Reach Us
+44-1522-440391

Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA

- *Corresponding Author:
- He H

Department of Mathematics, Louisiana State University

Baton Rouge, LA 70803, USA

**Tel:**1 225-578-1665

**E-mail:**[email protected]

**Received Date**: January 19, 2017; **Accepted Date:** January 25, 2017; **Published Date**: February 03, 2017

**Citation: **He H (2017) Invariant Tensor Product. J Generalized Lie Theory Appl 11: 252. doi:10.4172/1736-4337.1000252

**Copyright:** © 2017 He H. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

In this paper, we define invariant tensor product and study invariant tensor products associated with discrete series representations. Let G(*V _{1}*)×G(

Representation theory; Tomography; Topology; Algebra

Various forms of invariant tensor products appeared in the literature implicitly, for example, in Schur’s orthogonality for finite groups [1]. In many cases, they are employed to study the space Hom_{G}(π_{1}, π_{2}) where one of the representations π_{1} and π_{2} is irreducible. In this paper, we formulate the concept of invariant tensor product uniformly. We also study the invariant tensor functor associated with discrete series representations for classical groups. For motivations and applications [2-4].

**Definition 1**

Let G be a locally compact tomography group and dg be a left invariant Haar measure. Let (π,H_{π}) and (π_{1},H_{π1}) be two unitary representations of G. Let V and V_{1} be two dense subspaces of H_{π} and H_{π1} . Formally, define the averaging operator

as follows, ∀u,v∈V, u_{1},v_{1}∈V_{1},

(1)

(2)

Suppose that is well-defined. The image of will be called the invariant tensor product. It will be denoted by . Whenever we use the notation , we assume V_{G}V_{1} is well-defined, that is, the integral (1) converges for all . Denote . Define

For any unitary representation of G, let be the same unitary representation of G equipped with the conjugate linear multiplication. If V is a subspace of , let V^{c} be the corresponding subspace of .

**Lemma 1.1**

Let G be a unimodular group. Suppose that is well-defined. Then the form (,)G is a well-defined Hermitian form on .

The main result proved in this paper is as follows.

**Theorem 1.1:** Let G(*m+n*) be a classical group of type I with m>n. Let *(G(n),G(m))* be diagonally embedded in G (see Def. 2). Suppose that *(π,H _{π})* is a discrete series representation of G(

**Example I**

Let *G* be a compact group. Let (π_{1},Hπ_{1}) and be two unitary representations of G. Then is always well-defined. Suppose that π_{1} is irreducible. Then the dimension of is the the multiplicty of occuring in H_{π}.

**Example II**

Let G be a real reductive group. Let π and π1 be two discrete series representations. Then is always well-defined. It is one dimensional if and only if . Otherwise, it is zero dimensional.

**Theorem 2.1:** Let 1 be an irreducible unitary representation of G. Suppose that V_{1} and V are both closed under the action of G. Suppose that is well-defined. Then induces an injection from to , the space of G-equivariant homomorphisms from Vc to the Hermitian dual .

Proof: For each define as follows:

We have for every

We see that is in the Hermitian dual of V_{1}. In addition, is G-equivariant:

(3)

(4)

(5)

(6)

(7)

(8)

Here *dh* is a left invariant measure if *G* is not unimodular. Now it is easy to see that for every u if and only if . So

is an injection.

**Corollary 2.1:** Under the same assumption as in Theorem 2.1, let G be a real reductive group and K a maximal compact Lie group of G. Suppose that V and V_{1} are both smooth and K-finite. Then induces an injection from into .

**Proof:** When V is K-finite, will land in the K-finite subspaces of V

_{1} which is isomorphic to V_{1}.

Let G be a Lie algebra group and dg a left invariant Haar measure. Let X be a manifold with a continuous free (right) *G* action. Suppose that X/G is a smooth manifold. Let be a unitary representation of G. For any , define

Then is a -valued function on X. We shall see that it realizes in the following sense.

**Theorem 3.1:** Let G be a Lie group and *dg* a left invariant Haar measure. Let X be a manifold with a continuous free (right) G action such that the topological quotient X/G is a smooth manifold. Suppose there exist measures (X,μ) and (X/G,d[x]) such that

Let C_{c}(X)be the set of continuous functions with compact support. Let be a representation of G. Then where is the set of continuous compactly supported sections of the vector bundle

Furthermore,

and for every and

Proof: Let and . It is easy to see that is compactly supported in X/G. In addition

So . Observe that

(9)

(10)

(11)

(12)

(13)

(14)

(15)

Absolute convergence are guaranteed since f(g) is compactly supported. Notice that

We have

**Definition 2**

Let G be a classical group that preserves a nondegenerate sesquilinear form Ω. Write G=G(V,Ω) or simply G(V), where V is a vector field over equipped with the nondegenerate sesquilinear form Ω. Let V=V_{1}V_{2} such that Ω(V_{1},V_{2})=0. Let G_{1}=G(V_{1}, Ω|V_{1}) and G_{2}=G(V_{2}, Ω|V_{2}). For each g_{1}G_{1},g_{2}G_{2}, let (g_{1},g_{2}) acts on diagonally. We say that G_{1}×G_{2} is diagonally embedded in G.

**Definition 3**

Let (G1,G2) be diagonally embedded in G. Let be a unitary representation of G and be a unitary representation of G_{1}. Let V be a subspace of that is invariant under G_{2}. Let V_{1} be a subspace of such that is well-defined. Define a linear G2- representation as follows:

Since the Lie group action of G_{2} commutes with the integration over G_{1}, the action of G_{2} on is well-defined.

The linear representation is not neccessarily continuous because no topology has been defined on

**Lemma 4.1**

The form on is G_{2}-invariant.

Proof: Let and Write . Then

(16)

(17)

(18)

(19)

(20)

(21)

Hence is G_{2}-invariant.

Let G(m+n) be a classical group preserving a nondegenerate sesquilinear form. Let (G(n), G(m)) be diagonally embedded in G. For any irreducible unitary representation of G(m+n), let be the Frechet space of smooth vectors.

**Theorem 5.1:** Suppose that is a discrete series representation of G(m+n). Suppose that m>n and (π_{1},H_{1}) is a unitary representation of G(n). Then is well-defined. Suppose that . Then (,)_{G(n)} is positive definite. Furthermore, completes to a unitary representation of G(m).

The key of the proof is to realize as a subspace of the L^{2}-sections of the Hilbert bundle

**Proof:** Write G= G(m+n). Fix a maximal compact subgroup K of G such that

are maximal compact subgroups of G(m) and G(n) respectively. Let a be a maximal Abelian subalgebra in the orthogonal complement of k with respect to the Killing form (,)_{K}, such that

Let A be the analytic group generated by a. The function log:Aa is well-defined. Let for each

Since is a discrete series representation, without loss of generality, realize on a right K-finite subspace of L^{2}(G). So .

Let be Harish-Chandra’s basic spherical function. Let be the space of Harish-Chandra’s Schwartz space. It is well-known that every satisfies for some see for example Ch. 12.4 [?]). For every for a constant C_{h}. Observe that is G(m)-invariant.

Fix a positive root system in . Let A+ be the corresponding closed Weyl Chamber. Let be the half sum of positive roots. Let . Then for a positive constant C_{u,v} [5,6]. Notice that for ,

for some q0 and C>0. Let ρ(n) be the half sum of positive roots of the restricted root system . Let be the positive Weyl chamber of with respect to the root system . Since

.

. It follows that for every . Notice that is always bounded for . We see that

always converges. So is well-defined. Now suppose that

Notice that

is bounded by a multiple of . So . For each and , define to be the -valued function on G:

in the strong sense. Notice that for gG,h_{1}∈G(n),

(22)

(23)

(24)

(25)

(26)

So can be regarded as a section of the Hilbert bundle

In addition, we have

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

where G(n)\G is equipped with a right G invariant measure. Eqn. (31) is valid because the integrative Eqn. (29) converges absolutely. In fact, we have

To see this, recall that . In particular, for any N>0 and a∈A+, k_{1},k_{2}∈K, there exists C_{u,N}>0 such that:

.

Write for g= k_{1}ak_{2}. Then there also exists C_{v,N}>0 such that

Fix an N such that . In particular, Observe that the function

is bounded by a multiple of (L(h)WN(g),WN(g)), which, by a Theorem of Cowling-Haagerup-Howe [5], is bounded by a multiple of . Hence

Eqn. (29) converges absolutely. Therefore Eqn. (31) holds.

Now we have

It follows that . Realize as , which is a subspace of L^{2}-sections of the Hilbert bundle:

Clearly (,)G(n) is positive definite. Let be the completion of .

Since G(m) acts on and it commutes with G(n), G(m) acts on and it preserves (,)G(n). So the action of each g2∈G(m) can be extended into a unitary operator on . The group structure is kept in this completion essentially due to the fact that each extension is unique. Therefore completes to a unitary representation of G(m).

**Definition 4**

Let Π_{u}(G) be the unitary dual of G. Suppose that m>n. Let π be a discrete series representation of G(m+n). We denote the functor from π_{1} to the completion of by ITπ. If ITπ(1)0, ITπ(π1) is a unitary representation of G(m). Regarding the zero dimensional representation as a unitary representation, ITπ defines a functor from unitary representations of G(n) to unitary representations of G(m).

One natural question arises. That is, if π_{1} is irreducible, is ITπ(1) irreducible? This is beyond the scope of this paper. In fact, this problem is quite difficult. In general, ITπ(π_{1}) is not irreducible. However, for a certain holomorphic discrete series representation π, ITπ(π_{1}) will indeed be irreducible. For the time being, it is not clear which discrete series representation π has such a property. This question may be intrinsically related to the cohomology induction [7].

- Serre JP (1977) Linear Representations of Finite Groups. Springer-Verlag 42.
- Li JS (1990) Theta Lifting for Unitary Representations with Nonzero Cohomology.Duke Mathematical Journal 61: 913-937.
- He H (2000)Theta Correspondence I-Semistable Range: Construction and Irreducibility.Communications in Contemporary Mathematics2: 255-283.
- He H,On the Gan-Gross-Prasad Conjecture for U(p,q),to appear.
- Cowling M, Haagerup U, Howe R (1988) Almost L2 matrix coefficients.J Reine Angew Math 387: 97-110.
- He H (2009)Bounds on Smooth Matrix Coefficients of L2 spaces. Selecta Mathematica 15:419-433.
- Knapp A, Vogan D (1995) Cohomological induction and unitary representations. Princeton University Press, Princeton, NJ.

Select your language of interest to view the total content in your interested language

- Adomian Decomposition Method
- Algebra
- Algebraic Geometry
- Analytical Geometry
- Applied Mathematics
- Axioms
- Balance Law
- Behaviometrics
- Big Data Analytics
- Binary and Non-normal Continuous Data
- Binomial Regression
- Biometrics
- Biostatistics methods
- Clinical Trail
- Combinatorics
- Complex Analysis
- Computational Model
- Convection Diffusion Equations
- Cross-Covariance and Cross-Correlation
- Deformations Theory
- Differential Equations
- Differential Transform Method
- Fourier Analysis
- Fuzzy Boundary Value
- Fuzzy Environments
- Fuzzy Quasi-Metric Space
- Genetic Linkage
- Geometry
- Hamilton Mechanics
- Harmonic Analysis
- Homological Algebra
- Homotopical Algebra
- Hypothesis Testing
- Integrated Analysis
- Integration
- Large-scale Survey Data
- Latin Squares
- Lie Algebra
- Lie Superalgebra
- Lie Theory
- Lie Triple Systems
- Loop Algebra
- Matrix
- Microarray Studies
- Mixed Initial-boundary Value
- Molecular Modelling
- Multivariate-Normal Model
- Noether's theorem
- Non rigid Image Registration
- Nonlinear Differential Equations
- Number Theory
- Numerical Solutions
- Operad Theory
- Physical Mathematics
- Quantum Group
- Quantum Mechanics
- Quantum electrodynamics
- Quasi-Group
- Quasilinear Hyperbolic Systems
- Regressions
- Relativity
- Representation theory
- Riemannian Geometry
- Robust Method
- Semi Analytical-Solution
- Sensitivity Analysis
- Smooth Complexities
- Soft biometrics
- Spatial Gaussian Markov Random Fields
- Statistical Methods
- Super Algebras
- Symmetric Spaces
- Theoretical Physics
- Theory of Mathematical Modeling
- Three Dimensional Steady State
- Topologies
- Topology
- mirror symmetry
- vector bundle

- Total views:
**1041** - [From(publication date):

April-2017 - May 22, 2019] - Breakdown by view type
- HTML page views :
**957** - PDF downloads :
**84**

**Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals**

International Conferences 2019-20