Generalized Gauss decompositions of simple algebraic groups

N. L. Gordeev

Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 6, Tome 523 (2023), pp. 19-38

Résumé

Let $\mathcal G$ be a simple algebraic group which is defined and split over a field $K$ and which corresponds to an irreducible root system $R$. Further, let $G = \mathcal G(K)$ be the group of $K$-points. We say that the group $G$ has an $M$-decomposition, where $M \subset R$, if every element of the subset $\prod_{\beta \in R\setminus M} X_\beta\cdot T\cdot \prod_{\alpha\in M}X_\alpha$, where $X_\beta, X_\alpha$ are root subgroups and $T$ is the group of $K$-points of a maximal split torus, can be represented uniquely as products of eleements of root subgroups and the group $T$. Moreover, we assume here that the order of the multiplication of elements of groups $X_\beta$ and $X_\alpha$ is fixed. If such a decomposition holds for every fixed order of the multiplication of elements of groups $\{X_\beta\}_{\beta \in R\setminus M}, \{X_\alpha\}_{\alpha \in M}$, we say that the group $G$ has the universal $M$-decomposition. The important example of the universal $M$-decomposition является is the classical Gauss decomposition where $M = R^+$ is the set of positive roots. In this paper we consider the examples of $M$-decompositions, which appear when we deal with parabolic subgroups of $\mathcal G$. Moreover, for groups of types $A_2, B_2$ we construct the identities which are obstacles to a construction of universal $M$-decomposition for some subsets $M\subset R$.