Principal Series Representations Research Articles

Genuine pro-$p$ Iwahori–Hecke algebras, Gelfand–Graev representations, and some applications

We study the Iwahori-component of the Gelfand–Graev representation of a central cover of a split linear reductive group and utilize our results for three applications. In fact, it is advantageous to begin at the pro- p level. Thus to begin we study the structure of a genuine pro- p Iwahori–Hecke algebra, establishing a Bernstein presentation. With this structure theory we first describe the pro- p part of the Gelfand–Graev representation and then the more subtle Iwahori part. For the first application we relate the Gelfand–Graev representation to the metaplectic representation of Sahi–Stokman–Venkateswaran, which conceptually realizes the Chinta–Gunnells action from the theory of Weyl group multiple Dirichlet series. For the second we compute the Whittaker dimension of the constituents of regular unramified principal series representations; for the third we do the same for unitary unramified principal series representations.

Hua operators on homogeneous line bundles over non-tube type bounded symmetric domains

Let Ω=G/K be a bounded symmetric domain of non-compact type. In this paper the image of the Poisson transform on the degenerate principal series representations of G attached to the Shilov boundary of Ω is considered. We characterize the images in terms of the third-order Hua operators Uν and Wν. When Ω is the exceptional domain of type V, we give the explicit formulas for the operators Uν and Wν.

Spinor-helicity representations of particles of any mass in dS4 and AdS4 spacetimes

The spinor-helicity representations of massive and (partially) massless particles in four-dimensional (anti–)de Sitter (A)dS spacetime are studied within the framework of the dual pair correspondence. We show that the dual groups (also known as “little groups”) of the anti–de Sitter and de Sitter groups are, respectively, O(2N) and O*(2N). For N=1, the generator of the dual algebra so(2)≅so*(2)≅u(1) corresponds to the helicity operator, and the spinor-helicity representation describes massless particles in (A)dS4. For N=2, the dual algebra is composed of two ideals, s and mΛ. The former ideal s≅so(3) fixes the spin of the particle, while the mass is determined by the latter ideal mΛ, which is isomorphic to so(2,1), iso(2), or so(3) depending on the cosmological constant being positive, zero, or negative. In the case of a positive cosmological constant, namely dS4, the spinor-helicity representation contains all massive particles corresponding to the principal series representations and the partially massless particles corresponding to the discrete series representations leaving out only the light massive particles corresponding to the complementary series representations. The zero and negative cosmological constant cases, which had been addressed in earlier references, are also discussed briefly. Finally, we consider the multilinear form of helicity spinors invariant under (A)dS group, which can serve as the (A)dS counterpart of the scattering amplitude, and discuss technical differences and difficulties of the (A)dS cases compared to the flat spacetime case. Published by the American Physical Society 2024

Fourier-Poisson Transforms Associated with the Principal Series Representations of Sp(1, n)

Fourier-Poisson Transforms Associated with the Principal Series Representations of Sp(1, n)

Fonctions d'une variable p-adique et représentations de [formula omitted]

We extend the dictionary between Fontaine rings and p-adic functionnal analysis, and we give a refinement of the p-adic local Langlands correspondence for principal series representations of GL2(Qp).

Parameters of Hecke algebras for Bernstein components of [formula omitted]-adic groups

Let G be a reductive group over a non-archimedean local field F. Consider an arbitrary Bernstein block Rep(G)s in the category of complex smooth G-representations. In earlier work the author showed that there exists an affine Hecke algebra H(O,G) whose category of right modules is closely related to Rep(G)s. In many cases this is in fact an equivalence of categories, like for Iwahori-spherical representations.In this paper we study the q-parameters of the affine Hecke algebras H(O,G). We compute them in many cases, in particular for principal series representations of quasi-split groups and for classical groups.Lusztig conjectured that the q-parameters are always integral powers of the cardinality of the residue field of F, and that they coincide with the q-parameters coming from some Bernstein block of unipotent representations. We reduce this conjecture to the case of absolutely simple p-adic groups, and we prove it for most of those.

Kato’s irreducibility criterion for Kac-Moody groups over local fields

In 2014, Braverman, Kazhdan, Patnaik and Bardy-Panse, Gaussent and Rousseau associated Iwahori-Hecke algebras to Kac-Moody groups over non-Archimedean local fields. In a previous paper, we defined and studied their principal series representations. In 1982, Kato provided an irreducibility criterion for these representations, in the reductive case. We had obtained partially this criterion in the Kac-Moody case. In this paper, we prove this criterion in the Kac-Moody case.

Slopes in eigenvarieties for definite unitary groups

We generalize bounds of Liu–Wan–Xiao for slopes in eigencurves for definite unitary groups of rank $2$ to slopes in eigenvarieties for definite unitary groups of any rank. We show that for a definite unitary group of rank $n$ , the Newton polygon of the characteristic power series of the $U_p$ Hecke operator has exact growth rate $x^{1+2/{n(n-1)}}$ , times a constant proportional to the distance of the weight from the boundary of weight space. The proof goes through the classification of forms associated to principal series representations. We also give a consequence for the geometry of these eigenvarieties over the boundary of weight space.

On the structure of principal series representations of SL2(𝔽̄p)

Let [Formula: see text] be a prime number and [Formula: see text] be the algebraic closure of the finite field [Formula: see text] with [Formula: see text] elements. Let [Formula: see text] be an algebraically closed field of characteristic [Formula: see text]. Let [Formula: see text] and [Formula: see text] be the standard Borel subgroup of [Formula: see text]. For any character [Formula: see text] of [Formula: see text], define [Formula: see text], where [Formula: see text] is the group algebra of [Formula: see text]. In this paper, we prove two interesting properties of [Formula: see text]: (a) [Formula: see text] is either of finite length or residually finite; (b) Any nontrivial quotient of [Formula: see text] is finite dimensional.

Rigid vectors in p-adic principal series representations

Rigid vectors in p-adic principal series representations

Estimate of the Weights of the Jacquet Module of the Principal Series Representations of $ \operatorname{GL}_{n}(𝕉) $

Estimate of the Weights of the Jacquet Module of the Principal Series Representations of $ \operatorname{GL}_{n}(𝕉) $

Two-point correlator of twist-2 light-ray operators in N=4 SYM in BFKL approximation

We generalize local operators of the leading twist-2 of N=4 SYM theory to the case of complex Lorentz spin j using principal series representation of sl(2,R). We give the direct computation of correlation function of two such non-local operators in the BFKL regime when j→1. The correlator appears to have the expected conformal coordinate dependence governed by the anomalous dimension of twist-2 operator in NLO BFKL approximation predicted by Kotikov and Lipatov.

From celestial correlators to AdS, and back

We present a general relation between celestial correlation functions in d-dimensions and Witten diagrams in (d + 1)-dimensional Euclidean anti-de Sitter (EAdS) space, to all orders in perturbation theory. Contact diagram processes are proportional to contact Witten diagrams and particle exchanges can be recast as a continuum of particle exchanges in EAdS where the exchanged particles carrying unitary Principal Series representations of SO(d + 1, 1). One can then try to import familiar EAdS techniques to study the properties of celestial correlators. In this work we use this relation to infer the analytic structure of the spectral density in the conformal partial wave expansion of celestial correlators which, at least perturbatively, should be a meromorphic function of the spectral parameter. We also discuss non-perturbative constraints from unitarity in Euclidean Conformal Field Theory, which requires positivity of the spectral density. This extends similar relations recently uncovered between boundary correlation functions in de Sitter space and Witten diagrams in EAdS, suggesting that EAdS could play a central role in efforts towards holography for all lambdas.

Ladder symmetries of black holes and de Sitter space: love numbers and quasinormal modes

In this note, we present a synopsis of geometric symmetries for (spin 0) perturbations around (4D) black holes and de Sitter space. For black holes, we focus on static perturbations, for which the (exact) geometric symmetries have the group structure of SO(1,3). The generators consist of three spatial rotations, and three conformal Killing vectors obeying a special melodic condition. The static perturbation solutions form a unitary (principal series) representation of the group. The recently uncovered ladder symmetries follow from this representation structure; they explain the well-known vanishing of the black hole Love numbers. For dynamical perturbations around de Sitter space, the geometric symmetries are less surprising, following from the SO(1,4) isometry. As is known, the quasinormal solutions form a non-unitary representation of the isometry group. We provide explicit expressions for the ladder operators associated with this representation. In both cases, the ladder structures help connect the boundary condition at the horizon with that at infinity (black hole) or origin (de Sitter space), and they manifest as contiguous relations of the hypergeometric solutions.

Epsilon Dichotomy for Linear Models: The Archimedean Case

Abstract Let $G=\textrm {GL}_{2n}({\mathbb {R}})$ or $G=\textrm {GL}_n({\mathbb {H}})$ and $H=\textrm {GL}_n({\mathbb {C}})$ regarded as a subgroup of $G$. Here, ${\mathbb {H}}$ is the quaternion division algebra over ${\mathbb {R}}$. For a character $\chi $ on ${\mathbb {C}}^\times $, we say that an irreducible smooth admissible moderate growth representation $\pi $ of $G$ is $\chi _H$-distinguished if $\operatorname {Hom}_H(\pi , \chi \circ \det _H)\neq 0$. We compute the root number of a $\chi _H$-distinguished representation $\pi $ twisted by the representation induced from $\chi $. This proves an Archimedean analogue of the conjecture by Prasad and Takloo-Bighash (J. Reine Angew. Math., 2011). The proof is based on the analysis of the contribution of $H$-orbits in a flag manifold of $G$ to the Schwartz homology of principal series representations. A large part of the argument is developed for general real reductive groups of inner type. In particular, we prove that the Schwartz homology $H_\ast (H, \pi \otimes \chi )$ is finite-dimensional and hence it is Hausdorff for a reductive symmetric pair $(G, H)$ and a finite-dimensional representation $\chi $ of $H$.

On Petersson norms of generic cusp forms and special values of adjoint L -functions for GSp4

We prove an explicit formula for the Petersson norms of some normalized generic cuspidal newforms on ${\rm GSp}_4$ whose archimedean components belong to either discrete series representations or spherical principal series representations. Our formula expresses the Petersson norms in terms of special values of adjoint $L$-functions and some elementary constants depending only on local representations.

The shadow formalism of Galilean CFT2

In this work, we develop the shadow formalism for two-dimensional Galilean conformal field theory (GCFT2). We define the principal series representation of Galilean conformal symmetry group and find its relation with the Wigner classification, then we determine the shadow transform of local operators. Using this formalism we derive the OPE blocks, Clebsch-Gordan kernels, conformal blocks and conformal partial waves. A new feature is that the conformal block admits additional branch points, which would destroy the convergence of OPE for certain parameters. We establish another inversion formula different from the previous one, but get the same result when decomposing the four-point functions in the mean field theory (MFT). We also construct a continuous series of bilocal actions of MFT, and an exceptional series of local actions, one of which is the BMS free scalar model. We notice that there is an outer automorphism of the Galilean conformal symmetry, and the GCFT2 can be regarded as null defect in higher dimensional CFTs.

Rankin-Selberg convolutions for GL(n)×GL(n) and GL(n)×GL(n−1) for principal series representations

Rankin-Selberg convolutions for GL(n)×GL(n) and GL(n)×GL(n−1) for principal series representations

On a matrix element representation of the GKZ hypergeometric functions

We develop a representation theory approach to the study of generalized hypergeometric functions of Gelfand, Kapranov and Zelevisnky (GKZ). We show that the GKZ hypergeometric functions may be identified with matrix elements of non-reductive Lie algebras {mathfrak L}_N of oscillator type. The Whittaker functions associated with principal series representations of mathfrak {gl}_{ell +1}({mathbb {R}}) being special cases of GKZ hypergeometric functions thus admit along with a standard matrix element representations associated with reductive Lie algebra mathfrak {gl}_{ell +1}({mathbb {R}}), another matrix element representation in terms of {mathfrak L}_{ell (ell +1)}.

Local Coefficients and Gamma Factors for Principal Series of Covering Groups

We consider an n n -fold Brylinski–Deligne cover of a reductive group over a p p -adic field. Since the space of Whittaker functionals of an irreducible genuine representation of such a cover is not one-dimensional, one can consider a local coefficients matrix arising from an intertwining operator, which is the natural analogue of the local coefficients in the linear case. In this paper, we concentrate on genuine principal series representations and establish some fundamental properties of such a local coefficients matrix, including the investigation of its arithmetic invariants. As a consequence, we prove a form of the Casselman–Shalika formula which could be viewed as a natural analogue for linear algebraic groups. We also investigate in some depth the behaviour of the local coefficients matrix with respect to the restriction of genuine principal series from covers of G L 2 GL_2 to S L 2 SL_2 . In particular, some further relations are unveiled between local coefficients matrices and gamma factors or metaplectic-gamma factors.