(\Theta(\pi)\) lies in the Jacquet–Langlands lift of the \(L\)-packet of \(\pi\).

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
20G05 Representation theory for linear algebraic groups
20G25 Linear algebraic groups over local fields and their integers
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References:

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