G-principal series of stocks in an algebra

1983 ◽  
pp. 240-244
Author(s):  
Francisco Poyatos
Keyword(s):  
Principal Series

scholarly journals R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

2021 ◽  
pp. 1-61
Author(s):  
Fan Gao
Keyword(s):  
Series Representation ◽  
Principal Series ◽  
Symplectic Groups ◽  
Principal Series Representation ◽  
Connected Group ◽  
Covering Group ◽  
Simply Connected

Abstract For a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such a principal series representation. Moreover, for certain saturated covers of a semisimple simply connected group, we also propose a simpler conjectural formula for such dimensions. This latter conjectural formula is verified in several cases, including covers of the symplectic groups.

scholarly journals Asymptotic symmetries and celestial CFT

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Laura Donnay ◽  
Sabrina Pasterski ◽  
Andrea Puhm
Keyword(s):  
Principal Series ◽  
Finite Energy ◽  
Continuous Series ◽  
Asymptotic Symmetry ◽  
Null Infinity ◽  
Conformal Dimension ◽  
Contour Integrals ◽  
Celestial Sphere ◽  
Goldstone Modes ◽  
Asymptotic Symmetries

Abstract We provide a unified treatment of conformally soft Goldstone modes which arise when spin-one or spin-two conformal primary wavefunctions become pure gauge for certain integer values of the conformal dimension ∆. This effort lands us at the crossroads of two ongoing debates about what the appropriate conformal basis for celestial CFT is and what the asymptotic symmetry group of Einstein gravity at null infinity should be. Finite energy wavefunctions are captured by the principal continuous series ∆ ∈ 1 + iℝ and form a complete basis. We show that conformal primaries with analytically continued conformal dimension can be understood as certain contour integrals on the principal series. This clarifies how conformally soft Goldstone modes fit in but do not augment this basis. Conformally soft gravitons of dimension two and zero which are related by a shadow transform are shown to generate superrotations and non-meromorphic diffeomorphisms of the celestial sphere which we refer to as shadow superrotations. This dovetails the Virasoro and Diff(S2) asymptotic symmetry proposals and puts on equal footing the discussion of their associated soft charges, which correspond to the stress tensor and its shadow in the two-dimensional celestial CFT.

scholarly journals Wavefunctions in dS/CFT revisited: principal series and double-trace deformations

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Hiroshi Isono ◽  
Hoiki Madison Liu ◽  
Toshifumi Noumi
Keyword(s):  
Principal Series ◽  
Light Fields ◽  
Quadratic Term ◽  
De Sitter Spacetime ◽  
Cosmological Horizon ◽  
De Sitter ◽  
Complementary Series ◽  
Point Function ◽  
Sitter Spacetime ◽  
Simple Scaling

Abstract We study wavefunctions of heavy scalars on de Sitter spacetime and their implications to dS/CFT correspondence. In contrast to light fields in the complementary series, heavy fields in the principal series oscillate outside the cosmological horizon. As a consequence, the quadratic term in the wavefunction does not follow a simple scaling and so it is hard to identify it with a conformal two-point function. In this paper, we demonstrate that it should be interpreted as a two-point function on a cyclic RG flow which is obtained by double-trace deformations of the dual CFT. This is analogous to the situation in nonrelativistic AdS/CFT with a bulk scalar whose mass squared is below the Breitenlohner-Freedman (BF) bound. We also provide a new dS/CFT dictionary relating de Sitter two-point functions and conformal two-point functions in the would-be dual CFT.

scholarly journals III. A critical study of spectral series.— Part I. The alkalies H and He

1911 ◽  
Vol 210 (459-470) ◽  
pp. 57-111 ◽  
Keyword(s):  
Principal Series ◽  
Atomic Weight ◽  
Spectral Lines ◽  
Critical Study ◽  
Spectral Series ◽  
Observational Error ◽  
Present Communication ◽  
Present Purpose

The following investigation was commenced some years ago, at a time when the discussion as to the atomic weight of Radium raised the question as to the dependence on their atomic weights of the wave-lengths of corresponding spectral lines of different elements of the same group. The arguments were necessarily vague and unconvincing in the absence of any exact knowledge as to the connection of wave­-length with atomic weight, even supposing such connection existed. Our knowledge of series spectra is chiefly—one might say almost wholly—due to the sets of very exact measurements of Kayser and Runge, and of Runge and Paschen, supplemented by extensions to longer and shorter wave-lengths by Bergmann, Konen and Hagenbach, Lehmann, Ram age, and Saunders. These have been only quite recently added to by Paschen and by the remarkable extension of the Sodium Principal series up to 48 terms by Wood. A most valuable feature of Kayser’s work was the publication of possible errors of observation. This has rendered it possible to test with certainty whether any relation suggesting itself is true within limits of observational error or not. In fact, without this, the investigation, of which the present communication forms a first part, could not have been carried out. So far as the author knows, Saunders is the only other observer who has accompanied his observations with estimates of this kind. Others have given probable errors—practically estimates of the exactness with which they can repeat readings of that feature of a line which they take to be the centre—an estimate of little value for the present purpose. In deducing data from a set of lines it is thus possible to express their errors in terms of the original errors in the observations, and limits to the latter give limiting variations to the former. We therefore know with certainty what latitude in inferences is permissible, and are often enabled to say that such inference is not justifiable.

scholarly journals Hyperfine structure in the arc spectrum of cæsium and nuclear rotation

1928 ◽  
Vol 121 (787) ◽  
pp. 432-447 ◽  
Keyword(s):  
Hyperfine Structure ◽  
Nuclear Spin ◽  
High Frequency ◽  
Principal Series ◽  
Very High Frequency ◽  
Nuclear Rotation ◽  
Higher Temperature ◽  
The Difference ◽  
Mechanical Moment ◽  
Very High

The arc spectrum of cæsium was investigated with the object of finding whether any of its lines possessed hyperfine structure, resulting from a nuclear magnetic moment, due to a quantised nuclear spin. The lines belonging to the principal series should, owing to the greater degree of penetration of the electron in the (1 s or 6 1 ) orbit, and the correspondingly greater interaction, show the greatest effect. The lines of the principal series are very easily broadened if the vapour pressure of the metal becomes high, so that great care had to be used in obtaining the spectrum of cæsium at a sufficiently low temperature. The most satisfactory method of excitation was found to be the application by means of external electrodes of a very high frequency alternating current to a tube filled with helium at about 2 mm. pressure containing a small quantity of cæsium. The tube required slight heating to bring out the cæsium lines; without this the helium spectrum was very much stronger than the metallic spectrum. At a very low vapour pressures of cæsium the discharge was blue in colour. Under these conditions the lines of the principal series showed no broadening greater than that due to thermal agitation, but at a slightly higher temperature the colour of the discharge became purple and the lines broadened. The lines belonging to the principal series were found to be very close doublets with very nearly constant frequencies differences. A theory is worked out which explains the origin of these doublets, assuming a nuclear spin of one half quantum; by correlating the difference in the separation of the hyperfine structure doublets in the 1 s — m 2 p 3/2 lines and the 1 s — m 2 p 1/2 lines, it is shown that a ratio of the magnetic to the mechanical moment of the nucleus about twice as great as the corresponding ratio for the electron would account for the observed frequency differences. The spectral notation used throughout is that of Hund. The results are compared with those found for the hyperfine structure of some of the bismuth lines by Back and Goudsmid, and are found to be in satisfactory agreement. A selection principle is found which applies both to the bismuth and the cæsium spectrum.

scholarly journals REGULAR FUNCTIONS ON THE SHILOV BOUNDARY

2005 ◽  
Vol 04 (06) ◽  
pp. 613-629 ◽  
Author(s):  
OLGA BERSHTEIN
Keyword(s):  
Symmetric Domain ◽  
Principal Series ◽  
Bounded Symmetric Domain ◽  
Bounded Symmetric Domains ◽  
Shilov Boundary ◽  
Generators And Relations ◽  
Regular Functions ◽  
Degenerate Principal Series

In this paper a *-algebra of regular functions on the Shilov boundary S(𝔻) of bounded symmetric domain 𝔻 is constructed. The algebras of regular functions on S(𝔻) are described in terms of generators and relations for two particular series of bounded symmetric domains. Also, the degenerate principal series of quantum Harish–Chandra modules related to S(𝔻) = Un is investigated.

scholarly journals Instability of complex CFTs with operators in the principal series

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Dario Benedetti
Keyword(s):  
Principal Series ◽  
Conformal Group ◽  
Point Of View ◽  
Field Theories ◽  
Operator Product ◽  
Quantum Field Theories ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Primary Operator ◽  
Tensor Models

Abstract We prove the instability of d-dimensional conformal field theories (CFTs) having in the operator-product expansion of two fundamental fields a primary operator of scaling dimension h = $$ \frac{d}{2} $$ d 2 + i r, with non-vanishing r ∈ ℝ. From an AdS/CFT point of view, this corresponds to a well-known tachyonic instability, associated to a violation of the Breitenlohner-Freedman bound in AdSd+1; we derive it here directly for generic d-dimensional CFTs that can be obtained as limits of multiscalar quantum field theories, by applying the harmonic analysis for the Euclidean conformal group to perturbations of the conformal solution in the two-particle irreducible (2PI) effective action. Some explicit examples are discussed, such as melonic tensor models and the biscalar fishnet model.

scholarly journals Kirillov models for distinguished representations

1989 ◽  
Vol 116 ◽  
pp. 89-110 ◽  
Author(s):  
Courtney Moen
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Automorphic Forms ◽  
Principal Series ◽  
Weil Representation ◽  
Local Factors ◽  
Distinguished Representations ◽  
Group A ◽  
Local Representations ◽  
Covering Groups

In the theory of automorphic forms on covering groups of the general linear group, a central role is played by certain local representations which have unique Whittaker models. A representation with this property is called distinguished. In the case of the 2-sheeted cover of GL2, these representations arise as the the local components of generalizations of the classical θ-function. They have been studied thoroughly in [GPS]. The Weil representation provides these representations with a very nice realization, and the local factors attached to these representations can be computed using this realization. It has been shown [KP] that only in the case of a certain 3-sheeted cover do we find other principal series of covering groups of GL2 which have a unique Whittaker model. It is natural to ask if these distinguished representations also have a realization analgous to the Weil representation.

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