We study degenerate principal series representations of the split real group $G_{2(2)}$ induced from a character of a maximal parabolic subgroup whose unipotent radical is a Heisenberg group. Using the Lie algebra action on the space of $K$-finite vectors, we find the points of reducibility and the complementary series. The minimal representation and a limit of discrete series are identified as kernel of the corresponding Knapp-Stein intertwining operator. Moreover, we show that some quaternionic discrete series representations occur as the subrepresentation on which the family of intertwining operators vanishes of order two.
In this paper we construct conformally invariant systems of first order and second order differential operators associated to a homogeneous line bundle $\Cal{L}_{s} \to G_0/Q_0$ with $Q_0$ a maximal parabolic subgroup of quasi-Heisenberg type. This generalizes the results by Barchini, Kable, and Zierau.To do so we use techniques different from ones used by them.
Let $G$ be an irreducible Hermitian Lie group and $D=G/K$ its bounded symmetric domain in $\mathbb C^d$ of rank $r$. Each $γ$ of the Harish-Chandra strongly orthogonal roots $\{γ_1, \cdots, γ_r\}$ defines a Heisenberg parabolic subgroup $P=MAN$ of $G$. We study the principal series representations $\Ind_P^G(1\otimes e^ν\otimes 1)$ of $G$ induced from $P$. We find the complementary series, reduction points, and unitary subrepresentations in this family of representations.
In this paper, we study the restriction of an irreducible unitary representation $π$ of the universal covering $\widetilde{Sp}_{2n}(\mb R)$ to a Heisenberg maximal parabolic group $\tilde P$. We prove that if $π|_{\tilde P}$ is irreducible, then $π$ must be a highest weight module or a lowest weight module. This is in sharp constrast with the $GL_n(\mathbb R)$ case. In addition, we show that for a unitary highest or lowest weight module, $π|_{\tilde P}$ decomposes discretely. We also treat the groups $U(p,q)$ and $O^*(2n)$.
We study the structure of minimal parabolic subgroups of the classical infinite dimensional real simple Lie groups, corresponding to the classical simple direct limit Lie algebras. This depends on the recently developed structure of parabolic subgroups and subalgebras that are not necessarily direct limits of finite dimensional parabolics. We then discuss the use of that structure theory for the infinite dimensional analog of the classical principal series representations. We look at the unitary representation theory of the classical lim--compact groups $U(\infty)$, $SO(\infty)$ and $Sp(\infty…
Let $X$ be a homogeneous space of a real reductive Lie group $G$. It was proved by T. Kobayashi and T. Oshima that the regular representation $C^{\infty}(X)$ contains each irreducible representation of $G$ at most finitely many times if a minimal parabolic subgroup $P$ of $G$ has an open orbit in $X$, or equivalently, if the number of $P$-orbits on $X$ is finite. In contrast to the minimal parabolic case, for a general parabolic subgroup $Q$ of $G$, we find a new example that the regular representation $C^{\infty}(X)$ contains degenerate principal series representations induced from $Q$ with i…
In \cite{MR3284482} and \cite{MR3658191}, the twisted standard $\mathcal{L}$-function $\mathcal{L}(s,π,χ,st)$ of a cuspidal representation $ π$ of the exceptional group of type $G_2$ was shown to be represented by a family of new-way Rankin-Selberg integrals. These integrals connect the analytic behaviour of $\mathcal{L}(s,π,χ,st)$ with that of a family of degenerate Eisenstein series $\mathcal{E}_E(χ, f_s, s, g)$ on quasi-split forms $H_E$ of $Spin_8$, induced from Heisenberg parabolic subgroups. The analytic behaviour of the series $\mathcal{E}_E(χ, f_s, s, g)$ in the right half-plane $Re(s)…
We give a complete combinatorial classification of those parabolic Verma modules in the principal block of the parabolic category $\mathcal{O}$ associated to a minimal or a maximal parabolic subalgebra of the special linear Lie algebra for which the answer to Kostant's problem is positive.
We study representations of the classical infinite dimensional real simple Lie groups $G$ induced from factor representations of minimal parabolic subgroups $P$. This makes strong use of the recently developed structure theory for those parabolic subgroups and subalgebras. In general parabolics in the infinite dimensional classical Lie groups are are somewhat more complicated than in the finite dimensional case, and are not direct limits of finite dimensional parabolics. We extend their structure theory and use it for the infinite dimensional analog of the classical principal series representa…
In this paper, we show that, under certain assumptions, a parabolic induction $Ind_B^Gλ$ from the Borel subgroup $B$ of a (real or $p$-adic) reductive group $G$ decomposes into a direct sum of the form: \[ Ind_B^Gλ= \left(Ind_P^G St_M\otimes χ_0\right) \oplus \left(Ind_P^G \mathbf{1}_M\otimes χ_0\right), \] where $P$ is a parabolic subgroup of $G$ with Levi subgroup $M$ of semi-simple rank $1$, $\mathbf{1}_M$ is the trivial representation of $M$, $St_M$ is the Steinberg representation of $M$ and $χ_0$ is a certain character of $M$. We construct examples of this phenomenon for all simply-connec…
Let $G$ be a simple real Lie group with maximal parabolic subgroup $P$ whose nilradical is abelian. Then $X=G/P$ is called a symmetric $R$-space. We study the degenerate principal series representations of $G$ on $C^\infty(X)$ in the case where $P$ is not conjugate to its opposite parabolic. We find the points of reducibility, the composition series and all unitarizable constituents. Among the unitarizable constituents we identify some small representations having as associated variety the minimal nilpotent $K_{\mathbb{C}}$-orbit in $\mathfrak{p}_{\mathbb{C}}^*$, where $K_{\mathbb{C}}$ is the …
In this article we connect topics from convex and integral geometry with well known topics in representation theory of semisimple Lie groups by showing that the $Cos^\lamda$ and $Sin^λ$-transforms on the Grassmann manifolds $Gr_p(K)=SU (n+1,K)/S (U (p,K)\times U (n+1-p,K))$ are standard intertwining operators between certain generalized principal series representations induced from a maximal parabolic subgroup $P_p$ of $SL (n+1,K)$. The index ${}_p$ indicates the dependence of the parabolic on p. The general results of Knapp and Stein and Vogan and Wallach then show that both transforms have m…
In the present article, we combine some techniques in the harmonic analysis together with the geometric approach given by modules over sheaves of rings of twisted differential operators ($\mathcal{D}$-modules), and reformulate the composition series and branching problems for objects in the Bernstein-Gelfand-Gelfand parabolic category $\mathcal{O}^\mathfrak{p}$ geometrically realized on certain orbits in the generalized flag manifolds. The general framework is then applied to the scalar generalized Verma modules supported on the closed Schubert cell of the generalized flag manifold $G/P$ for $…
We study the structure of the $(\mathfrak{g},K)$-modules of the principal series representations of $SL(3,\mathbb{R})$ and $Sp(2,\mathbb{R})$ induced from minimal parabolic subgroups, in the case when the infinitesimal character is nonsingular. The composition factors of these modules are known by Kazhdan-Lusztig-Vogan conjecture. In this paper, we give complete descriptions of the socle filtrations of these modules.
Given a regular supercuspidal representation $ρ$ of the Levi subgroup $M$ of a standard parabolic subgroup $P=MN$ in a connected reductive group $G$ defined over a non-archimedean local field $F$, we serve you a Rodier type structure theorem which provides us a geometrical parametrization of the set $JH(Ind^G_P(ρ))$ of Jordan--H{ö}lder constituents of the Harish-Chandra parabolic induction representation $Ind^G_P(ρ)$, vastly generalizing Rodier structure theorem for $P=B=TU$ Borel subgroup of a connected split reductive group about 40 years ago. Our novel contribution is to overcome the essent…
For p odd, the Lie group SO_0(p+1,p+1) has a family of unitary degenerate principal series representations realized on the space of real (p+1) by (p+1) skew symmetric matrices, similar to the Stein's complementary series for SL(2n,C) or Speh's representation for SL(2n,R). We consider their restriction on the subgroup G= SO(p+1,p) and prove that they are still irreducible and is equivalent to (a unitarization of) the principal series representation of G, and also irreducible under a maximal parabolic subgroup of G.
We obtain an irreducibility criterion for generalized principal series, extending known and frequently employed results for principal series. Our approach rests on a newly observed semi-direct product decomposition of the relative Weyl group and its action on generalized principal series, in conjunction with the theory of Jacquet module. We thus circumvent the obstacle that the Weyl group of a general Levi subgroup is not a Coxeter group. Our statements on irreducibility are formulated in terms of a subgroup of the Knapp--Stein R-group, which arises naturally from our decomposition of the Weyl…
Let $G$ be a split $p$-adic reductive group with connected centre and simply connected derived subgroup. We show that certain "chains" of principal series of $G$ do not exist and we establish several properties of the Breuil-Herzig construction $Π(ρ)^\mathrm{ord}$. In particular, we obtain a natural characterization of the latter and we prove a conjecture of Breuil-Herzig. In order to do so, we partially compute Emerton's $δ$-functor $\mathrm{H^\bullet Ord}_P$ of derived ordinary parts with respect to a parabolic subgroup on a principal series. We formulate a new conjecture on the extensions b…
We provide a uniform construction of $L^2$-models for all small unitary representations in degenerate principal series of semisimple Lie groups which are induced from maximal parabolic subgroups with abelian nilradical. This generalizes previous constructions to the case of a maximal parabolic subgroup which is not necessarily conjugate to its opposite, and hence the previously used Jordan algebra methods have to be generalized to Jordan pairs. The crucial ingredients for the construction of the $L^2$-models are the Lie algebra action and the spherical vector. Working in the so-called Fourier …
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…
This paper studies Emerton's Jacquet module functor for locally analytic representations of p-adic reductive groups. When P is a parabolic subgroup whose Levi factor M is not commutative, we show that passing to an isotypical subspace for the derived subgroup of M gives rise to essentially admissible locally analytic representations of the torus Z(M), which have a natural interpretation in terms of rigid geometry. We use this to extend Emerton's representation-theoretic construction of eigenvarieties by constructing eigenvarieties interpolating automorphic representations whose local component…
We derive a Givental-type stationary phase integral representation for the specified $\Gr_{m,N}$-Whittaker function introduced in \cite{GLO2}, which presumably describes the $S^1\times U_N$-equivariant Gromov-Witten invariants of Grassmann variety $\Gr_{m,N}$. Our main tool is a generalization of Whittaker model for principal series $\CU(\frak{gl}_N)$-modules. In particular, our construction includes a representation theory interpretation of the Batyrev--Ciocan-Fontanine--Kim--van Straten toric degeneration of Grassmannian, providing a direct connection between this toric degeneration of $\Gr_…
We define a category of planar diagrams whose Grothendieck group contains an integral version of the infinite rank Heisenberg algebra, thus yielding a categorification of this algebra. Our category, which is a q-deformation of one defined by Khovanov, acts naturally on the categories of modules for Hecke algebras of type A and finite general linear groups. In this way, we obtain a categorification of the bosonic Fock space. We also develop the theory of parabolic induction and restriction functors for finite groups and prove general results on biadjointness and cyclicity in this setting.
In the spirit of geometric quantisation we consider representations of the Heisenberg(--Weyl) group induced by hypercomplex characters of its centre. This allows to gather under the same framework, called p-mechanics, the three principal cases: quantum mechanics (elliptic character), hyperbolic mechanics and classical mechanics (parabolic character). In each case we recover the corresponding dynamic equation as well as rules for addition of probabilities. Notably, we are able to obtain whole classical mechanics without any kind of semiclassical limit h->0. Keywords: Heisenberg group, Kirillov'…
We calculate the homology of three families of 2-step nilpotent Lie (super)algebras associated with the symplectic, orthogonal, and general linear groups. The symplectic case was considered by Getzler and the main motivation for this work was to complete the calculations started by him. In all three cases, these algebras can be realized as the nilpotent radical of a parabolic subalgebra of a simple Lie algebra, and our first approach relies on a theorem of Kostant, but is otherwise elementary and involves combinatorics of Weyl groups and partitions which may be of independent interest. Our sec…
