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Ext-multiplicity theorem for standard representations of \((\textrm{GL}_{n+1},\textrm{GL}_n)\)

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Let \(\pi _1\) be a standard representation of \(\textrm{GL}_{n+1}(F)\) and let \(\pi _2\) be the smooth dual of a standard representation of \(\textrm{GL}_n(F)\). When F is non-Archimedean, we prove that \(\textrm{Ext}^i_{\textrm{GL}_n(F)}(\pi _1, \pi _2)\) is \(\cong \mathbb {C}\) when \(i=0\) and vanishes when \(i \ge 1\). The main tool of the proof is a notion of left and right Bernstein–Zelevinsky filtrations. An immediate consequence of the result is to give a new proof on the multiplicity at most one theorem. Along the way, we also study an application of an Euler–Poincaré pairing formula of D. Prasad on the coefficients of Kazhdan–Lusztig polynomials. When F is an Archimedean field, we use the left–right Bruhat-filtration to prove a multiplicity result for the equal rank Fourier–Jacobi models of standard principal series.

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Acknowledgements

The author would like to thank Dragan Miličić, Dipendra Prasad, Gordan Savin and Peter Trapa for useful communications. The author would like to thank the referee for useful remarks and suggestions to improve expositions of the article.

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  1. Department of Mathematics, The University of Hong Kong, Hong Kong, China

    Kei Yuen Chan

  2. Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China

    Kei Yuen Chan

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Chan, K.Y. Ext-multiplicity theorem for standard representations of \((\textrm{GL}_{n+1},\textrm{GL}_n)\). Math. Z. 303, 45 (2023). https://doi.org/10.1007/s00209-022-03198-y

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