The Bernstein-Sato b-function for the complement of the open $SL_n$-orbit on a triple flag variety

TL;DR

This paper computes Bernstein-Sato b-functions for specific invariant sections related to the complement of the open orbit in a triple flag variety, extending known calculations to a more complex geometric setting.

Contribution

It extends the calculation of Bernstein-Sato b-functions to the setting of triple flag varieties and their orbit complements, building on Kashiwara's work.

Findings

Explicit formulas for the b-functions are derived.

The method adapts Kashiwara's approach to a more complex geometric context.

Results provide new insights into the singularities of orbit complements.

Abstract

We calculate Bernstein-Sato b-functions for $f_{G^3}^\lambda$, a $SL_n$-invariant section of a line bundle on $SL_n/B \times SL_n/B \times \mathbb{P}^{n - 1}$ whose zero-set is the complement of the open $G$-diagonal orbit. The proof uses a similar calculation by Kashiwara of the b-function for $f^\lambda$, a $B^-$-semiinvariant section of a line bundle on $SL_n/B$ whose zero-set is the complement of the big Bruhat cell.