scholarly journals On the degrees of matrix coefficients of intertwining operators

2012 ◽  
Vol 260 (2) ◽  
pp. 433-456 ◽  
Author(s):  
Tobias Finis ◽  
Erez Lapid ◽  
Werner Müller
2018 ◽  
Vol 30 (2) ◽  
pp. 347-384
Author(s):  
Arnab Mitra ◽  
Steven Spallone

AbstractLet{G^{1}}be an orthogonal, symplectic or unitary group over a local field and let{P=MN}be a maximal parabolic subgroup. Then the Levi subgroupMis the product of a group of the same type as{G^{1}}and a general linear group, acting on vector spacesXandW, respectively. In this paper we decompose the unipotent radicalNofPunder the adjoint action ofM, assuming{\dim W\leq\dim X}, excluding only the symplectic case with{\dim W}odd. The result is a Weyl-type integration formula forNwith applications to the theory of intertwining operators for parabolically induced representations of{G^{1}}. Namely, one obtains a bilinear pairing on matrix coefficients, in the spirit of Goldberg–Shahidi, which detects the presence of poles of these operators at 0.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Adisorn Kittisopaporn ◽  
Pattrawut Chansangiam

AbstractThis paper proposes an effective gradient-descent iterative algorithm for solving a generalized Sylvester-transpose equation with rectangular matrix coefficients. The algorithm is applicable for the equation and its interesting special cases when the associated matrix has full column-rank. The main idea of the algorithm is to have a minimum error at each iteration. The algorithm produces a sequence of approximated solutions converging to either the unique solution, or the unique least-squares solution when the problem has no solution. The convergence analysis points out that the algorithm converges fast for a small condition number of the associated matrix. Numerical examples demonstrate the efficiency and effectiveness of the algorithm compared to renowned and recent iterative methods.


2014 ◽  
Vol 29 (03n04) ◽  
pp. 1430001 ◽  
Author(s):  
V. K. DOBREV

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.


1983 ◽  
Vol 3 (1) ◽  
pp. 129-133 ◽  
Author(s):  
Colin E. Sutherland

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.


2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Mirko Primc

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.


2021 ◽  
Vol 36 (6) ◽  
pp. 359-379
Author(s):  
Kirill M. Terekhov

Abstract This article is dedicated to the general finite-volume framework used to discretize and solve saddle-point problems of various physics. The framework applies the Ostrogradsky–Gauss theorem to transform a divergent part of the partial differential equation into a surface integral, approximated by the summation of vector fluxes over interfaces. The interface vector fluxes are reconstructed using the harmonic averaging point concept resulting in the unique vector flux even in a heterogeneous anisotropic medium. The vector flux is modified with the consideration of eigenvalues in matrix coefficients at vector unknowns to address both the hyperbolic and saddle-point problems, causing nonphysical oscillations and an inf-sup stability issue. We apply the framework to several problems of various physics, namely incompressible elasticity problem, incompressible Navier–Stokes, Brinkman–Hazen–Dupuit–Darcy, Biot, and Maxwell equations and explain several nuances of the application. Finally, we test the framework on simple analytical solutions.


2011 ◽  
Vol 63 (6) ◽  
pp. 1238-1253 ◽  
Author(s):  
Daniel Bump ◽  
Maki Nakasuji

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.


1976 ◽  
Vol 15 (1) ◽  
pp. 151-163 ◽  
Author(s):  
J. A. Bennett

A ray or quasi-optical approximation is developed, using complex rays. The ‘amplitude’ terms are matrices, rather than vectors that represent the wave polarization. Thus, the way the propagation resolves a source into various modes is described. The second term in the amplitude series is shown to include a type of inter-mode coupling. It is shown that initial values needed to integrate along the rays can be chosen so that the leading term of the approximation agrees with the far-field solution for localized sources in a homogeneous medium. By invoking the ‘canonical problem’ method, the result is extended to give an approximation for the Green's matrix in a slowly-varying medium.


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