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PreprintCC BY 4.0arXiv

Iwahori Whittaker functions for unramified quasi-split groups

  • Solomon Friedberg
  • Claire Frechette
  • Henrik P. A. Gustafsson
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Abstract

Let G be a connected unramified quasi-split reductive group defined over a nonarchimedean local field F. This paper concerns the Iwahori Whittaker functions attached to the unramified principal series of G(F). We introduce Demazure-Whittaker operators and show that in the non-split case they generate a Hecke algebra with unequal parameters that are determined by G. We then connect these operators to the Whittaker functions, establishing formulas for calculating all values for a basis of Iwahori Whittaker functions and giving a new proof of the Casselman-Shalika formula for the spherical Whittaker function. In doing so we build on the work for split groups of Brubaker, Bump and Licata; the new feature here is the appearance of a Hecke algebra with unequal parameters.

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Cite

BibTeXRIS
BibTeX
@misc{friedberg2026iwahori,
  title = {{Iwahori Whittaker functions for unramified quasi-split groups}},
  author = {Solomon Friedberg and Claire Frechette and Henrik P. A. Gustafsson},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37614},
  eprint = {2610.37614},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37614},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Iwahori Whittaker functions for unramified quasi-split groups
AU  - Solomon Friedberg
AU  - Claire Frechette
AU  - Henrik P. A. Gustafsson
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37614
UR  - https://arxiv.org/abs/2610.37614
AB  - Let G be a connected unramified quasi-split reductive group defined over a nonarchimedean local field F. This paper concerns the Iwahori Whittaker functions attached to the unramified principal series of G(F). We introduce Demazure-Whittaker operators and show that in the non-split case they generate a Hecke algebra with unequal parameters that are determined by G. We then connect these operators to the Whittaker functions, establishing formulas for calculating all values for a basis of Iwahori Whittaker functions and giving a new proof of the Casselman-Shalika formula for the spherical Whittaker function. In doing so we build on the work for split groups of Brubaker, Bump and Licata; the new feature here is the appearance of a Hecke algebra with unequal parameters.
ER  -