Unipotent Representations for Classical Groups

Unipotent representations of classical groups — symplectic $\mathrm{Sp}(2n)$, orthogonal $\mathrm{SO}(N)$, and metaplectic $\mathrm{Mp}(2n)$ — are a distinguished family of irreducible admissible representations of real or $p$-adic groups. They are closely related to the Arthur parameters and the geometric objects (nilpotent orbits, perverse sheaves) studied in the geometric Langlands program.

The tools are implemented using a TypeScript port of the unipotentrepn library written by Jia-Jun Ma:

• DRC diagrams — the combinatorial invariants (Decorated Row-Column tableaux) parametrizing unipotent representations.
• Local systems — the attached local systems on nilpotent orbits.
• Counting formulas — recursive formulas counting unipotent representations by signature.
• Barbasch–Vogan duality — a duality between nilpotent orbits in dual groups, refining the Barbasch–Vogan correspondence.
• Springer correspondence — the map from nilpotent orbits to Weyl group representations.

📄️Unipotent Representation Explorer

Explore DRC diagrams, local systems, counting formulas, Barbasch-Vogan duality, and Springer correspondence for classical groups