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On a result of Moeglin and Waldspurger in residual characteristic 2

Abstract

Let \(F\) be a \(p\)-adic field, \(\mathbf G\) a connected reductive group over \(F\), and \(\pi \) an irreducible admissible representation of \(\mathbf G(F)\). A result of Moeglin and Waldspurger states that, if the residual characteristic of \(F\) is different from \(2\), then the ‘leading’ coefficients in the character expansion of \(\pi \) at the identity element of \(\mathbf G(F)\) give the dimensions of certain spaces of degenerate Whittaker forms. In this paper, we extend their result to residual characteristic 2. The outline of the proof is the same as in the original paper of Moeglin and Waldspurger, but certain constructions are modified to accommodate the case of even residual characteristic.

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Acknowledgments

The author is grateful to Shiv Prakash Patel, Dr. T. Deshpande and Professors Dipendra Prasad and J-K Yu for useful conversations. He also thanks Professors S. DeBacker, W. T. Gan, R. Kottwitz, C. Moeglin, G. Savin and F. Shahidi for their encouragement. He owes a special debt to the referee for a careful reading, for having caught many inaccuracies, and for having made a number of suggestions that have made the exposition at various points more transparent and less unpleasant to read.

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Correspondence to Sandeep Varma.

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Varma, S. On a result of Moeglin and Waldspurger in residual characteristic 2. Math. Z. 277, 1027–1048 (2014). https://doi.org/10.1007/s00209-014-1292-8

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  • DOI: https://doi.org/10.1007/s00209-014-1292-8

Keywords

  • Reductive group
  • Character expansion
  • Nilpotent orbit
  • Degenerate Whittaker forms