Unitary Groups, Unitary Steinberg Groups, and Unitary K-Groups

We are concerned with finite-dimensional classical groups over arbitrary commutative fields. In an orthogonal group a Siegel transvection, that is, an element centralizing l⊥ for some totally singular 2-dimensional subspace l, plays the same role as a transvection in the linear, symplectic or unitary groups. The Main Theorem of this paper describes the possible embeddings of classical groups in classical groups such that (Siegel) transvections act as (Siegel) transvections.

By the methods used heretofore for the determination of the automorphisms of certain families of linear groups, for example, the (projective) unimodular, orthogonal, symplectic, and unitary groups (7, 8), it has been necessary to consider the various families separately and to give many case-by-case discussions, especially when the underlying vector space has few elements, even though the final results are very much the same for all of the groups. The purpose of this article is to give a completely uniform treatment of this problem for all the known finite simple linear groups (listed in §2 below). Besides the * ‘classical groups” mentioned above, these include the “exceptional groups,” considered over the complex field by Cartan and over an arbitrary field by Dickson, Chevalley, Hertzig, and the author (3, 4, 5, 6, 10, 15). The automorphisms of the latter groups are given here for the first time.

On imprimitive representations of finite reductive groups in non-defining characteristic

Classical groups over finite fields, including symplectic groups, unitary groups and orthogonal groups, are finite simple groups of crucial importance. The structure and order of classical groups are the basic problems in group theory. Geometry of classical groups has been found extremely useful in many fields, such as combinatorial mathematics, coding and cryptographic theory. In this paper, a class of quasi-classical groups are introduced, which partially generalize the concepts of symplectic groups and orthogonal groups. We determine the structure and order of quasi-classical groups. As an application, we discuss the relationship between quasi-classical groups and self-dual normal bases over finite fields.

The spectrum of a finite group is the set of its element orders. An arithmetic criterion determining whether a given natural number belongs to a spectrum of a given group is furnished for all finite special, projective general, and projective special linear and unitary groups.

Given an arbitrary set π of primes, we complete the description of the finite linear and unitary groups having the D π-property.

Let G be a finite group and let cd (G) be the set of irreducible character degrees of G. The degree graph Δ(G) is the graph whose set of vertices is the set of primes that divide degrees in cd (G), with an edge between p and q if pq divides a for some degree a ∈ cd (G). We determine the graph Δ(G) for the finite simple groups of types A ℓ(q) and 2 A ℓ (q 2), that is, for the simple linear and unitary groups.

We show a connection between Lusztig induction operators in finite general linear and unitary groups and parabolic induction in cyclotomic rational double affine Hecke algebras. Two applications are given: an explanation of a bijection result of Broue, Malle and Michel, and some results on modular decomposition numbers of finite general groups. 2010 AMS Subject Classification: 20C33 Dedicated to the memory of Robert Steinberg

Categories of two-colored pair partitions Part II: Categories indexed by semigroups

Maximal Overgroups of Singer Elements in Classical Groups

In this paper we deduce a graded version of Quillen–Suslin’s Local-Global Principle for the traditional classical groups, viz. general linear, symplectic and orthogonal groups and establish its equivalence of the normality property of the respective elementary subgroups. This generalizes previous result of Basu–Rao–Khanna. Then, as an application, we establish an analogue Local-Global Principle for the commutator subgroups of the special linear and symplectic groups. Finally, by using Swan–Weibel’s homotopy trick, we establish graded analogue of the Local-Global Principle for the transvection subgroups of the full automorphism groups of the linear, symplectic and orthogonal modules. This generalizes the previous result of Bak–Basu–Rao.

In this paper we prove isomorphisms between 5 Lie groups (of arbitrary dimension and fixed signatures) in Clifford algebra and classical matrix Lie groups - symplectic, orthogonal and linear groups. Also we obtain isomorphisms of corresponding Lie algebras.

We compute a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. In this special case, we require the given supercuspidal representation to contain a skew maximal simple stratum, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a canonical lifting of its underlying simple character, together with the base change of the level-zero component of its inducing cuspidal type, modified by a sign attached to a quadratic Gauss sum defined by the internal structure of the simple character. To obtain this result, we study the reducibility points of a parabolic induction and the corresponding module over the affine Hecke algebra, defined by the covering type over the product of types of the given supercuspidal representation and of a candidate of its base change.

By a classical group we mean one of the groups G L n ( R ) G{L_n}(R) , G L n ( C ) G{L_n}(C) , G L n ( H ) G{L_n}(H) , U ( p , q ) U(p,\,q) , O n ( C ) {O_n}(C) , O ( p , q ) O(p,\,q) , S O ∗ ( 2 n ) S{O^{\ast }}(2n) , S p 2 n ( C ) S{p_{2n}}(C) , S p 2 n ( R ) S{p_{2n}}(R) , or S p ( p , q ) Sp(p,\,q) . Let G G be a classical group and L L its Lie algebra. For each x ∈ L x \in L we determine the closure of the orbit G ⋅ x G \cdot x (for the adjoint action of G G on L L ). The problem is first reduced to the case when x x is nilpotent. By using the exponential map we also determine the closures of conjugacy classes of G G .

Let M denote the set of the simple 3-dimensional unitary groups U3 and the simple linear groups L2 over finite fields of odd characteristic.We prove that each periodic group saturated with groups in M is locally finite and isomorphic to either U3(Q) or L2(Q) for a suitable locally finite field Q of odd characteristic.