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Distinction of the Steinberg representation with respect to split symmetric subgroups

Guy Shtotland

TL;DR

This work analyzes when the Steinberg representation of a split reductive group $G$ is $H$-distinguished for a split symmetric subgroup $H$, considering both finite fields and non-archimedean local fields. Distinction is reframed via harmonic functions on hypergraphs attached to the symmetric space $X=G/H$, enabling a precise dichotomy: over finite fields, quasi-splitness of $X$ is necessary and sufficient for distinction; over local fields, quasi-splitness is necessary but not always sufficient, with a unique counterexample. A unifying dual-group perspective is developed: the Langlands parameter $\phi_{St}$ factors through the $X$-dual group $G^\vee_X$ via the map $\iota: SL_2\times G^\vee_X\to G^\vee$ exactly when $St_{\chi_0}$ is $H$-distinguished, tying harmonic-graph criteria to Langlands data. The paper also extends the analysis to general unramified twists $St_\chi$ and connects the distinction problem to the geometry of unipotent orbits in $G^\vee$, providing a coherent picture that links representation-theoretic distinction, hypergraph harmonicity, and Langlands duality in the symmetric-space setting.

Abstract

We study the distinction of the Steinberg representation of a split reductive group $G$ with respect to a split symmetric subgroup $H \subset G$. We do that both over a $p$ adic field and over a finite field. We relate these distinction problems to problems about determining the existence of a non zero harmonic function on certain hyper graphs related to $X = G/H$. Under these assumptions, we verify the relative local Langlands conjecture for the Steinberg representation by showing that over a $p$ adic field the Steinberg representation is $H$-distinguished if and only if its Langalnds parameter factors through the dual group of $X$.

Distinction of the Steinberg representation with respect to split symmetric subgroups

TL;DR

This work analyzes when the Steinberg representation of a split reductive group is -distinguished for a split symmetric subgroup , considering both finite fields and non-archimedean local fields. Distinction is reframed via harmonic functions on hypergraphs attached to the symmetric space , enabling a precise dichotomy: over finite fields, quasi-splitness of is necessary and sufficient for distinction; over local fields, quasi-splitness is necessary but not always sufficient, with a unique counterexample. A unifying dual-group perspective is developed: the Langlands parameter factors through the -dual group via the map exactly when is -distinguished, tying harmonic-graph criteria to Langlands data. The paper also extends the analysis to general unramified twists and connects the distinction problem to the geometry of unipotent orbits in , providing a coherent picture that links representation-theoretic distinction, hypergraph harmonicity, and Langlands duality in the symmetric-space setting.

Abstract

We study the distinction of the Steinberg representation of a split reductive group with respect to a split symmetric subgroup . We do that both over a adic field and over a finite field. We relate these distinction problems to problems about determining the existence of a non zero harmonic function on certain hyper graphs related to . Under these assumptions, we verify the relative local Langlands conjecture for the Steinberg representation by showing that over a adic field the Steinberg representation is -distinguished if and only if its Langalnds parameter factors through the dual group of .
Paper Structure (18 sections, 43 theorems, 12 equations, 3 figures, 1 table)

This paper contains 18 sections, 43 theorems, 12 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Methods of proof
  4. Structure of the paper
  5. Related works
  6. Acknowledgement
  7. The Steinberg representation
  8. The Steinberg representation over a finite field
  9. The Steinberg representation over a non archimedean local field
  10. Distinction of the Steinberg representations and harmonic functions
  11. $X$ being quasi split is a necessary condition for distinction
  12. Harmonic functions in the algebraically closed field case
  13. Harmonic functions in the finite field case
  14. Harmonic functions in the non archimedean local field case
  15. Steinberg representation of a general unramified character
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $F$ be a finite field. Under the assumption that both $G$ and $H$ are split over $F$, $St$ is $H$ distinguished if and only if $X$ is quasi split.

Figures (3)

  • Figure 1: On the left the hyper graph $\Gamma_{\Bar{F}}(PGL_3/PO_3)$ and on the right the hyper graph $\Gamma_{\Bar{F}}(PGL_3/PGL_2)$. $\alpha,\beta$ are the two simple roots of $PGL_3$
  • Figure 2: The hyper graph $\Gamma_{\Bar{F}}(PSp_4/PGL_2)$. $\alpha,\beta$ are the two simple roots of $PSp_4$.
  • Figure 3: The hyper graph $\Gamma_{\Bar{F}}(G_2/PGL_2\times SL_2)$. $\alpha,\beta$ are the two simple roots of $G_2$.

Theorems & Definitions (93)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 83 more