Principal Series Representations Research Articles

Conformal wave expansions for flat space amplitudes

The extended BMS algebra contains a conformal subgroup that acts on the celestial sphere as SO(1, 3). It is of interest to perform mode expansions of free fields in Minkowski spacetime that realize this symmetry in a simple way. In the present work we perform such a mode expansion for massive scalar fields using the unitary principal series representations of SO(1, 3) with a view to developing a holographic approach to gravity in asymptotically flat spacetime. These mode expansions are also of use in studying holography in three-dimensional de Sitter spacetime.

Rankin–Selberg periods for spherical principal series

By the unfolding method, Rankin-Selberg L-functions for ${\rm GL}(n)\times{\rm GL}(m)$ can be expressed in terms of period integrals. These period integrals actually define invariant forms on tensor products of the relevant automorphic representations. By the multiplicity-one theorems due to Sun-Zhu and Chen-Sun such invariant forms are unique up to scalar multiples and can therefore be related to invariant forms on equivalent principal series representations. We construct meromorphic families of such invariant forms for spherical principal series representations of ${\rm GL}(n,\mathbb{R})$ and conjecture that their special values at the spherical vectors agree in absolute value with the archimedean local L-factors of the corresponding L-functions. We verify this conjecture in several cases. This work can be viewed as the first of two steps in a technique due to Bernstein-Reznikov for estimating L-functions using their period integral expressions.

R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

AbstractFor a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such a principal series representation. Moreover, for certain saturated covers of a semisimple simply connected group, we also propose a simpler conjectural formula for such dimensions. This latter conjectural formula is verified in several cases, including covers of the symplectic groups.

Decompositions of principal series representations of Iwahori–Hecke algebras for Kac–Moody groups over local fields

Recently, Iwahori-Hecke algebras were associated to Kac-Moody groups over non-Archimedean local fields. In a previous paper, we introduced principal series representations for these algebras and partially generalized Kato's irreducibility criterion. In this paper, we study how some of these representations decompose when they are reducible and deduce information on the irreducible representations of these algebras.

Existence of invariant norms in p-adic representations of GL2(F) of large weights

Let F be a finite extension of Qp and let q be the cardinality of its residue field. The Breuil-Schneider conjecture for G=GLn(F)[BS07] gives a necessary and sufficient condition for the existence of an invariant norm on ρ⊗π, where ρ is an irreducible algebraic representation of G and π is an irreducible smooth representation of G over F‾. The conjecture is still open, even when n=2, if π is a principal series representation. In this case, assuming π is unramified and ρ=Symk⊗detm, it had been verified by Breuil [Bre03b] and De Ieso [DI13] when k<q. We extend their work to the range k<q2/2, imposing some technical conditions on π.

R-groups for unitary principal series of Spin groups

We study the unitary principal series of the split group Spinm(F), where F is a p-adic field. Let χ˜ be a unitary character of a maximal F-split torus T˜ of GSpinm(F), and let χ be its restriction to T=T˜∩Spinm(F). The R-groups Rχ˜ and Rχ of the corresponding principal series representations fit in the exact sequence 0→Rχ˜→Rχ→Rχ/Rχ˜→0. We give a complete answer to the question of splitting of this exact sequence. We also prove that the multiplicity is one when irreducible constituents in unitary principal series are restricted from GSpin(F) to Spin(F). Further, based on Keys' result, we prove Arthur's conjecture on R-groups for unitary principal series of Spin.

The degenerate principal series representations of exceptional groups of type E7 over p-adic fields

In this paper, we study the degenerate principal series of a split, simply-connected, simple p-adic group of type $E_7$. We determine the points of reducibility and the maximal semi-simple subrepresentation at each point.

A contraction of the principal series representations of SL(2,R)

A contraction of the principal series representations of <i>SL</i>(2,<b>R</b>)

Globally analytic principal series representation and Langlands base change

S. Orlik and M. Strauch have studied locally analytic principal series representation for general $p$-adic reductive groups generalizing an earlier work of P. Schneider for $GL(2)$ and related the condition of irreducibility of such locally analytic representation with that of a suitable Verma module. In this article, we take the case of $GL(n)$ and study the globally analytic principal series representation under the action of the pro-$p$ Iwahori subgroup of $GL(n,\mathbb{Z}_p)$, following the notion of globally analytic representations introduced by M. Emerton. Furthermore, we relate the condition of irreducibility of our globally analytic principal series to that of a Verma module. Finally, using Steinberg tensor product theorem, we construct Langlands base change of our globally analytic principal series to a finite unramified extension of $\mathbb{Q}_p$, generalizing an earlier work of Clozel for $GL(2)$.

An invitation to the principal series

Scalar unitary representations of the isometry group of dd-dimensional de Sitter space SO(1,d)SO(1,d) are labeled by their conformal weights \DeltaΔ. A salient feature of de Sitter space is that scalar fields with sufficiently large mass compared to the de Sitter scale 1/\ell1/ℓ have complex conformal weights, and physical modes of these fields fall into the unitary continuous principal series representation of SO(1,d)SO(1,d). Our goal is to study these representations in d=2d=2, where the relevant group is SL(2,\mathbb{R})SL(2,ℝ). We show that the generators of the isometry group of dS_22 acting on a massive scalar field reproduce the quantum mechanical model introduced by de Alfaro, Fubini and Furlan (DFF) in the early/late time limit. Motivated by the ambient dS_22 construction, we review in detail how the DFF model must be altered in order to accommodate the principal series representation. We point out a difficulty in writing down a classical Lagrangian for this model, whereas the canonical Hamiltonian formulation avoids any problem. We speculate on the meaning of the various de Sitter invariant vacua from the point of view of this toy model and discuss some potential generalizations.

English

In this note, we study formal deformations of derived representations of the principal series representations of SL(2, ℝ). In particular, we recover all the representations of the derived principal series by deforming one of them. Similar results are also obtained for SL(2, ℂ).

Positive Representations of Split Real Simply-Laced Quantum Groups

We construct the positive principal series representations for \mathcal U_q(\mathfrak g_\mathbb R) where \mathfrak g is of simply-laced type, parametrized by \mathbb R_{\geq 0}^r where r is the rank of \mathfrak g . We describe explicitly the actions of the generators in the positive representations as positive essentially self-adjoint operators on a Hilbert space, and prove the transcendental relations between the generators of the modular double. We define the modified quantum group \mathbf U_{\mathfrak q \tilde{\mathfrak q}}(\mathfrak g_\mathbb R) of the modular double and show that the representations of both parts of the modular double commute weakly with each other, there is an embedding into a quantum torus algebra, and the commutant contains its Langlands dual.

The degenerate principal series representations of exceptional groups of type E6 over p-adic fields

In this paper, we study the reducibility of degenerate principal series of the simple, simply-connected exceptional group of type E6. Furthermore, we calculate the maximal semi-simple subrepresentation and quotient of these representations.

Values of Iwahori-fixed vectors in the non-split Bessel model on GSp(2n)

We realize the non-split Bessel model of Novodvorsky and Piatetski-Shapiro in [22] as a generalized Gelfand-Graev representation of GSp(4), as suggested by Kawanaka in [17]. With uniqueness of the model already established in [22], we establish existence of a Bessel model for unramified principal series representations. We then connect the Iwahori-fixed vectors in the Bessel model to a linear character of the Hecke algebra of GSp(4) following the method outlined more generally in [3]. We use this connection to calculate the image of Iwahori-fixed vectors of unramified principal series in the model, and ultimately provide an explicit alternator expression for the spherical vector in the model. We show that the resulting alternator expression matches previous results of Bump, Friedberg, and Furusawa in [6]. We extend all results to GSp(2n), under the assumption that the aforementioned uniqueness property of the model holds for n > 2.

Yang-Baxter basis of Hecke algebra and Casselman's problem (extended abstract)

We generalize the definition of Yang-Baxter basis of type A Hecke algebra introduced by A.Lascoux, B.Leclerc and J.Y.Thibon (Letters in Math. Phys., 40 (1997), 75–90) to all the Lie types and prove their duality. As an application we give a solution to Casselman's problem on Iwahori fixed vectors of principal series representation of p-adic groups.

Multiplicities under basechange: Finite field case

A general proposition is proved relating multiplicities (for the restriction of a representation of a group to a subgroup) under basechange for groups over finite fields, and used to calculate certain multiplicities for cuspidal representations of general linear groups which become principal series representations under basechange for which the multiplicities can be calculated by geometric methods. For E/F quadratic extension of finite fields, we use this method to calculate which 1 dimensional representations of GLn(E) appear in a cuspidal representation of GL2n(F). We also calculate which 1 dimensional representations of GLn(F)×GLn(F) appear in a cuspidal representation of GL2n(F).

Symmetry breaking operators for real reductive groups of rank one

For a pair of real reductive groups G′⊂G we consider the space HomG′(π|G′,τ) of intertwining operators between spherical principal series representations π of G and τ of G′, also called symmetry breaking operators. Restricting to those pairs (G,G′) where dim⁡HomG′(π|G′,τ)<∞ and G and G′ are of real rank one, we classify all symmetry breaking operators explicitly in terms of their distribution kernels. This generalizes previous work by Kobayashi–Speh for (G,G′)=(O(1,n+1),O(1,n)) to the reductive pairs(G,G′)=(U(1,n+1;F),U(1,m+1;F)×F) with F=C,H,O and F<U(n−m;F). In most cases, all symmetry breaking operators can be constructed using one meromorphic family of distributions whose poles and residues we describe in detail. In addition to this family, there may occur some sporadic symmetry breaking operators which we determine explicitly.

Langlands parameters, functoriality and Hecke algebras

Let $G$ and $\tilde G$ be reductive groups over a local field $F$. Let $\eta : \tilde G \to G$ be a $F$-homomorphism with commutative kernel and commutative cokernel. We investigate the pullbacks of irreducible admissible $G$-representations $\pi$ along $\eta$. Following Borel, Adler--Korman and Xu, we pose a conjecture on the decomposition of the pullback $\eta^* \pi$. It is formulated in terms of enhanced Langlands parameters and includes multiplicities. This can be regarded as a functoriality property of the local Langlands correspondence. We prove this conjecture for three classes: principal series representation of split groups (over non-archimedean local fields), unipotent representations (also with $F$ non-archimedean) and inner twists of $GL_n, SL_n, PGL_n$. Our main techniques involve Hecke algebras associated to Langlands parameters. We also prove a version of the pullback/functoriality conjecture for those.

A note on quadratic twisting of epsilon factors for modular forms with arbitrary nebentypus

In this article, we investigate the variance of local $\varepsilon$-factor for a modular form with arbitrary nebentypus with respect to twisting by a quadratic character. We detect the type of the supercuspidal representation from that. For modular forms with trivial nebentypus, similar results are proved by Pacetti. Our method however is completely different from that of Pacetti and we use representation theory crucially. For ramified principal series (with $\infdiv{p}{N}$ and $p$ odd) and unramified supercuspidal representations of level zero, we relate these numbers with the Morita's $p$-adic Gamma function.

On the K-theory stable bases of the Springer resolution

Cohomological and K-theoretic stable bases originated from the study of quantum cohomology and quantum K-theory. Restriction formula for cohomological stable bases played an important role in computing the quantum connection of cotangent bundle of partial flag varieties. In this paper we study the K-theoretic stable bases of cotangent bundles of flag varieties. We describe these bases in terms of the action of the affine Hecke algebra and the twisted group algebra of KostantKumar. Using this algebraic description and the method of root polynomials, we give a restriction formula of the stable bases. We apply it to obtain the restriction formula for partial flag varieties. We also build a relation between the stable basis and the Casselman basis in the principal series representations of the Langlands dual group. As an application, we give a closed formula for the transition matrix between Casselman basis and the characteristic functions.