Metaplectic Tensor Products for Automorphic Representation of (r)

2016 ◽  
Vol 68 (1) ◽  
pp. 179-240 ◽  
Author(s):  
Shuichiro Takeda
Keyword(s):  
Tensor Product ◽  
Weyl Group ◽  
Tensor Products ◽  
Automorphic Representation ◽  
Group Element ◽  
Levi Subgroup ◽  
Automorphic Representations ◽  
Technical Assumption ◽  
Metaplectic Cover

AbstractLet M = GLr1 ✗ … × GLrk ⊆ GLr be a Levi subgroup of GLr, where r = r1 + … +rk, and its metaplectic preimage in the n-fold metaplectic cover r1 of GLr. For automorphic representations π1, …, πk of r1 (), … ,rk (), we construct (under a certain technical assumption that is always satisfied when n = 2) an automorphic representation π of () that can be considered as the “tensor product” of the representations π1, … , πk. This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place v, πv is equivalent to the local metaplectic tensor product of π1,v, … , πk,v defined by Mezo. Then we show that if all of the πi are cuspidal (resp. square-integrable modulo center), then the metaplectic tensor product is cuspidal (resp. square-integrable modulo center). We also show that (both locally and globally) the metaplectic tensor product behaves in the expected way under the action of a Weyl group element and show the compatibility with parabolic inductions.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

scholarly journals SUBPROJECTIVITY OF PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES OF CONTINUOUS FUNCTIONS

2021 ◽  
pp. 1-14
Author(s):  
R.M. CAUSEY
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Tensor Products ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Spaces Of Continuous Functions

Abstract Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, K n , $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each K i is scattered.

scholarly journals Casselman’s Basis of Iwahori Vectors and the Bruhat Order

2011 ◽  
Vol 63 (6) ◽  
pp. 1238-1253 ◽  
Author(s):  
Daniel Bump ◽  
Maki Nakasuji
Keyword(s):  
Weyl Group ◽  
Bruhat Order ◽  
Group Element ◽  
The Other ◽  
Intertwining Operators ◽  
Transition Matrices ◽  
Natural Basis ◽  
Spherical Representation ◽  
Langlands Parameters ◽  
Combinatorial Condition

AbstractW. Casselman defined a basis fu of Iwahori fixed vectors of a spherical representation of a split semisimple p-adic group G over a nonarchimedean local field F by the condition that it be dual to the intertwining operators, indexed by elements u of the Weyl group W. On the other hand, there is a natural basis , and one seeks to find the transition matrices between the two bases. Thus, let and . Using the Iwahori–Hecke algebra we prove that if a combinatorial condition is satisfied, then , where z are the Langlands parameters for the representation and α runs through the set S(u, v) of positive coroots (the dual root systemof G) such that with rα the reflection corresponding to α. The condition is conjecturally always satisfied if G is simply-laced and the Kazhdan–Lusztig polynomial Pw0v,w0u = 1 with w0 the long Weyl group element. There is a similar formula for conjecturally satisfied if Pu,v = 1. This leads to various combinatorial conjectures.

Extension of normal functionals on W*-tensor products

1975 ◽  
Vol 78 (2) ◽  
pp. 301-307 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
General Result ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
Hilbert Algebras ◽  
Special Cases

A deep result in the theory of W*-tensor products, the Commutation theorem, states that if M and N are W*-algebras faithfully represented as von Neumann algebras on the Hilbert spaces H and K, respectively, then the commutant in L(H ⊗ K) of the W*-tensor product of M and N coincides with the W*-tensor product of M′ and N′. Although special cases of this theorem were established successively by Misonou (2) and Sakai (3), the validity of the general result remained conjectural until the advent of the Tomita-Takesaki theory of Modular Hilbert algebras (6). As formulated, the Commutation theorem is a spatial result; that is, the W*-algebras in its statement are taken to act on specific Hilbert spaces. Not surprisingly, therefore, known proofs rely heavily on techniques of representation theory.

Noetherian Tensor Products

1972 ◽  
Vol 15 (2) ◽  
pp. 235-238
Author(s):  
E. A. Magarian ◽  
J. L. Motto
Keyword(s):  
Tensor Product ◽  
Jacobson Radical ◽  
Tensor Products ◽  
Minimum Condition ◽  
Ideal Structure ◽  
The Ideal

Relatively little is known about the ideal structure of A⊗RA' when A and A' are R-algebras. In [4, p. 460], Curtis and Reiner gave conditions that imply certain tensor products are semi-simple with minimum condition. Herstein considered when the tensor product has zero Jacobson radical in [6, p. 43]. Jacobson [7, p. 114] studied tensor products with no two-sided ideals, and Rosenberg and Zelinsky investigated semi-primary tensor products in [9].All rings considered in this paper are assumed to be commutative with identity. Furthermore, R will always denote a field.

scholarly journals Tensor Products and Bimorphisms

1976 ◽  
Vol 19 (4) ◽  
pp. 385-402 ◽  
Author(s):  
Bernhard Banaschewski ◽  
Evelyn Nelson
Keyword(s):  
Tensor Product ◽  
Commutative Ring ◽  
Tensor Products ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Bilinear Maps ◽  
Special Situation ◽  
Dual Spaces ◽  
Finite Dimensional

The binary tensor product, for modules over a commutative ring, has two different aspects: its connection with universal bilinear maps and its adjointness to the internal hom-functor. Furthermore, in the special situation of finite-dimensional vector spaces, the tensor product can also be described in terms of dual spaces and the internal hom-functor. The aim of this paper is to investigate these relationships in the setting of arbitrary concrete categories.

scholarly journals On the Structure of Monodromy Algebras and Drinfeld Doubles

1997 ◽  
Vol 09 (03) ◽  
pp. 371-395
Author(s):  
Florian Nill
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Tensor Products ◽  
Spin Chains ◽  
Coadjoint Action ◽  
Braided Group ◽  
Gauge Transformations ◽  
Loop Algebras ◽  
Algebraic Tensor Product ◽  
Quasitriangular Hopf Algebra

We give a review and some new relations on the structure of the monodromy algebra (also called loop algebra) associated with a quasitriangular Hopf algebra H. It is shown that as an algebra it coincides with the so-called braided group constructed by S. Majid on the dual of H. Gauge transformations act on monodromy algebras via the coadjoint action. Applying a result of Majid, the resulting crossed product is isomorphic to the Drinfeld double [Formula: see text]. Hence, under the so-called factorizability condition given by N. Reshetikhin and M. Semenov–Tian–Shansky, both algebras are isomorphic to the algebraic tensor product H ⊗ H. It is indicated that in this way the results of Alekseev et al. on lattice current algebras are consistent with the theory of more general Hopf spin chains given by K. Szlachányi and the author. In the Appendix the multi-loop algebras ℒm of Alekseev and Schomerus [3] are identified with braided tensor products of monodromy algebras in the sense of Majid, which leads to an explanation of the "bosonization formula" of [3] representing ℒm as H ⊗…⊗ H.

Simple Cn modules with multiplicities 1 and applications

1994 ◽  
Vol 72 (7-8) ◽  
pp. 326-335 ◽  
Author(s):  
D. J. Britten ◽  
J. Hooper ◽  
F. W. Lemire
Keyword(s):  
Weyl Group ◽  
Tensor Products ◽  
Weight Space ◽  
One Dimensional ◽  
Infinite Dimensional ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Recursion Formulas ◽  
Completely Reducible ◽  
Index 2

In this paper we show that there exist exactly two nonequivalent simple infinite dimensional highest weight Cn modules having the property that every weight space is one dimensional. The tensor products of these modules with any finite-dimensional simple Cn module are proven to be completely reducible and we provide an explicit decomposition for such tensor products. As an application of these decompositions, we obtain two recursion formulas for computing the multiplicities of simple finite dimensional Cn modules. These formulas involve a sum over subgroups of index 2 in the Weyl group of Cn.

scholarly journals NuclearJC-algebras and tensor products of types

1993 ◽  
Vol 16 (4) ◽  
pp. 717-723
Author(s):  
Fatmah B. Jamjoom
Keyword(s):  
Tensor Product ◽  
Tensor Products

This article is a continuation of [1], to which the reader is referred for the definition and properties of theJC-tensor product of twoJC-algebras. Our standard references for nuclear and postliminalC*-algebras are[2,3,4,5,6,7]. We extend the notion of nuclearity toJC-algebras and prove that postliminalJC-algebras are nuclear. In contrast with the situation which occurs forC*-algebras, theJC-tensor product of two postliminalJC-algebras turns out, in general, to be non-postliminal and can even be anitliminal.

An approximation property for operator systems

2019 ◽  
Vol 22 (01) ◽  
pp. 1950005
Author(s):  
Wei Wu
Keyword(s):  
Tensor Product ◽  
Convex Sets ◽  
Approximation Property ◽  
Tensor Products ◽  
Order Unit ◽  
Operator System ◽  
Operator Systems ◽  
Natural Sense

Motivated by an observation of Namioka and Phelps on an approximation property of order unit spaces, we introduce the [Formula: see text]-tensor product and the [Formula: see text]-tensor product of two compact matrix convex sets. We define a new approximation property for operator systems, and give a characterization using the [Formula: see text]- and [Formula: see text]-tensor products in the spirit of Grothendieck. Thus, an operator system has the operator system approximation property if and only if it is [Formula: see text]-nuclear in a natural sense.

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