Jantzen conjecture for graded affine Hecke algebras

Let $\mathcal H$ be a graded affine Hecke algebra, let $M$ be a standard $\mathcal H$ -module, and let $J^0M\supseteq J^1M\supseteq\cdots$ be its Jantzen filtration arising from a one-parameter deformation of the central character. The conjecture asserts that every layer $J^iM/J^{i+1}M$ is semisimple and that, for every irreducible constituent $L$ of $M$ , its multiplicities across the Jantzen layers are given by the full local intersection-cohomology multiplicities of the corresponding orbit closure: the graded multiplicity of $L$ in $\operatorname{gr}_J M$ equals the graded local intersection-cohomology multiplicity attached to that orbit closure.

References

Primary source

arXiv

Additional references

Jantzen filtrations for graded affine Hecke algebras — arXiv

graded-affine-hecke-algebras intersection-cohomology representation-theory

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new paper claims a proof in the equal-parameter case, but the result has not been independently verified and broader cases remain open.

The conjecture identifies the algebraic Jantzen filtration with geometric intersection-cohomological multiplicities for graded affine Hecke algebras. A recent paper claims this in the equal-parameter, real-central-character, dominant-direction setting; the full generality remains unsettled.

Known results

Type $A$ degenerate affine Hecke algebras: Jantzen filtrations and Rogawski’s multiplicity refinement were obtained in this setting (Suzuki, 1998).
A geometric standard-module multiplicity formula was established using constructible sheaves and intersection cohomology (Ciubotaru–Kato, 2021).

September 2026 claimed proof

The paper identifies the standard-to-contragredient map with a costalk-to-stalk map and uses Hard Lefschetz to determine elementary divisors, proving the stated restricted case. An August 2026 announcement says GPT 5.6 produced a broader proof, but the author explicitly sought gaps and the claim remains unverified; dominance appears necessary from a type- $A$ example.

Current status (as of September 2026): The equal-parameter, real-central-character, dominant-direction case is claimed proved, but independent verification is absent and the broader conjecture remains open.

Sources

No solutions have been posted yet.