Harish-Chandra Induction Research Articles

EXTENSIONS OF CHARACTERS IN TYPE D AND THE INDUCTIVE MCKAY CONDITION, I

AbstractThis is a contribution to the study of$\mathrm {Irr}(G)$as an$\mathrm {Aut}(G)$-set forGa finite quasisimple group. Focusing on the last open case of groups of Lie type$\mathrm {D}$and$^2\mathrm {D}$, a crucial property is the so-called$A'(\infty )$condition expressing that diagonal automorphisms and graph-field automorphisms ofGhave transversal orbits in$\mathrm {Irr}(G)$. This is part of the stronger$A(\infty )$condition introduced in the context of the reduction of the McKay conjecture to a question about quasisimple groups. Our main theorem is that a minimal counterexample to condition$A(\infty )$for groups of type$\mathrm {D}$would still satisfy$A'(\infty )$. This will be used in a second paper to fully establish$A(\infty )$for any type and rank. The present paper uses Harish-Chandra induction as a parametrization tool. We give a new, more effective proof of the theorem of Geck and Lusztig ensuring that cuspidal characters of any standard Levi subgroup of$G=\mathrm {D}_{ l,\mathrm {sc}}(q)$extend to their stabilizers in the normalizer of that Levi subgroup. This allows us to control the action of automorphisms on these extensions. From there, Harish-Chandra theory leads naturally to a detailed study of associated relative Weyl groups and other extendibility problems in that context.

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

In this paper, we begin with the classification of Harish-Chandra imprimitive representations in non-defining characteristic. We recall the connection of this problem to certain generalizations of Iwahori-Hecke algebras and show that Harish-Chandra induction is compatible with the Morita equivalence by Bonnafé and Rouquier, thus reducing the classification problem to quasi-isolated blocks. Afterwards, we consider imprimitivity of unipotent representations of certain classical groups. In the case of general linear and unitary groups, our reduction methods then lead to results for arbitrary Lusztig series.

Categorical actions on unipotent representations of finite unitary groups

Using Harish-Chandra induction and restriction, we construct a categorical action of a Kac-Moody algebra on the category of unipotent representations of finite unitary groups in non-defining characteristic. We show that the decategorified representation is naturally isomorphic to a direct sum of level 2 Fock spaces. From our construction we deduce that the Harish-Chandra branching graph coincides with the crystal graph of these Fock spaces, solving a recent conjecture of Gerber-Hiss-Jacon. We also obtain derived equivalences between blocks, yielding Broue’s abelian defect group conjecture for unipotent \(\ell \)-blocks at linear primes \(\ell \).

A VARIANT OF HARISH-CHANDRA FUNCTORS

Harish-Chandra induction and restriction functors play a key role in the representation theory of reductive groups over finite fields. In this paper, extending earlier work of Dat, we introduce and study generalisations of these functors which apply to a wide range of finite and profinite groups, typical examples being compact open subgroups of reductive groups over non-archimedean local fields. We prove that these generalisations are compatible with two of the tools commonly used to study the (smooth, complex) representations of such groups, namely Clifford theory and the orbit method. As a test case, we examine in detail the induction and restriction of representations from and to the Siegel Levi subgroup of the symplectic group $\text{Sp}_{4}$ over a finite local principal ideal ring of length two. We obtain in this case a Mackey-type formula for the composition of these induction and restriction functors which is a perfect analogue of the well-known formula for the composition of Harish-Chandra functors. In a different direction, we study representations of the Iwahori subgroup $I_{n}$ of $\text{GL}_{n}(F)$, where $F$ is a non-archimedean local field. We establish a bijection between the set of irreducible representations of $I_{n}$ and tuples of primitive irreducible representations of smaller Iwahori subgroups, where primitivity is defined by the vanishing of suitable restriction functors.

Characters of odd degree

We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verication of the inductive McKay condition for groups of Lie type and primes ‘ such that a Sylow ‘-subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction. Specics of characters of odd degree, namely, that most of them lie in the principal Harish-Chandra series, then allow us to deduce from it the McKay conjecture for the prime 2, hence for characters of odd degree.

Stabilizing bisets

Let G be a finite group and let R be a commutative ring. We analyze the ( G , G ) -bisets which stabilize an indecomposable RG-module. We prove that the minimal ones are unique up to equivalence. This result has the same flavor as the uniqueness of vertices and sources up to conjugation and resembles also the theory of cuspidal characters in the context of Harish-Chandra induction for reductive groups, but it is different and very general. We show that stabilizing bisets have rather strong properties and we explore two situations where they occur. Moreover, we prove some specific results for simple modules and also for p-groups.

Green Vertex Theory, Green Correspondence, and Harish-Chandra Induction

In modular representation theory of a finite group the theory of Green vertices plays an important role. In the representation theory of finite groups with split BN-pairs Harish-Chandra induction has become one of the most important tools in the last decades. This paper will show how the combination of both gives new insight into the modular representation theory of these groups.

On Harish-Chandra Induction and Restriction for Modules of Levi Subgroups

On Harish-Chandra Induction and Restriction for Modules of Levi Subgroups

A note on harish-chandra induction

R.B.Howlett and G.I.Lehrer obtained a detailed description of the endo-morphism algebra of an induced cuspidal representation of a reductive group over a finite field. They conjectured (and proved in many cases) that this algebra is always isomorphic to the group algebra of a certain finite group. This conjecture is equivalent to the statement that the induced representation contains an irreducible constituent with multi-plicty 1. G.Lusztig has shown that this is true for groups with connected centre. In this note, we characterize such an irreducible constituent with multiplicity I by its degree and deduce from this result, using regular embeddings and Clifford theory, that Howlett's and LehreFs conjecture holds in general.