Distinguished Representations of Symmetric Pairs

Definitions and mathematical statement

Let $\theta\colon G\to G$ be an involutive automorphism, so that $\theta^2=\operatorname{id}_G$, and let \[ H=G^\theta:=\{g\in G:\theta(g)=g\}. \] Here $H$ denotes the full fixed-point subgroup specified in the source. For a complex representation $\pi$ of $G$ on a nonzero space $V_\pi$, give ${\mathbb{C}}$ the trivial $H$-action. An $H$-invariant linear functional satisfies \[ \ell(\pi(h)v)=\ell(v)\qquad(h\in H,\ v\in V_\pi). \] The notation $\operatorname{Hom}_H(\pi,{\mathbb{C}})$ denotes the space of such functionals, with any continuity or other regularity requirements imposed by the intended representation category. The representation $\pi$ is called $H$-distinguished when \[ \operatorname{Hom}_H(\pi,{\mathbb{C}})\ne\{0\}. \] Classification question. For a symmetric pair $H=G^\theta$, determine which irreducible representations $\pi$ of $G$ are $H$-distinguished. Equivalently, once the group setting and representation category have been specified, characterize the isomorphism classes \[ \left\{[\pi]:\pi\text{ is an irreducible representation of }G, \ \operatorname{Hom}_H(\pi,{\mathbb{C}})\ne\{0\}\right\}. \] Unresolved scope. The supplied record does not specify the class of groups $G$, an underlying field or topology, or the representation category. It also does not specify whether invariant functionals must be continuous. Consequently, this is a conditional classification schema preserving the source question; it does not supply a fully specified classification problem in any particular setting.

Short English statement

For the full fixed-point subgroup $H=G^\theta$ of an involution, classify the irreducible representations of $G$ admitting a nonzero $H$-invariant complex linear functional. The group setting, representation category, and required regularity of the functional remain unspecified in the supplied record.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30001736$, title "Distinguished Representations of Symmetric Pairs," source code catalogue entry; collection: Oberwolfach Reports Open Problems.
• Supplied source citation: Automorphic Forms: New Directions (2011), Oberwolfach Reports, pp. 719--779, DOI 10.4171/owr/2011/14. https://doi.org/10.4171/owr/2011/14. The record attributes the question to the workshop-report collection and gives proposed year 2011, but names no individual proposer or precise location within the report.
• Supplied literature source, labeled "symmetric-variety dual groups": https://arxiv.org/abs/2308.15800. No authors, title, or specific supporting passage are supplied for this link.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes classification for general symmetric pairs as active, with conjectures concerning relations to symmetric-space dual groups. This is supplied dated curation, not an independently verified assessment; the record explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is a supplied verification claim; no independent source consultation is asserted here.