Varieties of Borel subalgebras for the Jacobson-Witt Lie algebras

Let W (n) {W(n)} be the Jacobson-Witt algebra over algebraic closed field with characteristic p > 2 {p>2}. In [K. Ou and B. Shu, Borel subalgebras of restricted Cartan-type Lie algebras, J. Algebra Appl. 21 2022, 11, Paper No. 2250210], we introduced the so-called B-subalgebra of W (n) {W(n)}, which serves as an analog of the Borel subalgebra of classical Lie algebras. As a sequel, we describe the structure of the variety consisting of all B-subalgebras of W (n) {W(n)} in this paper. This variety presents an analog of the flag variety for classical Lie algebras. It is shown that is related to the variety of all full flags in. Additionally, we provide a detailed description of the varieties for W (1) {W(1)} as an illustrative example. With the above setting-up, one may establish the Springer theory and geometric representations for the Jacobson-Witt algebras.