Levi Subgroup Research Articles

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

Finiteness theorems for orbits of algebraic groups

Let the reductive algebraic group G act linearly on the vector space V. In this paper we prove a number of theorems to the effect that the number of orbits of a subgroup H of G on a subvariety X of V is finite. For example, let L be a Levi subgroup of G and let S be the maximum torus of the centre of L. Then we show that for every non-zero weight λ of S on Lie(G), the number of orbits of L on the weight space Lie(G) λ is finite.

R.C.A. Institutes announces television courses

In this note, we prove some results on the classification of compact complex homogeneous spaces. We first consider the case of a parallelizable space M=G/Γ, where G is a complex connected Lie group and Γ is a discrete cocompact subgroup of G. Using a generalization of results in [M. Otte, J. Potters, Manuscripta Math. 10 (1973) 117–127; D. Guan, Trans. Amer. Math. Soc. 354 (2002) 4493–4504, see also Electron. Res. Announc. Amer. Math. Soc. 3 (1997) 90], it will be shown that, up to a finite covering, G/Γ is a torus bundle over the product of two such quotients, one where G is semisimple, the other where the simple factors of the Levi subgroups of G are all of type Al. In the general case of compact complex homogeneous spaces, there is a similar decomposition into three types of building blocks.