English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We prove the strong Macdonald conjecture of Hanlon and Feigin for reductivegroups G. In a geometric reformulation, we show that the Dolbeault cohomology$H^q(X;\Omega^p)$ of the loop Grassmannian X is freely generated by de Rham'sforms on the disk coupled to algebra generators of $H*(BG)$. Equating Eulercharacteristics of the two gives an identity, independently known to Macdonald[M], which generalises Ramanujan's_1\psi_1 sum. Simply laced root systems atlevel 1 are related to a `strong'_4\psi_4 sum. Failure of Hodge decompositionimplies the singularity of X, and of the algebraic loop groups.

The strong Macdonald conjecture and Hodge theory on the loop Grassmannian