Michele Carmassi | Searcharxiv

arXiv2021

The B-orbits on a Hermitian symmetric variety in characteristic 2

Let $G$ be a reductive linear algebraic group over an algebraically closed field $\mathbb{K}$ of characteristic $2$. Fix a parabolic subgroup $P$ such that the corresponding parabolic subgroup over $\mathbb{C}$ has abelian unipotent radical and fix a Levi subgroup $L\subseteq P$. We parametrize the orbits of a Borel $B\subseteq P$ over the Hermitian symmetric variety $G/L$ supposing the root system $Φ$ is irreducible. For $Φ$ simply laced we prove a combinatorial characterization of the Bruhat order over these orbits. We also prove a formula to compute the dimension of the orbits from combinatorial characteristics of their representatives.

math.AG↗

arXiv2021

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

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 $Φ(G)$ which we suppose irreducible. In particular, for $Φ(G)$ 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.

math.AG↗