Positive formulae for K-types of SL_3(R)-irreps and a Blattner formula for smooth K-orbit closures

TL;DR

This paper proves a version of Blattner's conjecture for certain irreducible subquotients of principal series representations of SL_3(R), providing positive formulae for K-types and analyzing smooth K-orbit closures.

Contribution

It establishes a new version of Blattner's conjecture for integral infinitesimal characters with smooth K-orbit closures, specifically applied to SL_3(R).

Findings

Proves Blattner's conjecture for specific irreducible subquotients.

Refines alternating-sum formulas to positive ones for SL_3(R).

Analyzes smooth K-orbit closures in the context of representation theory.

Abstract

We prove a version of Blattner's conjecture, for irreducible subquotients of principal series representations with integral infinitesimal character of a real reductive Lie group whose Beilinson-Bernstein D-module is supported on a K-orbit with smooth closure. (The cases usually considered are closed orbits, or their preimages along G/B -> G/P.) We apply this to G_R = SL_3(R), where all four K-orbits on G/B have smooth closure, and refine the resulting alternating-sum formulae to ones with only positive terms.