Invariant trilinear forms for SL(3,R)

TL;DR

This paper analyzes the orbit structure of the third power of the flag variety for SL(3,R), constructing invariant distributions and providing infinitely many linearly independent triple products of induced representations.

Contribution

It offers a detailed orbit classification and constructs new invariant distributions, filling a gap in the understanding of invariant trilinear forms for SL(3,R).

Findings

Identified 70 orbits plus a continuous family in the flag variety.

Constructed invariant distributions on the continuous family.

Provided infinitely many linearly independent triple products.

Abstract

We give a detailed analysis of the orbit structure of the third power of the flag variety attached to SL(3,R). It turns out that 36 generalized Schubert cells split into 70 orbits plus one continuous family of orbits. On the latter, we construct invariant distributions and thus fill a gap in the literature by giving infinitely many linearly independent triple products of induced representations.