Аннотация:
Let $K$ be an infinite field. There are many important examples of infinite projectively simple groups (i.e., groups without non-central normal subgroups) supplied by linear algebra. For example, $SL_n(K)$, $n>1$, $Sp_n(K,f)$ (where $Sp_n(K,f)$ are symplectic groups of alternating forms $f$), and etc.
A very useful extension of the range of examples of infinite projectively simple groups was the transition to linear algebraic groups, which led to new interesting conjectures and results. This approach made it possible to identify common properties that reflect the phenomenon of projective simplicity.
Let $G$ be a linear algebraic group defined over a field $K$, $G_K$ be the group of its $K$-rational points. Recall that a group $G$ is anisotropic over $K$ if it has no proper parabolic subgroups defined over $K$. Here a parabolic subgroup is a subgroup containing a Borel subgroup. Denote by $G_K^+$ the normal subgroup of $G_K$ generated by rational over $K$ elements of unipotent radicals of $K$-defined parabolic subgroups. In this situation, J. Tits established the following important fact (1964).
Theorem.
Let $K$ contain at least $4$ elements. Then any subgroup of $G_K$ normalized by the group $G_K^+$ either contains $G_K^+$ or is central. In particular, $G_K^+$ is projectively simple. Thus, a new class of projectively simple groups arises. It is natural to assume that the structure of the group $G_K$ is known if $G_K=G_K^+$. For special groups $G$ and many fields $K$ this fact was known by the time of the proof of the theorem, and therefore the following assumption seemed quite natural.
Conjecture (Kneser–Tits).For a simply connected simple group $G$, which is defined and isotropic over the field $K$, $G_K^+=G_K$.
Note that the Kneser–Tits conjecture is obviously true in the case when $K$ is algebraically closed. One also note that E. Cartan established the validity of the conjecture in the case when $K=\mathbb{R}$ and $G$ is a simple simply connected algebraic group. For a long time it was believed that the Kneser–Tits conjecture was true since it is confirmed in a number of special cases. However, in 1975, V.P. Platonov showed that in the general case the conjecture is false. The latter led to Tits's definition of groups of algebraic $K$-groups $W(K,G)=G_K/G_K^+$ (for further details see [1].
Let $G$ be a simply connected $K$-defined simple algebraic group. Then it belongs to one of the types $A_n$, $B_n$, $C_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$, and $G_2$. Among these types, the most interesting (and difficult to study) are groups of type $A_n$. The outer forms of groups of this type are limited to the following special unitary groups
$$
SU_m(D, f) = \{ u \in U_m(D, f) \colon Nrd_{M_m(D)}(a) = 1 \},
$$
where $D$ is a division algebra of index $d$ endowed with a unitary involution $\tau$ (i.e., with a nontrivial restriction on the center $D$), and $K$ coincides with the field of $\tau$-invariant elements of the center $D$, $f$ is a non-degenerate $m$-dimensional Hermitian form, $U_m(D,f)$ is a unitary group of the form $f$, and $n = md-1$.
In the isotropic case the form $f$ is isotropic and there is an extensive bibliography devoted to the calculation of such groups. Passing to the anisotropic situation, we note that the Hermitian form $f$ must be anisotropic. Despite the fact that the first papers on this topic date back to the early 2000s, the study of such groups is still difficult to approach. Since these groups will play a key role in the report, we will give their precise definition.
Definition.
The group $SUK_1^{an}(D,\tau) =SU_1(D, f)/U_1(D, f)'$, where $U_1(D, f)'$ is the commutant of the group $U_1(D, f)$, is called
reduced unitary Whitehead group for the anisotropic form $f$.
The first main results related to the calculation of non-trivial reduced Whitehead groups were obtained in frame of the class of Henselian division algebras and used the idea of reducing the problem of calculating these groups to the definition of some special subgroups of the multiplicative groups for their residue algebras. The structure of finite-dimensional central Henselian algebras was firstly obtained by Platonov and Yanchevski\u{i} in 1985.
In a recent paper by the speaker [2] a scheme was proposed for calculating the groups $SUK_1^{an}(D,\tau)$ for the so-called cyclic involutions $\tau$. The aim of the talk is to generalize the results from [2] related to the case of $G$-involutions for solvable groups $G$.
References. [1] P. Gille. Le probléme de Kneser–Tits. Séminaire Boubaki. Astérisque. Vol. 326. — 2009. — Vol. 2001/2008, no. 983. — x+409 p.
[2] V. I. Yanchevskiĭ. Henselian division algebras and reduced unitary Whitehead groups for outer forms of anisotropic algebraic groups of type $A_n$. Mat. Sb. — 2022. — Vol. 213, no. 8. — P. 83–148.