Lift of the Trivial Representation: The Nonsplit Type D Case
Abstract
Let G̃ be a nonlinear double cover of the real points of a connected, simply connected, semisimple complex group. In previous work, we introduced ∏ρ/2s(G̃), the set of genuine small representations with infinitesimal character ρ/2 in the simply laced setting, and showed that, when G is split, these representations are exhausted by the Kazhdan--Patterson lift of the trivial representation. In this paper, we consider nonsplit real Spin groups of type Dn. For each ρ/2-compatible genuine central character, we determine the irreducible constituents of LiftGG̃(ℂ). We also describe explicitly the relevant genuine small representations in terms of Langlands parameters. The nonsplit case exhibits a new phenomenon: the lift of the trivial representation need not exhaust ∏sρ/2(G̃). For the families considered here, equality holds only for Spiñ(2p+2,2p); in the other two families the lift is a proper subset, and in one case it is zero.
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BibTeX
@misc{tsai2026lift,
title = {{Lift of the Trivial Representation: The Nonsplit Type $D$ Case}},
author = {Wan-Yu Tsai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23675},
eprint = {2609.23675},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23675},
}
RIS
TY - UNPB
TI - Lift of the Trivial Representation: The Nonsplit Type D Case
AU - Wan-Yu Tsai
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23675
UR - https://arxiv.org/abs/2609.23675
AB - Let G̃ be a nonlinear double cover of the real points of a connected, simply connected, semisimple complex group. In previous work, we introduced ∏_ρ/2^s(G̃), the set of genuine small representations with infinitesimal character ρ/2 in the simply laced setting, and showed that, when G is split, these representations are exhausted by the Kazhdan--Patterson lift of the trivial representation. In this paper, we consider nonsplit real Spin groups of type D_n. For each ρ/2-compatible genuine central character, we determine the irreducible constituents of Lift_G^G̃(ℂ). We also describe explicitly the relevant genuine small representations in terms of Langlands parameters. The nonsplit case exhibits a new phenomenon: the lift of the trivial representation need not exhaust ∏^s_ρ/2(G̃). For the families considered here, equality holds only for Spiñ(2p+2,2p); in the other two families the lift is a proper subset, and in one case it is zero.
ER -