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.

Let $E$ be a (right) Hilbert module over a $C^{\ast }$-algebra $A$. If $E$ is equipped with a left action of a second $C^{\ast }$-algebra $B$, then tensor product with $E$ gives rise to a functor from the category of Hilbert $B$-modules to the category of Hilbert $A$-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in Clare et al. [Parabolic induction and restriction via $C^{\ast }$-algebras and Hilbert $C^{\ast }$-modules, Compos. Math.FirstView (2016), 1–33, 2].

This paper is about the reduced group $C^{\ast }$-algebras of real reductive groups, and about Hilbert $C^{\ast }$-modules over these $C^{\ast }$-algebras. We shall do three things. First, we shall apply theorems from the tempered representation theory of reductive groups to determine the structure of the reduced $C^{\ast }$-algebra (the result has been known for some time, but it is difficult to assemble a full treatment from the existing literature). Second, we shall use the structure of the reduced $C^{\ast }$-algebra to determine the structure of the Hilbert $C^{\ast }$-bimodule that represents the functor of parabolic induction. Third, we shall prove that the parabolic induction bimodule admits a secondary inner product, using which we can define a functor of parabolic restriction in tempered representation theory. We shall prove in a sequel to this paper that parabolic restriction is adjoint, on both the left and the right, to parabolic induction in the context of tempered unitary Hilbert space representations.

We show that the modules for the Frobenius kernel of a reductive algebraic group over an algebraically closed field of positive characteristic $p$ induced from the $p$-regular blocks of its parabolic subgroups can be $\mathbb{Z}$-graded. In particular, we obtain that the modules induced from the simple modules of $p$-regular highest weights are rigid and determine their Loewy series, assuming the Lusztig conjecture on the irreducible characters for the reductive algebraic groups, which is now a theorem for large $p$. We say that a module is rigid if and only if it admits a unique filtration of minimal length with each subquotient semisimple, in which case the filtration is called the Loewy series.

Let $F$ be a local non-archimedean field of characteristic zero. We prove that a representation of $GL\left( n,\,F \right)$ obtained from irreducible parabolic induction of supercuspidal representations is distinguished by an orthogonal group only if the inducing data is distinguished by appropriate orthogonal groups. As a corollary, we get that an irreducible representation induced from supercuspidals that is distinguished by an orthogonal group is metic.

Let F be a p-adic field. Consider a dual pair where SO(2n+1)+ is the split orthogonal group and is the metaplectic cover of the symplectic group Sp(2n) over F. We study lifting of characters between orthogonal and metaplectic groups. We say that a representation of SO(2n+1)+ lifts to a representation of if their characters on corresponding conjugacy classes are equal up to a transfer factor. We study properties of this transfer factor, which is essentially the character of the difference of the two halves of the oscillator representation. We show that the lifting commutes with parabolic induction. These results were motivated by the paper ‘Lifting of characters on orthogonal and metaplectic groups’ by Adams who considered the case F=ℝ.

We study basic properties of the category of smooth representations of a p-adic group G with coefficients in any commutative ring R in which p is invertible. Our main purpose is to prove that Hecke algebras are Noetherian whenever R is; this question arose naturally with Bernstein's fundamental work for R = ℂ, in which case he proved this Noetherian property. In a first step, we prove that Noetherianity would follow from a generalization of the so-called second adjointness property between parabolic functors, also due to Bernstein for complex representations. Then, to attack this second adjointness, we introduce and study ‘parahoric functors’ between representations of groups of integral points of smooth integral models of G and of their ‘Levi’ subgroups. Applying our general study to Bruhat-Tits parahoric models, we get second adjointness for minimal parabolic groups. For non-minimal parabolic subgroups, we have to restrict to classical and linear groups, and use smooth models associated with Bushnell-Kutzko and Stevens semi-simple characters. The same strategy should apply to ‘tame’ groups, using Yu's smooth models and generic characters.

For the groups $G={\mathrm{Sp}}(p,q),\ \mathrm{SO}^\ast(2n)$, and $\mathrm{U}(m,n)$, we consider degenerate principal series whose infinitesimal character coincides with a finite-dimensional representation of $G$. We prove that each irreducible constituent of maximal Gelfand–Kirillov dimension is a derived functor module. We also show that at an appropriate ‘most singular’ parameter, each irreducible constituent is weakly unipotent and unitarizable. Conversely we show that any weakly unipotent representation associated to a real form of the corresponding Richardson orbit is unique up to isomorphism and can be embedded into a degenerate principal series at the most singular integral parameter (apart from a handful of very even cases in type D). We also discuss edge-of-wedge-type embeddings of derived functor modules into degenerate principal series.

We obtain a decomposition formula of a representation of Sp(p, q) or SO*(2n) unitarily induced from a derived functor module, which enables us to reduce the problem of irreducible decompositions to the study of derived functor modules. In particular, we show that such an induced representation is decomposed into a direct sum of irreducible unitarily induced modules from derived functor modules under some regularity condition on the parameters. In particular, representations of SO*(2n) and Sp(p, q) induced from one-dimensional unitary representations of their parabolic subgroups are irreducible.

We study the structure of generalized Verma modules over a semi-simple complex finite-dimensional Lie algebra, which are induced from simple modules over a parabolic subalgebra. We consider the case when the annihilator of the starting simple module is a minimal primitive ideal if we restrict this module to the Levi factor of the parabolic subalgebra. We show that these modules correspond to proper standard modules in some parabolic generalization of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ and prove that the blocks of this parabolic category are equivalent to certain blocks of the category of Harish-Chandra bimodules. From this we derive, in particular, an irreducibility criterion for generalized Verma modules. We also compute the composition multiplicities of those simple subquotients, which correspond to the induction from simple modules whose annihilators are minimal primitive ideals.

We use some fundamental work of Bernstein to study parabolic induction in reductive p-adic groups. In particular, we determine when parabolic induction from a component of the Bernstein decomposition of a Levi subgroup to the corresponding component of the full group is an equivalence of categories.

Let G be a connected reductive group defined over a local non Archimedean field F with residue field F; let P be a parahoric subgroup with associated reductive quotient M. If σ is an irreducible cuspidal representation of M(F) it provides an irreducible representation of P by inflation. We show that the pair (P,σ) is an ${\mathfrak S}$-type as defined by Bushnell and Kutzko. The cardinality of${\mathfrak S}$ can be bigger than one; we show that if one replaces P by the full centraliser ${\hat P}$ of the associated facet in the enlarged affine building of G, and σ by any irreducible smooth representation ${\hat σ}$ of ${\hat P}$ which contains σ on restriction then (${\hat P}$,${\hat σ}$) is an ${\mathfrak s}$-type for a singleton set ${\mathfrak s}$. Our methods employ invertible elements in the associated Hecke algebra${\mathcal H}$ (σ) and they imply that the appropriate parabolic induction functor and its left adjoint can be realised algebraically via pullbacks from ring homomorphisms.