The authors study a family of parameters corresponding to unitary representations with cohomology, for the real points G of a connected reductive algebraic group \({\mathbb{G}}\) defined over \({\mathbb{R}}\). One wants analogues of results of D. Shelstad on functoriality with respect to L-groups [Compos. Math. 39, 11-45 (1979; Zbl 0431.22011), Ann. Sci. Éc. Norm. Supér., IV. Sér. 12, 1-31 (1979; Zbl 0433.22006), Can. J. Math. 33, 513-558 (1981; Zbl 0457.22006), Math. Ann. 259, 385-430 (1982; Zbl 0506.22014)] for a class of non-tempered representations.
By Langland’s classification, irreducible admissible representations of G are partitioned into finite sets called L-packets. Either every representation in a packet is tempered, or none are.
Let \(\Theta_{\pi}\) denote the character of the irreducible representation \(\pi\). A distribution is said to be stable if it is in the closure of the span of all sums \(\sum \Theta_{\pi}\), \(\pi\in \Pi\), over tempered L-packets \(\Pi\). As an analytic function, a stable distribution is invariant under conjugation by elements of G(\({\mathbb{C}})\), when this makes sense. Such distributions can be transferred to inner forms of G, and further, the span of \(\{\Theta_{\pi}|\pi\in \Pi \}\) for a tempered packet is also spanned by \(\{\Theta_ 0,\Theta_ 1,...,\Theta_ n\}\) where \(\Theta_ 0=\sum_{\pi \in \Pi}\Theta_{\pi}\) is a stable distribution for G and each \(\Theta_ i\), \(i\geq 1\), is the transfer via L-functoriality to G of a stable tempered distribution on a smaller group (inversion). For non-tempered packets, all of these results fail.
In [Lect. Notes Math. 1041, 1-49 (1984; Zbl 0541.22011)] J. Arthur conjectured that non-tempered L-packets can sometimes be enlarged to finite sets of irreducible unitary representations, so that the enlarged packets will satisfy stability and transfer via functoriality analogous to Shelstad’s results. The authors extend the notion of transfer to non- tempered stable distributions and verify Arthur’s conjectures that an averaged sum of characters in an enlarged packet is stable, and that transfer holds, in the form of a character identity, for unitary representations with cohomology. The remaining conjecture, inversion, fails in general, e.g., for Sp(2).
One should note that G is not assumed connected. The proofs involve cohomological techniques to reduce to the tempered case. The results of the authors are discussed in the notes [Unipotent automorphic representations: Conjectures, preprint] of J. Arthur.