Distinction of the Steinberg representation with respect to a symmetric pair
Abstract
Let $K$ be a non-archimedean local field of residual characteristic $p\neq 2$. Let $G$ be a connected reductive group over $K$, let $\theta$ be an involution of $G$ over $K$, and let $H$ be the connected component of $\theta$-fixed subgroup of $G$ over $K$. By realizing the Steinberg representation of $G$ as the $G$-space of complex smooth harmonic cochains following the idea of Broussous--Courtès, we study its space of distinction by $H$ as a finite dimensional complex vector space. We give an upper bound of the dimension, and under certain conditions, we show that the upper bound is sharp by explicitly constructing a basis using the technique of Poincaré series. Finally, we apply our general theory to the case where $G$ is a general linear group and $H$ a special orthogonal subgroup, which leads to a complete classification result.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2024
- DOI:
- 10.48550/arXiv.2410.03247
- arXiv:
- arXiv:2410.03247
- Bibcode:
- 2024arXiv241003247W
- Keywords:
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- Mathematics - Representation Theory;
- Mathematics - Number Theory;
- 22E50;
- 20E42;
- 11F70
- E-Print:
- 75 pages, 8 figures. Comments welcome!