scholarly journals Homological branching law for $$({\mathrm {GL}}_{n+1}(F), {\mathrm {GL}}_n(F))$$: projectivity and indecomposability

2021 ◽  
Author(s):  
Kei Yuen Chan
Keyword(s):  
Opposite Direction ◽  
Local Field ◽  
Main Tool ◽  
Irreducible Representations ◽  
Smooth Representation ◽  
Branching Law ◽  
Left And Right

AbstractLet F be a non-Archimedean local field. This paper studies homological properties of irreducible smooth representations restricted from $${\mathrm {GL}}_{n+1}(F)$$ GL n + 1 ( F ) to $${\mathrm {GL}}_n(F)$$ GL n ( F ) . A main result shows that each Bernstein component of an irreducible smooth representation of $${\mathrm {GL}}_{n+1}(F)$$ GL n + 1 ( F ) restricted to $${\mathrm {GL}}_n(F)$$ GL n ( F ) is indecomposable. We also classify all irreducible representations which are projective when restricting from $${\mathrm {GL}}_{n+1}(F)$$ GL n + 1 ( F ) to $${\mathrm {GL}}_n(F)$$ GL n ( F ) . A main tool of our study is a notion of left and right derivatives, extending some previous work joint with Gordan Savin. As a by-product, we also determine the branching law in the opposite direction.

scholarly journals Smooth representations of unit groups of split basic algebras over non-Archimedean local fields

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Carlos A. M. André ◽  
João Dias
Keyword(s):  
Local Field ◽  
Unit Group ◽  
Basic Algebra ◽  
Local Fields ◽  
Dimensional Representation ◽  
Smooth Representation ◽  
One Dimensional ◽  
Finite Dimensional

Abstract We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-dimensional split basic algebra 𝒜 over a non-Archimedean local field. In particular, we prove a version of Gutkin’s conjecture, namely, we prove that every irreducible smooth representation of 𝐺 is compactly induced by a one-dimensional representation of the unit group of some subalgebra of 𝒜. We also discuss admissibility and unitarisability of smooth representations of 𝐺.

scholarly journals Conjectures and results about parabolic induction of representations of $${\text {GL}}_n(F)$$

2020 ◽  
Vol 222 (3) ◽  
pp. 695-747
Author(s):  
Erez Lapid ◽  
Alberto Mínguez
Keyword(s):  
Linear Group ◽  
Local Field ◽  
General Linear Group ◽  
Irreducible Representations ◽  
General Linear ◽  
Parabolic Induction ◽  
Geometric Conditions ◽  
Irreducible Components ◽  
Special Cases

Abstract In 1980 Zelevinsky introduced certain commuting varieties whose irreducible components classify complex, irreducible representations of the general linear group over a non-archimedean local field with a given supercuspidal support. We formulate geometric conditions for certain triples of such components and conjecture that these conditions are related to irreducibility of parabolic induction. The conditions are in the spirit of the Geiss–Leclerc–Schröer condition that occurs in the conjectural characterization of $$\square $$ □ -irreducible representations. We verify some special cases of the new conjecture and check that the geometric and representation-theoretic conditions are compatible in various ways.

scholarly journals On the unramified spectrum of spherical varieties over p-adic fields

2008 ◽  
Vol 144 (4) ◽  
pp. 978-1016 ◽  
Author(s):  
Yiannis Sakellaridis
Keyword(s):  
Harmonic Analysis ◽  
Weyl Group ◽  
Reductive Group ◽  
Dual Group ◽  
Irreducible Representations ◽  
Spherical Varieties ◽  
Suitable Space ◽  
Borel Orbits ◽  
Left And Right ◽  
New Interpretation

AbstractThe description of irreducible representations of a group G can be seen as a problem in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G×G by left and right multiplication. For a split p-adic reductive group G over a local non-archimedean field, unramified irreducible smooth representations are in bijection with semisimple conjugacy classes in the ‘Langlands dual’ group. We generalize this description to an arbitrary spherical variety X of G as follows. Irreducible unramified quotients of the space $C_c^\infty (X)$ are in natural ‘almost bijection’ with a number of copies of AX*/WX, the quotient of a complex torus by the ‘little Weyl group’ of X. This leads to a description of the Hecke module of unramified vectors (a weak analog of geometric results of Gaitsgory and Nadler), and an understanding of the phenomenon that representations ‘distinguished’ by certain subgroups are functorial lifts. In the course of the proof, rationality properties of spherical varieties are examined and a new interpretation is given for the action, defined by Knop, of the Weyl group on the set of Borel orbits.

scholarly journals Simple Patch Antenna with Filtering Function Using Two U-Slots

2021 ◽  
Vol 21 (5) ◽  
pp. 425-429
Author(s):  
Youngje Sung
Keyword(s):  
Opposite Direction ◽  
Patch Antenna ◽  
Center Frequency ◽  
Impedance Bandwidth ◽  
Sharp Band ◽  
Left And Right ◽  
Filtering Function

In this study, two U-slots of different sizes are used to combine the filtering function with a patch antenna. The U-shaped slots are etched into the patch, and currents in the opposite direction exist around these slots. Therefore, the currents cancel each other out, and a radiation null is formed. As a result, two radiation nulls are implemented on the left and right sides of the passband. To demonstrate the novelty of the proposed concept, a filtering patch antenna with a center frequency of 3.21 GHz and a 10 dB impedance bandwidth of 19.9% is designed and fabricated. High suppression levels of 25.33 and 19.32 dB in the lower and higher stopbands, respectively, are achieved. Therefore, a sharp band skirt and good selectivity are exhibited in the boresight gain response. The two radiation nulls are located at 2.4 and 3.7 GHz and can be independently adjusted.

scholarly journals Combinatorial Reid's recipe for consistent dimer models

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Alastair Craw ◽  
Liana Heuberger ◽  
Jesus Tapia Amador
Keyword(s):  
Moduli Space ◽  
Hilbert Scheme ◽  
Abelian Subgroup ◽  
Main Tool ◽  
Derived Category ◽  
Irreducible Representations ◽  
Crepant Resolution ◽  
Mckay Correspondence ◽  
Dimer Model ◽  
Dimer Models

Reid's recipe for a finite abelian subgroup $G\subset \text{SL}(3,\mathbb{C})$ is a combinatorial procedure that marks the toric fan of the $G$-Hilbert scheme with irreducible representations of $G$. The geometric McKay correspondence conjecture of Cautis--Logvinenko that describes certain objects in the derived category of $G\text{-Hilb}$ in terms of Reid's recipe was later proved by Logvinenko et al. We generalise Reid's recipe to any consistent dimer model by marking the toric fan of a crepant resolution of the vaccuum moduli space in a manner that is compatible with the geometric correspondence of Bocklandt--Craw--Quintero-V\'{e}lez. Our main tool generalises the jigsaw transformations of Nakamura to consistent dimer models. Comment: 29 pages, published version

scholarly journals Gross–Prasad periods for reducible representations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
David Loeffler
Keyword(s):  
Local Field ◽  
Irreducible Representations ◽  
Total Degree ◽  
Field Extensions

Abstract We study GL 2 ⁡ ( F ) {\operatorname{GL}_{2}(F)} -invariant periods on representations of GL 2 ⁡ ( A ) {\operatorname{GL}_{2}(A)} , where F is a non-archimedean local field and A / F {A/F} a product of field extensions of total degree 3. For irreducible representations, a theorem of Prasad shows that the space of such periods has dimension ⩽ 1 {\leqslant 1} , and is non-zero when a certain ε-factor condition holds. We give an extension of this result to a certain class of reducible representations (of Whittaker type), extending results of Harris–Scholl when A is the split algebra F × F × F {F\times F\times F} .

scholarly journals Decoding of Human Movements Based on Deep Brain Local Field Potentials Using Ensemble Neural Networks

2017 ◽  
Vol 2017 ◽  
pp. 1-16 ◽  
Author(s):  
Mohammad S. Islam ◽  
Khondaker A. Mamun ◽  
Hai Deng
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Local Field ◽  
Local Field Potentials ◽  
Ensemble Classifier ◽  
Majority Voting ◽  
Final Decision ◽  
Field Potentials ◽  
Left And Right ◽  
Deep Brain

Decoding neural activities related to voluntary and involuntary movements is fundamental to understanding human brain motor circuits and neuromotor disorders and can lead to the development of neuromotor prosthetic devices for neurorehabilitation. This study explores using recorded deep brain local field potentials (LFPs) for robust movement decoding of Parkinson’s disease (PD) and Dystonia patients. The LFP data from voluntary movement activities such as left and right hand index finger clicking were recorded from patients who underwent surgeries for implantation of deep brain stimulation electrodes. Movement-related LFP signal features were extracted by computing instantaneous power related to motor response in different neural frequency bands. An innovative neural network ensemble classifier has been proposed and developed for accurate prediction of finger movement and its forthcoming laterality. The ensemble classifier contains three base neural network classifiers, namely, feedforward, radial basis, and probabilistic neural networks. The majority voting rule is used to fuse the decisions of the three base classifiers to generate the final decision of the ensemble classifier. The overall decoding performance reaches a level of agreement (kappa value) at about0.729±0.16for decoding movement from the resting state and about0.671±0.14for decoding left and right visually cued movements.

scholarly journals Multiplicity Free Jacquet Modules

2012 ◽  
Vol 55 (4) ◽  
pp. 673-688 ◽  
Author(s):  
Avraham Aizenbud ◽  
Dmitry Gourevitch
Keyword(s):  
Finite Field ◽  
Natural Number ◽  
Parabolic Subgroup ◽  
Local Field ◽  
Classical Method ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Unipotent Subgroup ◽  
Multiplicity Free

AbstractLet F be a non-Archimedean local field or a finite field. Let n be a natural number and k be 1 or 2. Consider G := GLn+k(F) and let M := GLn(F) × GLk(F) < G be a maximal Levi subgroup. Let U < G be the corresponding unipotent subgroup and let P = MU be the corresponding parabolic subgroup. Let be the Jacquet functor, i.e., the functor of coinvariants with respect toU. In this paper we prove that J is a multiplicity free functor, i.e., dim HomM(J(π), ρ) ≤ 1, for any irreducible representations π of G and ρ of M. We adapt the classical method of Gelfand and Kazhdan, which proves the “multiplicity free” property of certain representations to prove the “multiplicity free” property of certain functors. At the end we discuss whether other Jacquet functors are multiplicity free.

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