We organize the nilpotent orbits in the exceptional complex Lie algebras into series and show that within each series the dimension of the orbit is a linear function of the natural parameter a = 1, 2, 4, 8, respectively for f4, e6, e7, e8. We observe similar regularities for the centralizers of nilpotent elements in a series and graded components in the associated grading of the ambient Lie alg...

In their famous paper [6], Kazhdan and Lusztig introduced the concept of equivalence classes such as left cell, right cell and two-sided cell in a Coxeter group W . We inherit the notations 6 L , 6 R , 6 LR , ∼ L , ∼ R and ∼ LR in [6]. Thus w ∼ LR y (resp. w ∼ L y, resp. w ∼ R y) means that the elements w, y ∈ W are in the same two-sided cell (resp. left cell, resp. right cell) of W , etc. Conc...

Let k be an arbitrary field, let G be a (smooth) linear algebraic group over k, and let U be a vector group over k on which G acts by automorphisms of algebraic groups. The action of G on U is said to be linear if there is a G-equivariant isomorphism of algebraic groups U ' Lie(U). Suppose that G is connected and that the unipotent radical of G is defined over k. If the G-module Lie(U) is simpl...

The purpose of this paper is to describe subgroups of a Chevalley group G of type B l , C l or F 4 , generated by a pair of unipotent short root subgroups. In fact we do somewhat more: namely, we classify orbits of a Chevalley group acting by simultaneous conjugation on pairs of unipotent short root subgroups. In the sequel of this paper we consider analogous problems for the case of the Cheval...

Let G be a possibly disconnected algebraic group over an algebraically closed field k of characteristic p > 0, such that its neutral connected component, H = G0, is a unipotent group. We recall that an algebraic group over k is defined to be a smooth group scheme of finite type over k. Let us fix a prime number l 6= p. If X is a k-scheme, we use D(X) to denote the bounded derived category of Ql...

In this paper, which is a sequel of PSV], we develop a completely elementary approach to calculations in Chevalley groups G = G((; R) of types = E 6 and E 7 over a commutative ring R using only the weight diagrams (alias, crystal graphs) of their minimal modules. After an elementary construction of a crystal base we explicitly describe action of root subgroups and of the extended Weyl group, mu...