Bhama Srinivasan

Brauer trees in classical groups

where the prime Y for the modular representation theory is distinct from the prime p dividing q and p is odd. The possible graphs occurring as Brauer trees of finite classical groups were described by Feit [S]. In this paper we complete his description by identifying the vertices with characters. An explicit description of Brauer trees for CL,(q) was given in [S]. We may suppose r > 2, since the trees have trivial structure for r = 2. The Jordan decomposition of characters is compatible with blocks and induces graph isomorphisms of Brauer trees for cyclic blocks. This reduces the problem to one of constructing the projective indecomposable characters in a cyclic block B where the non-exceptional characters of B are unipotent characters in the sense of Deligne and Lusztig. The projective indecomposable characters in such unipotent cyclic blocks are most readily constructed by Frobenius induction from proper subgroups. In the context of classical groups, this effectively means Harish-Chandra induction from subparabolic subgroups. Despite this relative paucity of means, the tree of B can be determined using combinatorial arguments on the partitions or symbols labeling the unipotent characters in B. In every case the tree has the form M....-.(YJ-*...OL 02 ai T, 72 1,

On the Steinberg Character of a Finite Simple Group of Lie Type

Let K be an algebraically closed field of characteristic ρ >0. If G is a connected, simple connected, semisimple linear algebraic group defined over K and σ an endomorphism of G onto G such that the subgroup G σ of fixed points of σ is finite, Steinberg ([6] [7]) has shown that there is a complex irreducible character χ of G σ with the following properties. χ vanishes at all elements of G σ which are not semi- simple, and, if x ∈ G is semisimple, χ(x) = ± n ( x ) where n ( x )is the order of a Sylow p-subgroup of ( Z G ( x )) σ ( Z G ( x ) is the centraliser of x in G ). If G is simple he has, in [6], identified the possible groups G σ they are the Chevalley groups and their twisted analogues over finite fields, that is, the ‘simply connected’ versions of finite simple groups of Lie type. In this paper we show, under certain restrictions on the type of the simple algebraic group G an on the characteristic of K , that χ can be expressed as a linear combination with integral coefficients of characters induced from linear characters of certain naturally defined subgroups of G σ . This expression for χ gives an explanation for the occurence of n ( x ) in the formula for χ ( x ), and also gives an interpretation for the ± 1 occuring in the formula in terms of invariants of the reductive algebraic group Z G ( x ).

Generalized Harish-Chandra theory for unipotent characters of finite classical groups

The Harish-Chandra theory for a finite group G of Lie type states the following: If p is an irreducible character of G, then there is a parabolic subgroup P of G, a Levi decomposition Lb’ of P, and a cuspidal character II/ of the Levi subgroup such that p is a constituent of the induced character indg(

Isometries in finite groups of Lie type

) of the pullback

Jordan decomposition and real-valued characters of finite reductive groups with connected center

of

Modular Representations, Old and New

to P. Furthermore, the pair (L,

Note from the book reviews editor

) is uniquely determined by p up to conjugacy in G. In the case where G is a classical group, that is, a general linear, unitary, symplectic, or orthogonal group over cFq, we generalize this theory relative to an odd prime r different from the defining characteristic. There is an integer e and a polynomial 4(X) of the form x’1 or Y’+ 1 with r dividing c

The blocks of finite general linear and unitary groups

(q) such that the following holds: To each unipotent character p of G corresponds a pair (L, Ic/), where L is a regular subgroup of G of the form a product of a classical group and cyclic tori of order 4(q), Ic, is a unipotent character of L of degree divisible by the full power of r dividing I L: Z(L)l, and p is a constituent of the virtual character Rf(IC/). Moreover, the pair (L,

The blocks of finite classical groups.

) is determined by p up to conjugacy in G. The Harish-Chandra theory is the case e = I. This work arose in an investigation by the authors of the Brauer r-blocks of G. A similar generalization of the Harish-Chandra theory has been proposed by R. Boyce.

The characters of the finite unitary groups

Publisher Summary This chapter discusses the isometries in finite groups of lie type. Let G be a connected semisimple linear algebraic group and σ a surjective endomorphism of G, such that the group of fixed points G σ = G is finite. Then G is a finite group of Lie type. Let T be a maximal torus of G fixed by σ and T = T, H = N G (T), H/T = W. Then T is an abelian subgroup of G. Any subgroup of G arising in this way is known as a “torus” of G. An irreducible character of T or an element of T is regular if it is not fixed by any non-identity element of W. It has been conjectured that there is a family of irreducible characters of T.