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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:
 2.2e+51 Representations of Lie and linear algebraic groups over local fields
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References:
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