Local newforms, Paškūnas-Stevens Whittaker functions, and explicit test vectors for GL(n)× GL(n)

@misc{groutides2026local,
  title = {{Local newforms, Paškūnas-Stevens Whittaker functions, and explicit test vectors for $\mathrmGL(n)\times \mathrmGL(n)$}},
  author = {Alexandros Groutides},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08633},
  eprint = {2610.08633},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08633},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Local newforms, Paškūnas-Stevens Whittaker functions, and explicit test vectors for GL(n)× GL(n)
AU  - Alexandros Groutides
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08633
UR  - https://arxiv.org/abs/2610.08633
AB  - Recently, Girsch-Kurinczuk have obtained new formulas for local newforms in depth-zero and minimax integral-depth cuspidal representations of p-adic GL(n). One of their main results is an averaging formula which provides an integral representation of the Whittaker newform in terms of the Paškūnas-Stevens Whittaker function. In the present paper, we first prove a reverse averaging formula by establishing an integral representation of the Paškūnas-Stevens Whittaker function in terms of the Whittaker newform. Together, these formulas give an explicit two-way passage between these two distinguished vectors. Finally, we use our averaging formula and earlier work of Kurinczuk-Matringe to give new explicit test vectors in terms of Whittaker newforms for such cuspidal GL(n)×GL(n) pairs, whenever their Rankin-Selberg L-factor is not identically equal to $1$.
ER  -