ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Tue, 04 Oct 2022 23:21:50 GMT2022-10-04T23:21:50Z50271- Degenerate principal series and local theta correspondence IIhttps://scholarbank.nus.edu.sg/handle/10635/103111Title: Degenerate principal series and local theta correspondence II
Authors: Lee, S.T.; Zhu, C.-B.
Abstract: Following our previous paper [LZ] which deals with the group U(n, n), we study the structure of certain Howe quotients Ωp,q and Ωp,q(1) which are natural Sp(2n, R) modules arising from the Oscillator representation associated with the dual pair (O(p, q), Sp(2n, R)), by embedding them into the degenerate principal series representations of Sp(2n, R) studied in [L2].
Wed, 01 Jan 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031111997-01-01T00:00:00Z
- Eigendistributions for orthogonal groups and representations of symplectic groupshttps://scholarbank.nus.edu.sg/handle/10635/103184Title: Eigendistributions for orthogonal groups and representations of symplectic groups
Authors: Howe, R.; Zhu, C.-B.
Abstract: We consider the action of H = O(p, q) on the matrix space Mp+q,n(ℝ). We study a certain orbit script O sign of H in the null cone N ⊆ Mp+q,n(ℝ) which supports an eigen-distribution dvscript O sign for H. Using some identities of Capelli type developed in the Appendix, we determine the structure of G̃ = Sp(2n, ℝ)̃ -cyclic module generated by dvscript O sign under the oscillator representation of double-struck G sigñ (the metaplectic cover of double-struck G sign = Sp(2n(p + q), ℝ)). Applications to local theta correspondence and a generalized Huygens' Principle are given.
Tue, 01 Jan 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031842002-01-01T00:00:00Z
- Degenerate principal series and local theta correspondence soohttps://scholarbank.nus.edu.sg/handle/10635/103113Title: Degenerate principal series and local theta correspondence soo
Authors: Lee, T.; Zhu, C.-B.
Abstract: ABSTRACT. In this paper we determine the structure of the natural Ũ(n, n) module Ωp,q(l) which is the Howe quotient corresponding to the determinant character detl of U(p,q). We first give a description of the tempered distributions on Mp+q,n(C) which transform according to the character det-l under the linear action of U(p,q). We then show that after tensoring with a character, Ωp,q(l) can be embedded into one of the degenerate series representations of U(n, n). This allows us to determine the module structure of Ωp,q(l). Moreover we show that certain irreducible constituents in the degenerate series can be identified with some of these representations Ωp,q(l) or their irreducible quotients. We also compute the Gelfand-Kirillov dimensions of the irreducible constituents of the degenerate series. © 1998 American Mathematical Society.
Thu, 01 Jan 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031131998-01-01T00:00:00Z
- Theta lifting of holomorphic discrete series: The case of U(n,n) × U(p,q)https://scholarbank.nus.edu.sg/handle/10635/104374Title: Theta lifting of holomorphic discrete series: The case of U(n,n) × U(p,q)
Authors: Nishiyama, K.; Zhu, C.-B.O.
Abstract: Let (G, G′) = (U(n, n),U(p, q)) (p + q ≤ n) be a reductive dual pair in the stable range. We investigate theta lifts to G of unitary characters and holomorphic discrete series representations of G′, in relation to the geometry of nilpotent orbits. We give explicit formulas for their K-type decompositions. In particular, for the theta lifts of unitary characters, or holomorphic discrete series with a scalar extreme K′-type, we show that the K structure of the resulting representations of G is almost identical to the KC-module structure of the regular function rings on the closure of the associated nilpotent KC-orbits in s, where g = t ⊕ s is a Cartan decomposition. As a consequence, their associated cycles are multiplicity free. © 2001 American Mathematical Society.
Mon, 01 Jan 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043742001-01-01T00:00:00Z
- Weyl's construction and tensor power decomposition for g2https://scholarbank.nus.edu.sg/handle/10635/104475Title: Weyl's construction and tensor power decomposition for g2
Authors: Huang, J.-S.; Zhu, C.-B.O.
Abstract: Let V be the 7-dimensional irreducible representations of G2. We decompose the tensor power V⊗n into irreducible representations of G2 and obtain all irreducible representations of G2 in the decomposition. This generalizes Weyl's work on the construction of irreducible representations and decomposition of tensor products for classical groups to the exceptional group G2. © 1999 American Mathematical Society.
Fri, 01 Jan 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1044751999-01-01T00:00:00Z
- Multiresolution on compact groupshttps://scholarbank.nus.edu.sg/handle/10635/131453Title: Multiresolution on compact groups
Authors: Lim, A.; Zhu, C.-B.
Abstract: Given a compact group M, we define the notion of multiresolution of L2 (M) with respect to an infinite sequence of subgroups G0 ⊆ G1 ⊆ G2 ⊆ ⋯ such that G = ∪∞ k=0 is a dense subgroup of M. We give characterizations of various axioms of multiresolution, demonstrate the existence and give the construction of a wavelet basis for L2 (M). We also construct stationary multiresolution and wavelets from cyclic vectors. An example of multiresolution on a non-abelian compact group is given for the infinite dihedral group, or isomorphically the real orthogonal group in dimension two. © 1999 Published by Elsevier Science Inc. All rights reserved.
Sat, 15 May 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1314531999-05-15T00:00:00Z
- A result on the gelfand-kirillov dimension of representations of classical groupshttps://scholarbank.nus.edu.sg/handle/10635/102751Title: A result on the gelfand-kirillov dimension of representations of classical groups
Authors: Zhu, C.-B.O.
Abstract: Let (G, G′) be the reductive dual pair (O(p,g),Sp(2n,R)). We show that if πT is a representation of Sp(2n, R) (respectively O(p, q)) obtained from duality correspondence with some representation of O(p, g) (respectively Sp(2n,K)), then its Gelfand-Kirillov dimension is less than or equal to (p + q)(2n - p+q-1/2) (respectively 2n(p + q - 2n+1/2)). © 1998 American Mathematical Society.
Thu, 01 Jan 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1027511998-01-01T00:00:00Z
- On the decay of matrix coefficients for exceptional groupshttps://scholarbank.nus.edu.sg/handle/10635/103785Title: On the decay of matrix coefficients for exceptional groups
Authors: Li, J.-S.; Zhu, C.-B.
Sat, 01 Jun 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1037851996-06-01T00:00:00Z
- Poincaré series of holomorphic representationshttps://scholarbank.nus.edu.sg/handle/10635/103939Title: Poincaré series of holomorphic representations
Authors: Tan, E.-C.; Zhu, C.-B.
Abstract: If V is a holomorphic representation of a Hermitian symmetric group G, we can define the Poincaré series of V by PV(t) = ∑λ (dimℂ Vλ)tλ where Vλ are the eigenspaces under the center of K, a maximal compact subgroup of G. We discuss properties of these formal power series, give explicit rational forms for some of the unitary holomorphic representations, and compute their Gelfand-Kirillov dimensions and Bernstein degrees (in the sense defined by Vogan).
Mon, 25 Mar 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1039391996-03-25T00:00:00Z
- On the (g, K)-cohomology of certain theta liftshttps://scholarbank.nus.edu.sg/handle/10635/103764Title: On the (g, K)-cohomology of certain theta lifts
Authors: Lee, S.T.; Zhu, C.-B.
Abstract: Let θp,q be the theta lift of Sp(4, ℝ) from the trivial representation of O(p, q), where p + q is even. We compute the (g, K)-cohomology of θp,q by embedding them into certain degenerate principal series representations of Sp(4, ℝ).
Tue, 01 May 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1037642001-05-01T00:00:00Z
- On certain distinguished unitary representations supported on null coneshttps://scholarbank.nus.edu.sg/handle/10635/103688Title: On certain distinguished unitary representations supported on null cones
Authors: Tan, E.-C.; Zhu, C.-B.
Abstract: Let double-struck F = ℂ or ℍ, and let G = U(p, q; double-struck F) be the isometry group of a double-struck F-hermitian form of signature (p, q). For 2n ≤ min (p, q), we consider the action of G on Vn, the direct sum of n copies of the standard module V = double-struck Fp+q, and the associated action of G on the regular part of the null cone, denoted by X00. We show that there is a commuting set of G-invariant differential operators acting on the space of C∞ functions on X00 which transform according to a distinguished GL(n, double-struck F) character, and the resulting kernel is an irreducible unitary representation of G. Our result can be interpreted as providing a geometric construction of the theta lift of the characters from the group G′ = U(n,n) or O*(4n). The construction and approach here follow a previous work of Zhu and Huang [Representation Theory 1 (1997)] where the group concerned is G = O(p, q) with p + q even.
Thu, 01 Oct 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036881998-10-01T00:00:00Z
- Representations with scalar K-types and applicationshttps://scholarbank.nus.edu.sg/handle/10635/104053Title: Representations with scalar K-types and applications
Authors: Zhu, C.-B.
Abstract: We discuss some results of Shimura on invariant differential operators and extend a folklore theorem about spherical representations to representations with scalar K-types. We then apply the result to obtain non-trivial isomorphisms of certain representations arising from local theta correspondence, many of which are unipotent in the sense of Vogan.
Wed, 01 Jan 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1040532003-01-01T00:00:00Z
- Uniqueness of Bessel Models: The Archimedean Casehttps://scholarbank.nus.edu.sg/handle/10635/104430Title: Uniqueness of Bessel Models: The Archimedean Case
Authors: Jiang, D.; Sun, B.; Zhu, C.-B.
Abstract: In the archimedean case, we prove uniqueness of Bessel models for general linear groups, unitary groups and orthogonal groups. © 2010 Springer Basel AG.
Fri, 01 Jan 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1044302010-01-01T00:00:00Z
- Local Theta Lifting of Generalized Whittaker Models Associated to Nilpotent Orbitshttps://scholarbank.nus.edu.sg/handle/10635/126652Title: Local Theta Lifting of Generalized Whittaker Models Associated to Nilpotent Orbits
Authors: Gomez, R.; Zhu, C.-B.
Abstract: Let (G, G̃) be a reductive dual pair over a local field (Formula presented.) of characteristic 0, and denote by V and (Ṽ the standard modules of G and G̃, respectively. Consider the set Max Hom (V, Ṽ) of full rank elements in Hom(V, (Ṽ), and the nilpotent orbit correspondence (Formula presented.) and (Formula presented.) induced by elements of Max Hom (V, Ṽ) via the moment maps. Let (Formula presented.) be a smooth irreducible representation of G. We show that there is a correspondence of the generalized Whittaker models of π of type (Formula presented.) and of Θ (π) of type (Formula presented.), where Θ (π) is the full theta lift of π. When (G, G̃) is in the stable range with G the smaller member, every nilpotent orbit (Formula presented.) is in the image of the moment map from Max Hom (V, Ṽ). In this case, and for (Formula presented.) non-Archimedean, the result has been previously obtained by Mglin in a different approach. © 2014 Springer Basel.
Wed, 01 Jan 2014 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1266522014-01-01T00:00:00Z
- Theta lifting of unitary lowest weight modules and their associated cycleshttps://scholarbank.nus.edu.sg/handle/10635/104376Title: Theta lifting of unitary lowest weight modules and their associated cycles
Authors: Nishiyama, K.; Zhu, C.-B.
Abstract: We consider a reductive dual pair (G, G′) in the stable range with G′ the smaller member and of Hermitian symmetric type. We study the theta lifting of(holomorphic) nilpotent K′ ℂ-orbits in relation to the theta lifting of unitary lowest weight representations of G′. We determine the associated cycles of all such representations. In particular, we prove that the multiplicity in the associated cycle is preserved under the theta lifting. We also develop a theory for the lifting of covariants arising from double fibrations by affine quotient maps.
Wed, 01 Dec 2004 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043762004-12-01T00:00:00Z
- Tensor product of degenerate principal series and local theta correspondencehttps://scholarbank.nus.edu.sg/handle/10635/104245Title: Tensor product of degenerate principal series and local theta correspondence
Authors: Li, J.-S.; Tan, E.-C.; Zhu, C.-B.
Abstract: We construct an intertwining map from the oscillator representation to the tensor product of two degenerate principal series of G and G1 which form a reductive dual pair in the sense of Howe. We derive results on local theta correspondence for the dual pairs (O(p, q), Sp(2q, ℝ)). (U(p, q), U(q, q)) and (Sp(p, q), O*(4q)). where p ≥ q. We also describe the correspondence completely for the pairs (O(p, q), SL(2, ℝ)), (U(p, q) U(1,1)), and (Sp(p, q), O*(4)). © 2001 Academic Press.
Sat, 10 Nov 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042452001-11-10T00:00:00Z
- Multiplicity one theorems: The Archimedean casehttps://scholarbank.nus.edu.sg/handle/10635/103588Title: Multiplicity one theorems: The Archimedean case
Authors: Sun, B.; Zhu, C.-B.
Abstract: Let G be one of the classical Lie groups GL n+1(R), GL n+1(C), U (p, q + 1), O(p, q + 1), O n+1(C), SO(p, q + 1), SO n+1(C), and let G' be re-spectively the subgroup GL n(R), GL n(C), U(p, q), O(p, q), O n(C), SO(p, q), SO n(C), embedded in G in the standard way. We show that every irreducible Casselman-Wallach representation of G' occurs with multiplicity at most one in every irreducible Casselman-Wallach representation of G. Similar results are proved for the Jacobi groups GL n(R)⋉H 2n+1(R), GL n(C)⋉ H 2n+1(C), U(p, q)⋉H 2p+2q+1(R), Sp 2n(R)⋉H 2n+1(R), Sp 2n(C)⋉H 2n+1(C), with their respective subgroups GL n(R), GL n(C), U(p, q), Sp 2n(R), and Sp 2n(C).
Sun, 01 Jan 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1035882012-01-01T00:00:00Z
- On Certain Rings of Highest Weight Vectorshttps://scholarbank.nus.edu.sg/handle/10635/103689Title: On Certain Rings of Highest Weight Vectors
Authors: Aslaksen, H.; Tan, E.C.; Zhu, C.B.
Abstract: Let Rm, n) be the ring of highest weight vectors of the action of Om × GLn on the polynomial algebra of m × n matrices. We determine Rm, 2 and find generators for Rm,3. In particular, the results about Rm,2 give branching rules and information about the structure of holomorphic representations of Sp4. © 1995 Academic Press. All rights reserved.
Mon, 01 May 1995 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036891995-05-01T00:00:00Z
- Degenerate principal series and local theta correspondence III: The case of complex groupshttps://scholarbank.nus.edu.sg/handle/10635/103112Title: Degenerate principal series and local theta correspondence III: The case of complex groups
Authors: Lee, S.T.; Zhu, C.-B.
Abstract: 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. © 2007 Elsevier Inc. All rights reserved.
Tue, 01 Jan 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031122008-01-01T00:00:00Z
- A general form of Gelfand-Kazhdan criterionhttps://scholarbank.nus.edu.sg/handle/10635/102652Title: A general form of Gelfand-Kazhdan criterion
Authors: Sun, B.; Zhu, C.-B.
Abstract: We formalize the notion of matrix coefficients for distributional vectors in a representation of a real reductive group, which consist of generalized functions on the group. As an application, we state and prove a Gelfand-Kazhdan criterion for a real reductive group in very general settings. © 2011 Springer-Verlag.
Thu, 01 Sep 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1026522011-09-01T00:00:00Z
- Theta lifting of nilpotent orbits for symmetric pairshttps://scholarbank.nus.edu.sg/handle/10635/104375Title: Theta lifting of nilpotent orbits for symmetric pairs
Authors: Nishiyama, K.; Ochiai, H.; Zhu, C.-B.
Abstract: We consider a reductive dual pair (G, G′) in the stable range with G′ the smaller member and of Hermitian symmetric type. We study the theta lifting of nilpotent K′ℂ-orbits, where K′ is a maximal compact subgroup of G′ and we describe the precise K ℂ-module structure of the regular function ring of the closure of the lifted nilpotent orbit of the symmetric pair (G, K). As an application, we prove sphericality and normality of the closure of certain nilpotent K ℂ-orbits obtained in this way. We also give integral formulas for their degrees. © 2005 American Mathematical Society.
Thu, 01 Jun 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043752006-06-01T00:00:00Z
- Uniqueness of Bessel Models: The Archimedean Casehttps://scholarbank.nus.edu.sg/handle/10635/53251Title: Uniqueness of Bessel Models: The Archimedean Case
Authors: Jiang, D.; Sun, B.; Zhu, C.-B.
Abstract: In the archimedean case, we prove uniqueness of Bessel models for general linear groups, unitary groups and orthogonal groups. © 2010 Springer Basel AG.
Fri, 01 Jan 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/532512010-01-01T00:00:00Z
- Fourier transform and rigidity of certain distributionshttps://scholarbank.nus.edu.sg/handle/10635/103291Title: Fourier transform and rigidity of certain distributions
Authors: Sun, B.; Zhu, C.-B.
Abstract: Let E be a finite-dimensional vector space over a local field, and F be its dual. For a closed subset X of E, and Y of F, consider the space D -ξ(E; X, Y) of tempered distributions on E whose support are contained in X and support of whose Fourier transform are contained in Y. We show that D-ξ(E; X, Y) possesses a certain rigidity property, for X, Y which are some finite unions of affine subspaces. © World Scientific Publishing Company.
Sat, 01 Dec 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1032912012-12-01T00:00:00Z
- Quivers and the Invariant Theory of Levi Subgroupshttps://scholarbank.nus.edu.sg/handle/10635/104012Title: Quivers and the Invariant Theory of Levi Subgroups
Authors: Aslaksen, H.; Tan, E.C.; Zhu, C.B.
Abstract: We develop a theory of invariants using the formalism of quivers, generalizing earlier results attributed to Procesi. As an application, let H be the Levi component of a parabolic subgroup of a classical Lie group G with Lie algebra g. We describe a finite set of generators for P[g]H, the space of H-invariant polynomials on g, as well as the H-invariants in the universal enveloping algebra, U(g)H, thus generalizing the results of Klink and Ton-That, and Zhu. © 1994 Academic Press. All rights reserved.
Tue, 15 Feb 1994 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1040121994-02-15T00:00:00Z
- Uniqueness of ginzburg-rallis models: The archimedean casehttps://scholarbank.nus.edu.sg/handle/10635/104431Title: Uniqueness of ginzburg-rallis models: The archimedean case
Authors: Binyong, D.; Jiang, S.; Zhu, C.B.
Abstract: In this paper we prove the uniqueness of Ginzburg-Rallis models in the Archimedean case. As a key ingredient, we introduce a new descent argument based on two geometric notions attached to submanifolds, which we call metrical properness and unipotent ?-incompatibility. © 2010 American Mathematical Society Reverts to public domain 28 years from publication.
Sun, 01 May 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1044312011-05-01T00:00:00Z
- The explicit duality correspondence of (Sp(p,q), O*(2n))https://scholarbank.nus.edu.sg/handle/10635/104294Title: The explicit duality correspondence of (Sp(p,q), O*(2n))
Authors: Li, J.-S.; Paul, A.; Tan, E.-C.; Zhu, C.-B.
Abstract: We investigate the type I dual pairs over the quaternion algebra ℍ, namely the family of dual pairs (Sp(p,q),O*(2n)). We give a complete and explicit description of duality correspondence for p + q ≤ n as well as some of the cases for p + q > n, in terms of the Langlands parameters. © 2002 Elsevier Science (USA). All rights reserved.
Sat, 10 May 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042942003-05-10T00:00:00Z
- Invariant differential operators on symplectic grassmann manifoldshttps://scholarbank.nus.edu.sg/handle/10635/103443Title: Invariant differential operators on symplectic grassmann manifolds
Authors: Schwarz, G.; Zhu, C.-b.
Abstract: Let M 2n,r denote the vector space of real or complex 2n×r matrices with the natural action of the symplectic group Sp 2n, and let G=G n,r =Sp 2n ×M 2n,r denote the corresponding semi-direct product. For any integer p with 0≤p≤n-1, let H denote the subgroup G p,r ×Sp 2n-2p of G. We explicitly compute the algebra of left invariant differential operators on G/H, and we show that it is a free algebra if and only if r≤2n-2p+1. We also give orthogonal analogues of these results, generalizing those of Gonzalez and Helgason [3]. © 1994 Springer-Verlag.
Thu, 01 Dec 1994 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034431994-12-01T00:00:00Z