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The unitarizability of the Aubert dual of strongly positive square integrable representations. (English) Zbl 1178.22018
The following conjecture: “the involution defined by the reviewer in [Trans. Am. Math. Soc. 347, No. 6, 2179–2189 (1995; Zbl 0827.22005)] preserves unitarity” is the main subject of the paper. This conjecture was proved by Tadić for the general linear groups over local non-Archimedean fields, and by Barbasch and Moy in the case of representations of arbitrary \(p\)-adic groups which admit non-zero Iwahori fixed vectors.
The main result of the paper is the validity of the conjecture for all strongly positive square-integrable representations of classical \(p\)-adic groups assuming a basic assumption (which follows from a certain Arthur’s conjecture).
The strongly positive square-integrable representations serve as the “building blocks” for all the square-integrable representations, as it can be seen from the Moeglin-Tadić classification in [C. Moeglin and M. Tadić, J. Am. Math. Soc. 15, No. 3, 715–786 (2002; Zbl 0992.22015)]. They include generalized Steinberg representations and regular discrete series.
It is known (proved by Silberger) that for each self-dual supercuspidal irreducible representation \(\rho\) of \(\text{GL}(n,F)\) (\(F\) a local non-Archimedean field) and each supercuspidal irreducible representation \(\sigma\) of a classical group \(G\) over \(F\), there exists a unique \(\alpha_{\rho,\sigma}\geq 0\) such that the parabolically induced representation from the representation \(\nu^{\alpha_{\rho,\sigma}}\rho\otimes\sigma\) of the Levi subgroup \(\text{GL}(n,F)\times G\) reduces. (Here \(\nu\) denotes the composition of the determinant mapping with the normalized absolute value on \(F\).)
Then the basic assumption states that \(\alpha_{\rho,\sigma}-\alpha_{\rho,1}\in\mathbb Z\) (where \(1\) denotes the trivial representation of the trivial group). Shahidi has proved that \(\alpha_{\rho,1}\in\frac{1}{2}\mathbb Z\), and moreover, he proved that the assumption holds if \(\sigma\) is generic.

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
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