Degenerate Principal Series Representations 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.

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.

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.

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.

Generalized Whittaker Functions for Degenerate Principal Series of GL(4, $\mathbb R$)

For degenerate principal series representations of \operatorname{GL}(n, \mathbb R) , we show that the spaces of corresponding class one generalized Whittaker functions are characterized by explicit systems of differential operators. By using this characterization, we give detailed calculations on \operatorname{GL}(4, \mathbb R) . We examine the dimensions of the spaces of generalized Whittaker functions and give their generators in terms of hypergeometric functions of one and two variables. We show that generalized Whittaker functions have multiplicity one by using the theory of hypergeometric functions.

Degenerate principal series representations for quaternionic unitary groups

We give a complete description of all points of reducibility and the composition series of the degenerate principal series representations for quaternionic unitary groups which are induced from a character of the maximal parabolic subgroup with abelian unipotent radical. The case of even orthogonal groups is also included.

EXPLICIT DOUBLING INTEGRALS FOR Sp2(F) AND $\widetilde{{{\rm Sp}}_{2}}(F)$ USING "GOOD TEST VECTORS"

In this paper, we offer some explicit computations of a formulation of the doubling method of Piatetski-Shapiro and Rallis for the groups Sp 2(F) (the rank 2 symplectic group) and its metaplectic cover [Formula: see text] for F a finite extension of ℚp with p ≠ 2. We determine a set of "good test vectors" for the irreducible constituents of unramified principal series representations for these groups as well as a set of "good theta test sections" in a family of degenerate principal series representations of Sp 4(F) and [Formula: see text]. Determining "good test data" that produces a non-vanishing doubling integral should indicate the existence of a non-vanishing theta lifts for dual pairs of the type ( Sp 2(F), O (V)) (respectively [Formula: see text]) where V is a quadratic space of an even (respectively odd) dimension.

On Bessel integrals for reducible degenerate principal series representations

We discuss generalized Bessel integrals with nondegenerate characters, which are assigned to irreducible submodules of a reducible degenerate principal series representation of Sp ( n , R ) . Then we give sufficient conditions for their vanishings which are based on the signatures of the nondegenerate characters. This consequently suggests a reasonable correspondence between open GL n ( R ) -orbits in the set of real symmetric matrices of size n and irreducible submodules of the reducible principal series representations.

Degenerate principal series representations and their holomorphic extensions

Let X = H / L be an irreducible real bounded symmetric domain realized as a real form in an Hermitian symmetric domain D = G / K . The intersection S of the Shilov boundary of D with X defines a distinguished subset of the topological boundary of X and is invariant under H. It can be realized as S = H / P for certain parabolic subgroup P of H. We study the spherical representations Ind P H ( λ ) of H induced from P. We find formulas for the spherical functions in terms of the Macdonald F 1 2 hypergeometric function. This generalizes the earlier result of Faraut–Koranyi for Hermitian symmetric spaces D. We consider a class of H-invariant integral intertwining operators from the representations Ind P H ( λ ) on L 2 ( S ) to the holomorphic representations of G restricted to H. We construct a new class of complementary series for the groups H = SO ( n , m ) , SU ( n , m ) (with n − m > 2 ) and Sp ( n , m ) (with n − m > 1 ). We realize them as discrete components in the branching rule of the analytic continuation of the holomorphic discrete series of G = SU ( n , m ) , SU ( n , m ) × SU ( n , m ) and SU ( 2 n , 2 m ) respectively.

TWISTED MELLIN TRANSFORMS OF A REAL ANALYTIC RESIDUE OF SIEGEL–EISENSTEIN SERIES OF DEGREE 2

We study a residual form of a real analytic Siegel–Eisenstein series, which generates a certain derived functor module occurring in a degenerate principal series representation. We compute its Mellin transforms twisted by various Maass wave forms to get explicit formulas as our results. We apply them to prove meromorphic continuations together with functional equations which are satisfied by those twisted Mellin transforms.

Small principal series and exceptional duality for two simply laced exceptional groups

We use the notion of rank defined in an earlier paper (2007) to introduce and study two correspondences between small irreducible unitary representations of the split real simple Lie groups of types E n , where n ∈ {6, 7}, and two reductive classical groups. We show that these correspondences classify all of the unitary representations of rank two (in the sense of our earlier paper) of these exceptional groups. We study our correspondences for a specific family of degenerate principal series representations in detail.

Covariants of [formula omitted] and degenerate principal series of [formula omitted

Let m⩽n and let P(M2n,m)SLm be the algebra of all SLm(C) invariant polynomials on the space M2n,m of all 2n×m complex matrices. We determine the free generators of its subalgebra of Spn(C) covariants. These generators are given as pfaffians. They are used to study the degenerate principal series representations of GLn(H) induced from characters of a maximal parabolic subgroup. We determine the composition series and unitarity of these representations.

Degenerate principal series and local theta correspondence III: The case of complex groups

Let ( G ′ , G ) be one of the following complex dual pairs: ( O ( m , C ) , Sp ( 2 n , C ) ) , ( GL ( m , C ) , GL ( 2 n , C ) ) , ( Sp ( 2 m , C ) , O ( 4 n , C ) ) . We determine the relationship between the space of G ′ -coinvariants and certain degenerate principal series representations of G. We also show that the space of G ′ -invariant distributions is generated by the Dirac distribution as a G-representation.

Functional equations of prehomogeneous zeta functions and intertwining operators

We establish a relation between the gamma matrices of the functional equations satisfied by zeta functions associated with prehomogeneous vector spaces and certain integrals related to the intertwining operator of degenerate principal series representations of general linear groups.

REFINED ALGEBRAIC QUANTIZATION WITH THE TRIANGULAR SUBGROUP OFSL(2, ℝ)

We investigate refined algebraic quantization with group averaging in a constrained Hamiltonian system whose gauge group is the connected component of the lower triangular subgroup of SL (2, ℝ). The unreduced phase space is T*ℝp+qwith p≥1 and q≥1, and the system has a distinguished classical [Formula: see text] observable algebra. Group averaging with the geometric average of the right and left invariant measures, invariant under the group inverse, yields a Hilbert space that carries a maximally degenerate principal unitary series representation of O (p, q). The representation is nontrivial iff (p, q) ≠ (1, 1), which is also the condition for the classical reduced phase space to be a symplectic manifold up to a singular subset of measure zero. We present a detailed comparison to an algebraic quantization that imposes the constraints in the sense ĤaΨ = 0 and postulates self-adjointness of the [Formula: see text] observables. Under certain technical assumptions that parallel those of the group averaging theory, this algebraic quantization gives no quantum theory when (p, q) = (1, 2) or (2, 1), or when p≥2, q≥2 and p+q ≡1 ( mod 2).