On restriction of unitarizable representations of general linear groups and the non-generic local Gan-Gross-Prasad conjecture. (English) Zbl 1512.22015
Let \(\pi\) and \(\sigma\) be irreducible smooth representations of \(p\)-adic \(\mathrm{GL}_n\) and \(\mathrm{GL}_{n-1}\) respectively which are of Arthur type. The recent generalization of the celebrated Gan-Gross-Prasad conjecture [W. T. Gan et al., Compos. Math. 156, No. 11, 2298–2367 (2020; Zbl 1470.11126)] predicts that \(\mathrm{Hom}_{\mathrm{GL}_{n-1}}(\pi\vert_{\mathrm{GL}_{n-1}},\sigma)\neq 0\) if and only if the Arthur parameters of \(\pi\) and \(\sigma\) satisfy certain combinatorial conditions. In the paper under review, the author proves the “only if” direction of this conjecture (Theorem 5.6), and extends the conjecture’s statement to the class of all unitarizable representations (Theorem 5.7). He also proves the “if” direction of the conjecture when at least one of \(\pi\) and \(\sigma\) is generic (Theorem 5.10). The most striking novelty of the proof is to transfer the intricate structure of the Bernstein-Zelevinsky product into the representation theory of affine Hecke algebras and quantum affine algebras of type A. It should be mentioned that the conjecture is fully established in [K. Y. Chan, J. Reine Angew. Math. 783, 49–94 (2022; Zbl 1491.81021)] using rather different techniques.
Reviewer: Dongwen Liu (Zhejiang)
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 20C15 | Ordinary representations and characters |
| 20G42 | Quantum groups (quantized function algebras) and their representations |
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