The Knapp–Stein Intertwining Operators Revisited: Renormalization and K-spectrum

2018 ◽  
pp. 119-133
Author(s):  
Toshiyuki Kobayashi ◽  
Birgit Speh
Keyword(s):  
Intertwining Operators

scholarly journals NONRELATIVISTIC HOLOGRAPHY — A GROUP-THEORETICAL PERSPECTIVE

2014 ◽  
Vol 29 (03n04) ◽  
pp. 1430001 ◽  
Author(s):  
V. K. DOBREV
Keyword(s):  
Representation Theory ◽  
Theoretical Perspective ◽  
Explicit Construction ◽  
Intertwining Operators ◽  
Difference Operators ◽  
Semisimple Lie Groups ◽  
Boundary Operators ◽  
Boundary Fields ◽  
Theoretical Results ◽  
Schrödinger Algebra

We give a review of some group-theoretical results related to nonrelativistic holography. Our main playgrounds are the Schrödinger equation and the Schrödinger algebra. We first recall the interpretation of nonrelativistic holography as equivalence between representations of the Schrödinger algebra describing bulk fields and boundary fields. One important result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory, and that these operators and the bulk-to-boundary operators are intertwining operators. Further, we recall the fact that there is a hierarchy of equations on the boundary, invariant with respect to Schrödinger algebra. We also review the explicit construction of an analogous hierarchy of invariant equations in the bulk, and that the two hierarchies are equivalent via the bulk-to-boundary intertwining operators. The derivation of these hierarchies uses a mechanism introduced first for semisimple Lie groups and adapted to the nonsemisimple Schrödinger algebra. These require development of the representation theory of the Schrödinger algebra which is reviewed in some detail. We also recall the q-deformation of the Schrödinger algebra. Finally, the realization of the Schrödinger algebra via difference operators is reviewed.

On a construction of unitary cocycles and the representation theory of amenable groups

1983 ◽  
Vol 3 (1) ◽  
pp. 129-133 ◽  
Author(s):  
Colin E. Sutherland
Keyword(s):  
Representation Theory ◽  
Probability Space ◽  
Amenable Group ◽  
Unitary Representations ◽  
Intertwining Operators ◽  
Type I ◽  
Amenable Groups

AbstractIf K is a countable amenable group acting freely and ergodically on a probability space (Γ, μ), and G is an arbitrary countable amenable group, we construct an injection of the space of unitary representations of G into the space of unitary 1-cocyles for K on (Γ, μ); this injection preserves intertwining operators. We apply this to show that for many of the standard non-type-I amenable groups H, the representation theory of H contains that of every countable amenable group.

scholarly journals Combinatorial bases of modules for affine Lie algebra B 2(1)

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Mirko Primc
Keyword(s):  
Lie Algebra ◽  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Intertwining Operators ◽  
Type B ◽  
Highest Weight Module ◽  
Highest Weight ◽  
Affine Lie Algebra ◽  
Algebra Theory ◽  
Simple Currents

AbstractWe construct bases of standard (i.e. integrable highest weight) modules L(Λ) for affine Lie algebra of type B 2(1) consisting of semi-infinite monomials. The main technical ingredient is a construction of monomial bases for Feigin-Stoyanovsky type subspaces W(Λ) of L(Λ) by using simple currents and intertwining operators in vertex operator algebra theory. By coincidence W(kΛ0) for B 2(1) and the integrable highest weight module L(kΛ0) for A 1(1) have the same parametrization of combinatorial bases and the same presentation P/I.

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.

scholarly journals Intertwining operator in thermal CFTd

2017 ◽  
Vol 32 (02n03) ◽  
pp. 1750006 ◽  
Author(s):  
Satoshi Ohya
Keyword(s):  
Momentum Space ◽  
Recurrence Relations ◽  
Algebraic Approach ◽  
Space Time ◽  
Conformal Algebra ◽  
Linear Recurrence ◽  
Intertwining Operators ◽  
Intertwining Operator ◽  
Primary Operator ◽  
S Matrix

It has long been known that two-point functions of conformal field theory (CFT) are nothing but the integral kernels of intertwining operators for two equivalent representations of conformal algebra. Such intertwining operators are known to fulfill some operator identities — the intertwining relations — in the representation space of conformal algebra. Meanwhile, it has been known that the S-matrix operator in scattering theory is nothing but the intertwining operator between the Hilbert spaces of in- and out-particles. Inspired by this algebraic resemblance, in this paper, we develop a simple Lie-algebraic approach to momentum-space two-point functions of thermal CFT living on the hyperbolic space–time [Formula: see text] by exploiting the idea of Kerimov’s intertwining operator approach to exact S-matrix. We show that in thermal CFT on [Formula: see text], the intertwining relations reduce to certain linear recurrence relations for two-point functions in the complex momentum space. By solving these recurrence relations, we obtain the momentum-space representations of advanced and retarded two-point functions as well as positive- and negative-frequency two-point Wightman functions for a scalar primary operator in arbitrary space–time dimension [Formula: see text].

scholarly journals The Plancherel formula for the horocycle space and generalizations

1996 ◽  
Vol 61 (1) ◽  
pp. 42-56 ◽  
Author(s):  
Ronald L. Lipsman
Keyword(s):  
Harmonic Analysis ◽  
Regular Representation ◽  
Plancherel Formula ◽  
Intertwining Operators ◽  
Direct Integral ◽  
Regular Representations ◽  
Spectral Distributions ◽  
Theoretic Technique ◽  
Direct Integral Decomposition ◽  
Integral Decomposition

AbstractThe Plancherel formula for the horocycle space, and several generalizations, is derived within the framework of quasi-regular representations which have monomial spectrum. The proof uses only machinery from the Penney-Fujiwara distribution-theoretic technique; no special semisimple harmonic analysis is needed. The Plancherel formulas obtained include the spectral distributions and the intertwining operators that effect the direct integral decomposition of the quasi-regular representation.

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