Degenerate Principal Series Research Articles

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.

Resonances and residue operators for pseudo-Riemannian hyperbolic spaces

For any pseudo-Riemannian hyperbolic space X over R,C,H or O, we show that the resolvent R(z)=(□−zId)−1 of the Laplace–Beltrami operator −□ on X can be extended meromorphically across the spectrum of □ as a family of operators Cc∞(X)→D′(X). Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in D′(X) forms a representation of the isometry group of X, which we identify with a subrepresentation of a degenerate principal series. Our study includes in particular the case of even functions on de Sitter and Anti-de Sitter spaces.For Riemannian symmetric spaces analogous results were obtained by Miatello–Will and Hilgert–Pasquale. The main qualitative differences between the Riemannian and the non-Riemannian setting are that for non-Riemannian spaces the resolvent can have poles of order two, it can have a pole at the branching point of the covering to which R(z) extends, and the residue representations can be infinite-dimensional.

Derived functor modules as irreducible constituents of degenerate principal series of the maximal Gelfand-Kirillov dimension

Derived functor modules as irreducible constituents of degenerate principal series of the maximal Gelfand-Kirillov dimension

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.

Special functions associated with K -types of degenerate principal series of Sp ( n , C )

We study K -types of degenerate principal series of Sp ( n , C ) by using two realizations of these infinite-dimensional representations. The first model we use is the classical compact picture; the second model is conjugate to the noncompact picture via an appropriate partial Fourier transform. In the first case, we find a family of K -finite vectors that can be expressed as solutions of specific hypergeometric differential equations; the second case leads to a family of K -finite vectors whose expressions involve Bessel functions.

Degenerate principal series for classical and odd GSpin groups in the general case

Let G n G_n denote either the group S O ( 2 n + 1 , F ) SO(2n+1, F) , S p ( 2 n , F ) Sp(2n, F) , or G S p i n ( 2 n + 1 , F ) G{\mathrm {Spin}}(2n+1, F) over a non-archimedean local field of characteristic different from two. We determine all composition factors of degenerate principal series of G n G_n , using methods based on the Aubert involution and known results on irreducible subquotients of the generalized principal series of a particular type.

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.

The compact picture of symmetry-breaking operators for rank-one orthogonal and unitary groups

We present a method to calculate intertwining operators between the underlying Harish-Chandra modules of degenerate principal series representations of a semisimple Lie group $G$ and a semisimple subgroup $G'$, and between their composition factors. Our method describes the restriction of these operators to the $K'$-isotypic components, $K'\subseteq G'$ a maximal compact subgroup, and reduces the representation theoretic problem to an infinite system of scalar equations of a combinatorial nature. For rank one orthogonal and unitary groups and spherical principal series representations we calculate these relations explicitly and use them to classify intertwining operators. We further show that in these cases automatic continuity holds, i.e. every intertwiner between the Harish-Chandra modules extends to an intertwiner between the Casselman--Wallach completions, verifying a conjecture by Kobayashi. Altogether, this establishes the compact picture of the recently studied symmetry breaking operators for orthogonal groups by Kobayashi--Speh, gives new proofs of their main results and extends them to unitary groups.

Dimension of the space of intertwining operators from degenerate principal series representations

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 infinite multiplicity even when the number of $Q$-orbits on $X$ is finite.

Fourier supports of K-finite Bessel integrals on classical tube domains

Let [Formula: see text] be the symmetric tube domain associated with the Jordan algebra [Formula: see text], [Formula: see text], [Formula: see text], or [Formula: see text], and [Formula: see text] be its Shilov boundary. Also, let [Formula: see text] be a degenerate principal series representation of [Formula: see text]. Then we investigate the Bessel integrals assigned to functions in general [Formula: see text]-types of [Formula: see text]. We give individual upper bounds of their supports, when [Formula: see text] is reducible. We also use the upper bounds to give a partition for the set of all [Formula: see text]-types in [Formula: see text], that turns out to explain the [Formula: see text]-module structure of [Formula: see text]. Thus, our results concretely realize a relationship observed by Kashiwara and Vergne [[Formula: see text]-types and singular spectrum, in Noncommutative Harmonic analysis, Lecture Notes in Mathematics, Vol. 728 (Springer, 1979), pp. 177–200] between the Fourier supports and the asymptotic [Formula: see text]-supports assigned to [Formula: see text]-submodules in [Formula: see text].

Analytic Continuation of Equivariant Distributions

Abstract We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein’s theory of analytic continuation of holonomic distributions. We use this to construct H-equivariant functionals on principal series representations of G, where G is a real reductive group and H is an algebraic subgroup. We also deduce the existence of generalized Whittaker models for degenerate principal series representations. As a special case, this gives short proofs of existence of Whittaker models on principal series representations and of analytic continuation of standard intertwining operators. Finally, we extend our constructions to the p-adic case using a recent result of Hong and Sun.

Real double flag varieties for the symplectic group

In this paper we study a key example of a Hermitian symmetric space and a natural associated double flag variety, namely for the real symplectic group G and the symmetric subgroup L, the Levi part of the Siegel parabolic PS. We give a detailed treatment of the case of the maximal parabolic subgroups Q of L corresponding to Grassmannians and the product variety of G/PS and L/Q; in particular we classify the L-orbits here, and find natural explicit integral transforms between degenerate principal series of L and G.

Non-Siegel Eisenstein series for symplectic groups

We explicitly (in terms of Langlands parameters) describe the image of the degenerate Eisenstein series in the case of a symplectic group. We study these series for any maximal parabolic subgroup and any Grossencharacter. We do that by an explicit analysis of the constant term of the Eisenstein series. Using already available information on the constituents of the local degenerate principal series, we further analyze the action of the local intertwining operators and determine their images. The local components of the images are given explicitly, in terms of their Langlands parameters. In this way, we have a complete description of the images of Eisenstein series in the right half-plane, for any Grossencharacter of the adeles over \(\mathbb {Q}.\) Thus, we complete previously known results obtained by different techniques for the trivial character and extend them to the case of the non-trivial characters.

Degenerate Whittaker functions for $\Sp_n(\R)$

In this paper, we construct Whittaker functions with exponential growth for the degenerate principal series of the symplectic group of genus n induced from the Siegel parabolic subgroup. This is achieved by explicitly constructing a certain Goodman–Wallach operator which yields an intertwining map from the degenerate principal series to the space of Whittaker functions, and by evaluating it on weight- l standard sections. We define a differential operator on such Whittaker functions which can be viewed as generalization of the ξ -operator on harmonic Maass forms for \SL2(\R) .

A functional equation and degenerate principal series

A functional equation between the $\zeta $ distributions can be obtained from the theory of prehomogeneous vector spaces. We show that the functional equation can be extended from the Schwartz space to certain degenerate principal series.

Bessel operators on Jordan pairs and small representations of semisimple Lie groups

We provide a uniform construction of L2-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 L2-models are the Lie algebra action and the spherical vector. Working in the so-called Fourier transformed picture of the degenerate principal series, the Lie algebra action is given in terms of Bessel operators on Jordan pairs. We prove that precisely for those parameters of the principal series where small quotients occur, the Bessel operators are tangential to certain submanifolds. Further, we show that the small quotients are unitarizable if and only if these submanifolds carry equivariant measures. In this case we can express the spherical vectors in terms of multivariable K-Bessel functions, and some delicate estimates for these Bessel functions imply the existence of the L2-models.

Invariant trilinear forms for spherical degenerate principal series of complex symplectic groups

We compute generalized Bernstein–Reznikov integrals associated with standard complex symplectic forms by studying Knapp–Stein intertwining operators between spherical degenerate principal series of [Formula: see text].

C*-algebraic intertwiners for degenerate principal series of special linear groups

The author constructs unitary intertwiners for degenerate C*-algebraic universal principal series of SL(n + 1) over a local field by explicitely normalizing standard intertwining integrals at the level of Hilbert modules.

Hecke duality of Ikeda lifts

Ikeda lifts form a distinguished subspace of Siegel modular forms. In this paper we prove several global and local results concerning this space. We find that degenerate principal series representations (for the Siegel parabolic) of the symplectic group Sp2n of even degree satisfy a Hecke duality relation which has applications to Ikeda lifts and leads to converse theorems. Moreover we apply certain differential operators to study pullbacks of Ikeda lifts.

Structure of the degenerate principal series on symmetric R-spaces and small representations

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∞(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 KC-orbit in pC⁎, where KC is the complexification of a maximal compact subgroup K⊆G and g=k+p the corresponding Cartan decomposition.