On Representations of GL_n(D) admitting a generalized linear period

Abstract

Let D be a quaternion division algebra over a non-Archimedean local field F of characteristic zero, and let G_n=GL_n(D). Let H_1,n−1={diag(g₁,g₂)∈G₁, g₂∈G_n−1} and, for s ∈ ℝ, define χ_s(diag(g₁,g₂))=ν(g₁)^2sν(g₂)^−2s. We classify, for n=3,4, the irreducible smooth representations of G_n admitting a generalized linear period with respect to (H_1,n−1,χ_s). Motivated by these results, we conjecture a complete classification for all n>2. Assuming this conjecture, we characterize such representations in terms of Langlands parameters: an irreducible smooth representation π of G_n admits such a period if and only if L(π) contains a Weil-Deligne subrepresentation isomorphic to L(ν^−2s), the (2n−4)-dimensional parameter of ν^−2s on G_n−2, and the four-dimensional quotient is the Langlands parameter of either the trivial representation of G₂, with s=±(n−2)/2, or an irreducible infinite-dimensional H_1,1-distinguished representation of G₂. We also verify the Lapid-Prasad conjecture in this setting: the L-packet of an irreducible representation of G_n admitting a linear period with respect to H_1,n−1 is invariant under ρ↦ρ̃^θ.

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BibTeX RIS

BibTeX

@misc{dagar2026representations,
  title = {{On Representations of $\mathrmGL_n(\mathrmD)$ admitting a generalized linear period}},
  author = {Prem Dagar and Hariom Sharma},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.34307},
  eprint = {2609.34307},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.34307},
}

RIS

TY  - UNPB
TI  - On Representations of GL_n(D) admitting a generalized linear period
AU  - Prem Dagar
AU  - Hariom Sharma
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.34307
UR  - https://arxiv.org/abs/2609.34307
AB  - Let D be a quaternion division algebra over a non-Archimedean local field F of characteristic zero, and let G_n=GL_n(D). Let H_1,n−1={diag(g_1,g_2)∈G_1, g_2∈G_n−1} and, for s ∈ ℝ, define χ_s(diag(g_1,g_2))=ν(g_1)^2sν(g_2)^−2s. We classify, for n=3,4, the irreducible smooth representations of G_n admitting a generalized linear period with respect to (H_1,n−1,χ_s). Motivated by these results, we conjecture a complete classification for all n>2. Assuming this conjecture, we characterize such representations in terms of Langlands parameters: an irreducible smooth representation π of G_n admits such a period if and only if L(π) contains a Weil-Deligne subrepresentation isomorphic to L(ν^−2s), the (2n−4)-dimensional parameter of ν^−2s on G_n−2, and the four-dimensional quotient is the Langlands parameter of either the trivial representation of G_2, with s=±(n−2)/2, or an irreducible infinite-dimensional H_1,1-distinguished representation of G_2. We also verify the Lapid-Prasad conjecture in this setting: the L-packet of an irreducible representation of G_n admitting a linear period with respect to H_1,n−1 is invariant under ρ↦ρ̃^θ.
ER  -

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