A Variant of Harish-Chandra Functors

Abstract

ABSTRACT. 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 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 In of GLnpFq, where F is a non-archimedean local field. We establish a bijection between the set of irreducible representations of In and tuples of primitive irreducible representations of smaller Iwahori subgroups, where primitivity is defined by the vanishing of suitable restriction functors.

Combinatorial Hopf algebras from representations of families of wreath products

“…The cocycles appearing in [7,Theorem 3.14] are trivial in this instance. All of these identifications being made, the functor i appearing in [7,Theorem 3.6] is the functor i K L K,L , while the functor i ϕ U(ϕ),V(ϕ) appearing in [7,Theorem 3.14] is the usual induction functor Rep(Aut…”

“…The induction functors that we shall consider below are, in this example, the analogues of the functors used to study the representations of GL n (Z/p k Z) in [7], [9], and [8].…”

“…to designate the object called G (etc. ) in [7,Section 3].) We then have, still in the notation of [7],…”

“…. Pasting together the two commuting diagrams from [7,Theorems 3.6 & 3.14] then yields the commuting diagram (10).…”

“…Observing that Y,H with a tensor product of copies of the Hopf algebra Sym Z of symmetric functions, indexed by the set of primitive irreducible elements of M δ Y,H . (Here irreducible elements are, by definition, elements of the distinguished basis (7).) The set of primitive irreducibles can readily be identified from the formula (13) for the comultiplication δ M .…”