On the Bruhat $\mathcal{G}$-order between local systems on the B-orbits of a Hermitian symmetric variety

TL;DR

This paper characterizes the Bruhat G-order on rank 1 local systems over B-orbits in Hermitian symmetric varieties, revealing a combinatorial structure dependent on the root system type.

Contribution

It provides a combinatorial description of the Bruhat G-order for local systems on B-orbits, extending previous work to the setting of Hermitian symmetric varieties with irreducible root systems.

Findings

For simply laced root systems, local systems are either trivial or only maximum rank orbits have non-trivial local systems.

The Hasse diagram splits into two components: trivial local systems and maximum rank non-trivial local systems.

Within each component, the order matches the classical Bruhat order.

Abstract

Following Lusztig and Vogan, we study the Bruhat $G$-order on the set $\mathcal{D}$ of rank $1$ local systems on $B$-orbits over an Hermitian symmetric variety $G/L$. The main aim is to give a combinatorial characterization similar to the one on the Bruhat order given by Gandini and Maffei. The results depend on the type of the root system $\Phi(G)$ which we suppose irreducible. In particular, for $\Phi(G)$ simply laced either all the local systems are trivial or only a specific subset of orbits (the orbits of maximum rank) admit a non-trivial local system. In this last case the Hasse diagram of the order admit two connected components: all the orbits with trivial local systems and the orbits of maximum rank with the (unique) non-trivial local system. On every connected component the order coincides with the Bruhat order.