Aizenbud–Gourevitch regularity conjecture

Aizenbud and Gourevitch proposed that every symmetric pair is regular in 2008; this would imply van Dijk’s conjecture for complex symmetric pairs.

Known results

• Aizenbud and Gourevitch (2008) proved regularity for several classical families, including $(\operatorname{GL}(V),\operatorname{O}(V))$, $(\operatorname{GL}(V),\operatorname{U}(V))$, and orthogonal and unitary block pairs.
• A later preprint (2022) proved the Gelfand property for eight exceptional complex symmetric pairs and $(\operatorname{Spin}_{4q+2},\operatorname{Spin}_{4q+1})$.
• The 2022 preprint reduced the remaining problem to DIII, CII, Spin-block, and EVII families, while describing their regularity as work in progress.

August 2026 claimed completion

A preprint dated August 19, 2026 claims to treat the remaining DIII, CII, Spin-block, and EVII families and derive consequences for Gelfand–Kazhdan pairs, thereby settling the conjecture and implying van Dijk’s conjecture. The retrieved evidence provides no independent verification or resolution of possible gaps.

Current status (as of August 2026): The conjecture has a new unverified preprint claim of resolution; absent independent confirmation, the remaining families and the full theorem remain mathematically unsettled.