Distinction of the Steinberg representation and the dual group of a symmetric space

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 relate this distinction problem to a problem about the existence of a non-zero harmonic function on a certain hyper-graph related to $X = G/H$. 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 Langlands parameter factors through the dual group of $X$.

💡 Research Summary

The paper investigates the distinction problem for the Steinberg representation of a split reductive group G over a field F with respect to a split symmetric subgroup H⊂G. A representation π of G is called H‑distinguished if Hom_H(π,ℂ)≠0. The authors focus on the Steinberg representation S_t (or its unramified twists S_t^χ) and aim to determine precisely when it is H‑distinguished.

Main definitions and necessary condition.
Let σ:G→G be an algebraic involution defined over F, H=G^σ, and X=G/H the symmetric space. X is called quasi‑split if there exists a Borel subgroup B⊂G such that σ(B)∩B is a maximal torus of B. The authors prove that quasi‑splitness of X is a necessary condition for the Steinberg representation to be H‑distinguished (Theorem 1.2 and Theorem 1.5(1)).

Finite‑field case.
When F is a finite field and both G and H are split, the quasi‑split condition is also sufficient: S_t is H‑distinguished if and only if X is quasi‑split (Theorem 1.2). The proof uses Frobenius reciprocity, the fact that S_t has a one‑dimensional B‑fixed line, and a description of the Hecke algebra action, translating the distinction problem into the existence of a non‑zero harmonic function on a combinatorial hyper‑graph Γ_F(X) (Definition 1.8).

p‑adic case and additional obstructions.
For non‑archimedean local fields, quasi‑splitness alone does not guarantee distinction. The authors exhibit a counterexample: G=GL_{2n+1}(F), H=GL_n(F)×GL_{n+1}(F). Here X is quasi‑split but S_t^χ is never H‑distinguished (Proposition 1.3). To obtain a complete criterion they introduce two extra conditions:

1. No σ‑invariant simple factor G_i of G/Z is isomorphic to PGL_{2n+1} with σ‑fixed subgroup P(GL_n×GL_{n+1}) (Theorem 1.5(2)).
2. For a given unramified character χ, the twist χ·χ₀^{−1} must be trivial on a certain finite subgroup Ω_H⊂Ω (Theorem 1.5(3)). Here χ₀ is a quadratic unramified character defined via the sign of the permutation induced by the homomorphism ω:G→Ω (Definition 1.4).

When χ=χ₀ the third condition is automatic, and the authors prove that the Steinberg representation S_t^{χ₀} is H‑distinguished precisely under conditions 1 and 2 (Theorem 1.5, “if” direction). The converse also holds (Remark 1.6).

Hyper‑graph formulation.
The distinction problem is translated into the existence of non‑zero harmonic functions on two hyper‑graphs:

• Γ_F(X): vertices are B‑orbits on X, hyper‑edges are labeled by simple roots α∈Δ, connecting B‑orbits that share the same P_α‑orbit.
• Γ_aff,F(X): vertices are Iwahori‑orbits on X, hyper‑edges labeled by affine simple reflections ˜Δ, connecting I‑orbits that share the same parahoric‑orbit.

A function f on the vertices is harmonic if for every hyper‑edge the sum of its values on that edge is zero. Proposition 1.11 shows that  dim Hom_H(S_t,1)=dim H(Γ_F(X)) and dim Hom_H(S_t^{χ₀},1)=dim H(Γ_aff,F(X)). Thus, constructing a non‑zero harmonic function yields distinction.

Construction of harmonic functions.
The authors first work over an algebraic closure (\bar F) of a finite field, constructing a non‑zero harmonic function on Γ_{\bar F}(X) under the quasi‑split assumption. They then descend to the original finite field using the splitness of H, ensuring Ω_H‑invariance. For the p‑adic case, they extend the construction to Γ_aff,F(X) by exploiting the action of the length‑zero subgroup Ω⊂W_aff on chambers of the Bruhat–Tits building, and by verifying the extra conditions of Theorem 1.5.

Dual group interpretation and relative Langlands conjecture.
Let G^∨ be the Langlands dual group of G and G_X^∨ the dual group attached to the symmetric space X (as in Sakellaridis–Venkatesh). There is a natural homomorphism  ι : G_X^∨ × SL₂ → G^∨. The conjectural relative local Langlands correspondence predicts that an H‑distinguished representation should have its Langlands parameter factor through ι. Theorem 1.7 confirms this for the Steinberg representation: S_t^{χ₀} is H‑distinguished if and only if (i) the unipotent variety U_X of G_X^∨ meets the open G^∨‑orbit of the unipotent variety U of G^∨, and (ii) the Langlands parameter φ_{St} of the Steinberg representation factors through G_X^∨. This provides a complete verification of the relative local Langlands conjecture for Steinberg representations.

Methodological contributions.
The paper introduces a novel combinatorial framework (hyper‑graphs derived from B‑ and Iwahori‑orbits) to translate representation‑theoretic distinction into a harmonic analysis problem on discrete structures. It also blends Bruhat–Tits building theory, Hecke algebra actions, and dual‑group geometry, offering a unified approach that works uniformly for finite fields and p‑adic fields.

Relation to prior work.
Previous studies on Steinberg distinction focused on Galois symmetric spaces (E/F quadratic extensions) and non‑split groups. This work differs by treating split symmetric subgroups, providing exact necessary and sufficient conditions, and by explicitly linking the distinction to the dual group of the symmetric space. The authors also compare their results with recent work of W. Zhang and others, emphasizing that while earlier papers gave bounds on Hom‑spaces, the present paper determines the exact dimension (0 or 1) and connects it to Langlands parameters.

In summary, the paper delivers a complete classification of when the Steinberg representation (and its unramified twists) of a split reductive group is distinguished by a split symmetric subgroup, establishes the equivalence with the existence of non‑zero harmonic functions on naturally defined hyper‑graphs, and confirms the relative local Langlands conjecture for this class of representations.