Jordan algebras, exceptional groups, and higher composition laws

Rational orbits of the space of pairs of exceptional Jordan algebras

Let $k$ be a field of characteristic not equal to $2,3$, $\mathbb{O}$ an octonion over $k$ and $\mathcal{J}$ the exceptional Jordan algebra defined by $\mathbb{O}$. We consider the prehomogeneous vector space $(G,V)$ where $G=GE_6\times \mathrm{GL}(2)$ and $V=\mathcal{J}\oplus \mathcal{J}$. We prove that generic rational orbits of this prehomogeneous vector space are in bijective correspondence with $k$-isomorphism classes of pairs $(\mathcal{M},\mathfrak{n})$ where $\mathcal{M}$'s are isotopes of $\mathcal{J}$ and $\mathfrak{n}$'s are cubic étale subalgebras of $\mathcal{M}$. Also we prove th…

Rational forms of exceptional dual pairs

We show that every exceptional Lie algebra over a number field can be obtained by Tits' construction from an octonion algebra O and a cubic Jordan algebra J. In particular, the exceptional Lie algebra contains a dual pair which is the direct sum of the derivation algebras of O and J. We determine rational forms of this dual pair.

通过秩为一的对称空间的普朗切尔公式考虑例外的θ对应关系

Exceptional theta correspondences via Plancherel formulas for rank one symmetric spaces

本文通过秩为一的对称空间的普朗切尔公式，研究了例外θ对应关系中共形群的最小表示及其与$G$和$G'$的对应关系。

Globally maximal arithmetic groups

We begin with a review of the structure of simple, simply-connected complex Lie groups and their Lie algebras, describe the Chevalley lattice and the associated split group over the integers. This gives us a hyperspecial maximal compact subgroup of the p-adic Lie group and we describe the other maximal parahoric subgroups and their Lie algebras starting from the hyperspecial one. We then consider the Killing form on the Chevalley lattice and show that it is divisible by 2 times the dual Coxeter number. The same holds for the Lie algebras of the other maximal parahorics. We compute the discrimi…

Q上F4型各向异性例外群的一级自守表示

Level one automorphic representations of an anisotropic exceptional group over $\mathbb{Q}$ of type $\mathrm{F}_{4}$

本文研究Q上F4型各向异性例外群的一级自守表示，给出按阿基米德分量计数的显式公式，并在Arthur-Langlands猜想下细化计数，进而得到GL26上Sato-Tate群为F4(R)的代数尖点一级自守表示数量的显式猜想公式。