# Real double flag varieties for the symplectic group

## TL;DR

This paper investigates the structure of double flag varieties associated with the real symplectic group, classifies orbits, and constructs explicit integral transforms between principal series representations.

## Contribution

It provides a detailed classification of orbits and develops explicit integral transforms for the case of maximal parabolic subgroups related to Grassmannians.

## Key findings

- Classification of L-orbits on the double flag variety
- Explicit integral transforms between principal series of G and L
- Analysis of the geometric structure of the double flag variety

## Abstract

In this paper we study a key example of a Hermitian symmetric space and a natural associated double flag variety, namely for the real symplectic group $G$ and the symmetric subgroup $L$, the Levi part of the Siegel parabolic $P_S$. We give a detailed treatment of the case of the maximal parabolic subgroups $Q$ of $L$ corresponding to Grassmannians and the product variety of $G/P_S$ and $L/Q$; in particular we classify the $L$-orbits here, and find natural explicit integral transforms between degenerate principal series of $L$ and $G$.