Standardness and Standard Automorphisms of Chevalley Groups,Ⅰ:the Case of Rank at Least Two

The author studies the linkage between the standardness and the standard automorphisms of ChevMley groups over rings. It is proved that if H is any standard subgroup of G（R）, then each of its automorphisms can be extended to an automorphism of G（R, I）, restricted to an automorphism of E（R, I）, and an automorphism of E（R, I） can be extended to one of G（R, I）. The case of Chevalley groups of rank at least two is treated in this paper. Further results about the case of Chevalley groups of rank one, the case of nomcommutative ground ring and some others exceptions will appear elsewhere.

A NEW TYPE OF PRONORMAT SUBGROUPS IN CHEVALLEY GROUPS

Let L be a simple Lie algebra with irreducible root system. having roots of different length, F be a field of charaCteristic different from 2, G = L(F) be a Chevalley group of type L over F. Denote by Φl the set of all long roots in Φl Set Gl = <xr (t); f∈EΦl,t∈F>. It is a subgroup of G generated by all the long root subgroups. This paper determines the pronormality of Gl in G when L is not of type G2.

Buildings and Groups Ⅱ

This is the second part of a pedagogical introduction to the theory of buildings of Jacques Tits.We de ne a(B,N)pair and construct a building out of it.Then we give a description of Chevalley groups,their(B,N)pair and the associated buildings.We illustrates this theory with many examples from classical groups.

HOMOMORPHISMS BETWEEN CHEVALLEY GROUPS OF TYPES C_nAND G_2 OVER FINITE FIELDS

It is known that there exists an isogeny sort of Chevalley groups G (Σ, F) associated to any indecomposable root system Σ and any field F . In this paper the author determines all nontrivial homomorphisms from G(Σ, k) to G(Σ, K) when the root system Σ is of type C n or G 2 , and the fields k and K are finite fields of characteristic p .

ON THE HARISH－CHANDRA HOMOMORPHISM FOR THE CHEVALLEY GROUPS OVER P－ADIC FIELD

The Harish-Chandra homomorphism for the higher congruence spherical functions algebra of Chevalley groups over p-adic fiealds is given in the case of the Levi-component of a (rational)parabolic subgroup. It is a generalization for the Harish-Chandra homomorphism for the higher congurence spherical functions algebra of the groups CLn over p-adic field in the same case.

Isomorphism Between Gal(E/F) and Gal(L(E)/L(F))

Let E be a field of finite extension of a perfect field F. We show that Gal(E/F) is isomorphic to Gal(L(E)/L(F)).