Let $F$ be any non archimedean locally compact field of residual characteristic $p$, let $G$ be any reductive connected $F$-group and let $K$ be any special parahoric subgroup of $G(F)$. We choose a parabolic $F$-subgroup $P$ of $G$ with Levi decomposition $P=MN$ in good position with respect to $K$. Let $C$ be an algebraically closed field of characteristic $p$. We choose an irreducible smooth $C$-representation $V$ of $K$. We investigate the natural intertwiner from the compact induced representation $\ind_{K}^{G(F)}V$ to the parabolically induced representation $\Ind_{P(F)}^{G(F)}(\ind_{M(F) \cap K}^{M(F)}V_{N(F)\cap K})$. Under a regularity condition on $V$, we show that the intertwiner becomes an isomorphism after a localisation at a specific Hecke operator. When $F$ has characteristic 0, $G$ is $F$-split and $K$ is hyperspecial, the result was essentially proved by Herzig. We define the notion of $K$-supersingular irreducible smooth $C$-representation of $G(F)$ which extends Herzig's definition for admissible irreducible representations and we give a list of $K$-supersingular irreducible representations which are supercuspidal and conversely a list of supercuspidal representations which are $K$-supersingular.
We study deformations of smooth mod $p$ representations (and their duals) of a $p$-adic reductive group $G$. Under some mild genericity condition, we prove that parabolic induction with respect to a parabolic subgroup $P=LN$ defines an isomorphism between the universal deformation rings of a supersingular representation $\barσ$ of $L$ and of its parabolic induction $\barπ$. As a consequence, we show that every Banach lift of $\barπ$ is induced from a unique Banach lift of $\barσ$.
Suppose $p \geq 5$ is a prime number, and let $G = \textrm{SL}_2(\mathbb{Q}_p)$. We calculate the derived functors $\textrm{R}^n\mathcal{R}_B^G(π)$, where $B$ is a Borel subgroup of $G$, $\mathcal{R}_B^G$ is the right adjoint of smooth parabolic induction constructed by Vignéras, and $π$ is any smooth, absolutely irreducible, mod $p$ representation of $G$.
Let G be the group of rational points of a reductive connected group over a finite field (resp. nonarchimedean local field of characteristic p) and R a commutative ring. The unipotent (resp. pro-p Iwahori) invariant functor takes a smooth representation of G to a module over the unipotent (resp. pro-p Iwahori) Hecke R-algebra H of G. We prove that these functors for G and for a Levi subgroup of G commute with the parabolic induction functors, as well as with the right adjoints of the parabolic induction functors. However, they do not commute with the left adjoints of the parabolic induction fu…
We give new criteria for the irreducibility of parabolic induction on the general linear group and its inner forms over a local non-archimedean field. In particular, we give a necessary and sufficient condition when the inducing data is of the form $π\otimesσ$ where $π$ is a ladder representation and $σ$ is an arbitrary irreducible representation. As an application we simplify the proof of the classification of the unitary dual.
In 1980 Zelevinsky introduced commuting varieties whose irreducible components classify complex, irreducible representations of the general linear group over a non-archimedean local field with a given supercuspidal support. We formulate geometric conditions for certain triples of such components and conjecture that these conditions are related to irreducibility of parabolic induction. The conditions are in the spirit of the Geiss--Leclerc--Schröer condition that occurs in the conjectural characterization of $\square$-irreducible representations. We verify some special cases of the new conjectu…
We prove a result which provides a link between the decomposition of parabolically induced representations and the Bushnell--Kutzko theory of typical representations. As an application, we show that there exists a well-defined inertial Langlands correspondence which respects the monodromy action of L-parameters, under some standard conjectures regarding the local Langlands correspondence. To allow for potential applications of this inertial Langlands correspondence, we also provide a complete construction of the set of typical representations, giving a parametrization of these in terms of the …
In this paper we study a generalization of the Jacquet module of a parabolic induction and construct a filtration on it. The successive quotient of the filtration is written by using the twisting functor.
We prove that the parabolic induction functor on BGG-category $\mathcal{O}$ associated to a complex reductive Lie algebra is gradable, that is, lifts to graded category $\mathcal{O}$ as constructed by Beilinson-Ginzburg-Soergel. Graded category $\mathcal{O}$ is equivalent to a category of stratified mixed Tate motives on a corresponding flag variety as recently defined by Soergel-Wendt. The graded version of parabolic induction is induced by a geometric parabolic induction functor we construct on the level of stratified mixed Tate motives. We also describe the effect of parabolic induction on …
Let $\mathbf{G}$ be a connected reductive group defined over a locally compact non-archimedean field $F$, let $\mathbf{P}$ be a parabolic subgroup with Levi $\mathbf{M}$ and compatible with a pro-$p$ Iwahori subgroup of $G := \mathbf{G}(F)$. Let $R$ be a commutative unital ring. We introduce the parabolic pro-$p$ Iwahori--Hecke $R$-algebra $\mathcal{H}_R(P)$ of $P := \mathbf{P}(F)$ and construct two $R$-algebra morphisms $Θ^P_M\colon \mathcal{H}_R(P)\to \mathcal{H}_R(M)$ and $Ξ^P_G\colon \mathcal{H}_R(P) \to \mathcal{H}_R(G)$ into the pro-$p$ Iwahori--Hecke $R$-algebra of $M := \mathbf{M}(F)$ …
We study the structure of parabolic inductions of a pro-$p$-Iwahori Hecke algebra. In particular, we give a classification of irreducible modulo $p$ representations of pro-$p$-Iwahori Hecke algebra in terms of supersingular representations. Since supersingular representations are classified by Ollivier and Vigneras, it completes the classification of irreducible modulo $p$ representations.
The classical parabolic induction functor is a fundamental tool on the representation theoretic side of the Langlands program. In this article, we study its derived version. It was shown by the second author that the derived category of smooth $G$-representations over $k$, $G$ a $p$-adic reductive group and $k$ a field of characteristic $p$, is equivalent to the derived category of a certain differential graded $k$-algebra $H_G^\bullet$, whose zeroth cohomology is a classical Hecke algebra. This equivalence predicts the existence of a derived parabolic induction functor on the dg Hecke algebra…
Given a supercuspidal representation $σ$ of a parabolic subgroup $P$ of reductive group $G$, we discover a universal hierarchical structure of reducibility of the parabolic induction $Ind^G_P(σ)$, i.e. always irreducible from some Levi-level up. As its applications, we provide a new simple proof of the generic irreducibility property of parabolic induction, and prove Clozel's finiteness conjecture of special exponents under some conditions. Indeed, those conditions are predicted by two conjectures of Shahidi which in some sense are proved for classical groups by Arthur in his monumental book--…
Let $G$ be a $p$-adic reductive group. We determine the extensions between admissible smooth mod $p$ representations of $G$ parabolically induced from supersingular representations of Levi subgroups of $G$, in terms of extensions between representations of Levi subgroups of $G$ and parabolic induction. This proves for the most part a conjecture formulated by the author in a previous article and gives some strong evidence for the remaining part. In order to do so, we use the derived functors of the left and right adjoints of the parabolic induction functor, both related to Emerton's $δ$-functor…
In this note we construct a "restriction" map from the cocenter of a reductive group G over a local non-archimedean field F to the cocenter of a Levi subgroup. We show that the dual map corresponds to parabolic induction and deduce that parabolic induction preserves stability. We also give a new (purely geometric) proof that the character of normalized parabolic induction does not depend on a parabolic subgroup. In the appendix, we use a similar argument to extend a theorem of Lusztig-Spaltenstein on induced unipotent classes to all infinite fields.
We study some aspects of the functor of parabolic induction within the context of reduced group C*-algebras and related operator algebras. We explain how Frobenius reciprocity fits naturally within the context of operator modules, and examine the prospects for an operator algebraic formulation of Bernstein's reciprocity theorem (his second adjoint theorem).
Given a (complex, smooth) irreducible representation $π$ of the general linear group over a non-archimedean local field and an irreducible supercuspidal representation $σ$ of a classical group, we show that the (normalized) parabolic induction $π\rtimesσ$ is reducible if there exists $ρ$ in the supercuspidal support of $π$ such that $ρ\rtimesσ$ is reducible. In special cases we also give irreducibility criteria for $π\rtimesσ$ when the above condition is not satisfied.
Ariki and Ginzburg, after the previous work of Zelevinsky on orbital varieties, proved that multiplicities in a total parabolically induced representations are given by the value at q=1 of Kazhdan-Lusztig Polynomials associated to the symmetric groups. In my thesis I explore the geometry of orbital varieties and I essentially obtain two important results: - first I prove a conjecture of Zelevinksy on a property of independence of total parabolically induced representations. - More crucially I give a strategy to compute multiplicities in general parabolically induced representations using the p…
Let G be a reductive group and L a Levi subgroup. Parabolic induction and restriction are a pair of adjoint functors between Ad-equivariant derived categories of either constructible sheaves or (not necessarily holonomic) D-modules on G and L, respectively. Bezrukavnikov and Yom Din proved, generalizing a classic result of Lusztig, that these functors are exact. In this paper, we consider a special case where L=T is a maximal torus. We give explicit formulas for parabolic induction and restriction in terms of the Harish-Chandra D-module on G x T. We show that this module is flat over D(T), whi…
We construct algebra homomorphisms from affine Yangians to the current algebras of rectangular $W$-algebras both in type A. The construction is given via the coproduct and the evaluation map for the affine Yangians. As a consequence, we show that parabolic inductions for representations of the rectangular $W$-algebras can be regarded as tensor product representations of the affine Yangians under the homomorphisms. The same method is applicable also to the super setting.
This paper is about the reduced group C*-algebras of real reductive groups, and about Hilbert C*-modules over these C*-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*-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*-algebra to determine the structure of the Hilbert C*-bimodule that represents the functor of parabolic induction. Third, we sha…
For a complex reductive Lie group G with Lie algebra g, Cartan subalgebra h and Weyl group W, we describe the category of perverse sheaves on h/W smooth w.r.t the natural stratification. The answer is given in terms of mixed Bruhat sheaves, which are certain mixed sheaf-cosheaf data on cells of a natural cell decomposition of h/W. Using the parabolic Bruhat decomposition, we relate mixed Bruhat sheaves with the properties of various procedures of parabolic induction and restriction that connect different Levi subgroups in G.
A W-algebra is an associative algebra constructed from a semisimple Lie algebra and its nilpotent element. This paper concentrates on the study of 1-dimensional representations of these algebras. Under some conditions on a nilpotent element (satisfied by all rigid elements) we obtain a criterium for a finite dimensional module to have dimension 1. It is stated in terms of the Brundan-Goodwin-Kleshchev highest weight theory. This criterium allows to compute highest weights for certain completely prime primitive ideals in universal enveloping algebras. We make an explicit computation in a specia…
In arXiv:2011.03313, the author has constructed a category of abstractly automorphic representations for $\mathrm{GL}(2)$ over a function field $F$. This is a symmetric monoidal Abelian category, constructed with the goal of having the irreducible automorphic representations as its simple objects. The goal of this paper is to systematically study this category. We will prove several structural theorems about this category. We will show that it admits an adjoint pair $(r^\mathrm{aut},i^\mathrm{aut})$ of automorphic parabolic restriction and induction functors, respectively. This will allow us t…
Let $G$ be a connected semisimple Lie group with a finite center, with a maximal compact subgroup $K$. There are two general methods to construct representations of $G$, namely, ordinary parabolic induction and cohomological parabolic induction. We define Eisenstein integrals relative to cohomological inductions which generalize Flensted-Jensen's fundamental functions for discrete series. They are analogous to Harish-Chandra's Eisenstein integrals related to ordinary inductions. We introduce the notion of Li positivity of a $K$ type in a representation of $G$ which is extremely useful in the s…
Let $F$ be a non archimedean local field, and $n_1$ and $n_2$ two positive even integers. We prove that if $π_1$ and $π_2$ are two smooth representations of $GL(n_1,F)$ and $GL(n_2,F)$ respectively, both admitting a Shalika period, then the normalised parabolically induced representation $π_1\times π_2$ also admits a Shalika period. Combining this with the results of \cite{M-localBF}, we obtain as a corollary the classification of generic representations of $GL(n,F)$ admitting a Shalika period when $F$ has characteristic zero. This result is relevant to the study of the Jacquet-Shalika exterio…
(Affine) $\mathcal{W}$-algebras are a family of vertex algebras defined by the generalized Drinfeld-Sokolov reductions associated with a finite-dimensional reductive Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, a nilpotent element $f$ in $[\mathfrak{g},\mathfrak{g}]$, a good grading $Γ$ and a symmetric invariant bilinear form $κ$ on $\mathfrak{g}$. We introduce free field realizations of $\mathcal{W}$-algebras by using Wakimoto representations of affine Lie algebras, where $\mathcal{W}$-algebras are described as the intersections of kernels of screening operators. We call these Wakimoto free …
We study the reduction modulo $l$ of some elliptic representations; for each of these representations, we give a particular lattice naturally obtained by parabolic induction in giving the graph of extensions between its irreducible sub-quotient of its reduction modulo $l$. The principal motivation for this work, is that these lattices appear in the cohomology of Lubin-Tate towers.
Following Lusztig, we consider a Coxeter group $W$ together with a weight function. Geck showed that the Kazhdan-Lusztig cells of $W$ are compatible with parabolic subgroups. In this paper, we generalize this argument to some subsets of $W$ which may not be parabolic subgroups. We obtain two applications: we show that under specific technical conditions on the parameters, the cells of a certain finite parabolic subgroup of $W$ are cells in the whole group, and we decompose the affine Weyl group $\tilde{G}_{2}$ into left and two-sided cells for a whole class of weight functions.
Let $p$ be a prime number. We show in this short note that mod-$p$ maximal compact inductions of a $p$-adic split reductive group do not have irreducible admissible subrepresentations.
We use geometric parabolic induction functors and the adjoint functors for the supergroups Osp(2m+1,2n) (where m and n vary) to categorify the action of the infinite-dimensional Clifford algebra on the Fock space of semi-infinite forms.
We introduce parabolic induction and restriction functors for rational Cherednik algebras, and study their basic properties. Then we discuss applications of these functors to representation theory of rational Cherednik algebras. In particular, we prove the Gordon-Stafford theorem about Morita equivalence of the rational Cherednik algebra for type A and its spherical subalgebra, without the assumption that c is not a half-integer, which was required up to now. Also, we classify representations from category O over the rational Cherednik algebras of type A which do not contain an S_n-invariant v…
We study cells with respect to the $p$-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see arXiv:1510.01556, arXiv:1901.02323) and their compatibility with standard parabolic subgroups. We show that after induction to the surrounding bigger Coxeter group the cell module of a right $p$-cell in a standard parabolic subgroup decomposes as a direct sum of cell modules. Along the way, we state some new positivity properties of the $p$-canonical basis.
We prove an Induction Equivalence and a Kashiwara Equivalence for coadmissible equivariant D-modules on rigid analytic spaces. This allows us to completely classify such objects with support in a single orbit of a classical point with co-compact stabiliser. As an application, we use the locally analytic Beilinson-Bernstein equivalence to construct new examples of large families of topologically irreducible locally analytic representations of certain compact semisimple p-adic Lie groups.
In this article, we consider the links between parabolic induction and the local Langlands correspondence. We enunciate a conjecture about the (enhanced) Langlands parameters of supercuspidal representation of split reductives $p$-adics groups. We are able to verify this in those known cases of the local Langlands correspondence for linear groups and classical groups. Furthermore, in the case of classical groups, we can construct the "cuspidal support" of an enhanced Langlands parameter and get a decomposition of the set of enhanced Langlands parameters a la Bernstein. We check that these cons…
Let G be a connected reductive group defined over a finite field F_q and let L be the Levi subgroup (defined over F_q) of a parabolic subgroup P of G. We define a linear map from class functions on L(F_q) to class functions on G(F_q). This map is independent of the choice of P. We show that for large q this map coincides with the known cohomological induction (whose definition involves a choice of P).
We construct a new family of irreducible representations of $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ and its modular double by quantizing the classical parabolic induction corresponding to arbitrary parabolic subgroups, such that the generators of $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ act by positive self-adjoint operators on a Hilbert space. This generalizes the well-established positive representations which corresponds to induction by the minimal parabolic (i.e. Borel) subgroup. We also study in detail the special case of type $A_n$ acting on $L^2(\mathbb{R}^n)$ with minimal functional d…
In a previous work, we have shown that a representation of a $p$-adic group obtained by (normalized) parabolic induction from an irreducible supercuspidal representation $σ$ of a Levi subgroup $M$ contains a subquotient which is square integrable, if and only if Harish-Chandra's $μ$-function has a pole in $σ$ of order equal to the parabolic rank of $M$. The aim of the present article is to interpret this result in terms of Langlands' functoriality principle.
Let G be a compact Lie group. We present two induction theorems for certain generalized G-equivariant cohomology theories. The theory applies to G-equivariant K-theory K_G, and to the Borel cohomology associated to any complex oriented cohomology theory. The coefficient ring of K_G is the representation ring R(G) of G. When G is a finite group the induction theorems for K_G coincide with the classical Artin and Brauer induction theorems for R(G).
We establish duality theory of p-adic unitary Banach representations of locally profinite groups. This is an extension of Iwasawa theory for profinite groups by P. Schneider and J. Teitelbaum. We also establish a criterion for an irreducibility of a unitary representation by certain simpleness of the Iwasawa module dual to it. Through the duality, the continuous induction of a unitary Banach representation of a closed subgroup is interpreted as a certain induction of Iwasawa module. It gives an explicit description of the dual of a continuous parabolic induction on GL_n(Q_p).
Irreducibility results for parabolic induction of representations of the general linear group over a local non-archimedean field can be formulated in terms of Kazhdan--Lusztig polynomials of type $A$. Spurred by these results and some computer calculations, we conjecture that certain alternating sums of Kazhdan--Lusztig polynomials known as parabolic Kazhdan--Lusztig polynomials satisfy properties analogous to those of the ordinary ones.
In this paper, we shall establish the Mackey formulas in the following two set ups: (i) on the tensor induction and restriction functors on the modules over cyclotomic Hecke algebras (Ariki-Koike algebras) and their standard subalgebras of parabolic subgroups. (ii) on the Bezrukavnikov-Etingof induction and restriction functors among categories O of rational Cherednik algebras for the complex reflection group of type G(r, 1, n) and their parabolic subgroups.
If $G$ is a group acting on a tree $X$, and ${\mathcal S}$ is a $G$-equivariant sheaf of vector spaces on $X$, then its compactly-supported cohomology is a representation of $G$. Under a finiteness hypothesis, we prove that if $H_c^0(X, {\mathcal S})$ is an irreducible representation of $G$, then $H_c^0(X, {\mathcal S})$ arises by induction from a vertex or edge stabilizing subgroup. If $G$ is a reductive group over a nonarchimedean local field $F$, then Schneider and Stuhler realize every irreducible supercuspidal representation of $G(F)$ in the degree-zero cohomology of a $G(F)$-equivariant …
