Representations induced by characters of a subgroup of the Cartan subgroup

1973 ◽  
pp. 23-73
Author(s):  
Floyd L. Williams
Keyword(s):  
Cartan Subgroup

On the ergodic properties of Cartan flows in ergodic actions of ${\bf SL}_{\bf 2}{\bf (R)}$ and ${\bf SO}{\bf ({\bi n},1)}$

1997 ◽  
Vol 17 (6) ◽  
pp. 1371-1382
Author(s):  
ALEX FURMAN ◽  
BENJAMIN WEISS
Keyword(s):  
Riemann Surface ◽  
Geodesic Flow ◽  
Probability Space ◽  
Compact Riemann Surface ◽  
Cartan Subgroup ◽  
Ergodic Properties

Let $G={\rm SL_2({\bf R})}$ (or $G={\rm SO}(n,1)$) act ergodically on a probability space $(X,m)$. We consider the ergodic properties of the flow $(X,m,\{g_t\})$, where $\{g_t\}$ is a Cartan subgroup of $G$. The geodesic flow on a compact Riemann surface is an example of such a flow; here $X=G/\Gamma$ is a transitive $G$-space, $G={\rm SL_2({\bf R})}$ and $\Gamma\subset G$ is a lattice. In this case the flow is Bernoullian.For the general ergodic $G$-action, the flow $(X,m,\{g_t\})$ is always a $K$-flow, however there are examples in which it is not Bernoullian.

scholarly journals Invariant measures for the continual Cartan subgroup

2008 ◽  
Vol 255 (9) ◽  
pp. 2661-2682 ◽  
Author(s):  
A.M. Vershik
Keyword(s):  
Invariant Measures ◽  
Cartan Subgroup

scholarly journals On the action of the Hua subgroups in special Moufang sets

2008 ◽  
Vol 144 (1) ◽  
pp. 77-84 ◽  
Author(s):  
YOAV SEGEV ◽  
RICHARD M. WEISS
Keyword(s):  
Invariant Subgroup ◽  
Cartan Subgroup ◽  
Root Group

AbstractWe show that in a special Moufang set, either the root groups are elementary abelian 2-groups, or the Hua subgroup H (= the Cartan subgroup) acts “irreducibly” on U, i.e. the only non-trivial H-invariant subgroup of a root group normalized by H is the whole root group.

IRREDUCIBLE UNITARY REPRESENTATIONS OF SUp,q II: THE CONTINUOUS SERIES

2001 ◽  
Vol 12 (01) ◽  
pp. 37-47 ◽  
Author(s):  
RAJ WILSON ◽  
ELIZABETH TANNER
Keyword(s):  
Kronecker Product ◽  
Discrete Series ◽  
Unitary Representations ◽  
Continuous Series ◽  
Maximal Dimension ◽  
Cartan Subgroup ◽  
Integrable Functions ◽  
Irreducible Unitary Representations ◽  
Square Integrable Functions ◽  
Vector Part

A class of irreducible unitary representations belonging to the continuous series of SUp,q is explicitly determined in a space composed of the Kronecker product of three spaces of square integrable functions. The continuous series corresponds to a Cartan subgroup whose vector part has maximal dimension. These representations are distinguished by a parameter r = 1, 2, …, p for p ≤ q in SUp,q. For r = 0, one obtains the representations in the discrete series as in [5], and all representations in the continuous series, for r ≠ 0, are obtained explicitly.

scholarly journals Ellipticity and discrete series

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Bernhard Krötz ◽  
Job J. Kuit ◽  
Eric M. Opdam ◽  
Henrik Schlichtkrull
Keyword(s):  
Discrete Series ◽  
Series Representation ◽  
Reductive Group ◽  
Cartan Subgroup ◽  
Discrete Series Representation ◽  
Spherical Spaces

Abstract We explain by elementary means why the existence of a discrete series representation of a real reductive group G implies the existence of a compact Cartan subgroup of G. The presented approach has the potential to generalize to real spherical spaces.

On Some q-Analogs of a Theorem of Kostant-Rallis

2000 ◽  
Vol 52 (2) ◽  
pp. 438-448 ◽  
Author(s):  
N. R. Wallach ◽  
J. Willenbring
Keyword(s):  
Conjugacy Class ◽  
Quantum Computation ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Cartan Subgroup ◽  
Semisimple Lie Groups ◽  
Semisimple Lie Group ◽  
Real Group ◽  
Qubit States

AbstractIn the first part of this paper generalizationsof Hesselink’s q-analog of Kostant’smultiplicity formula for the action of a semisimple Lie group on the polynomials on its Lie algebra are given in the context of the Kostant-Rallis theorem. They correspond to the cases of real semisimple Lie groups with one conjugacy class of Cartan subgroup. In the second part of the paper a q-analog of the Kostant-Rallis theorem is given for the real group SL(4, ) (that is SO(4) acting on symmetric 4 × 4 matrices). This example plays two roles. First it contrasts with the examples of the first part. Second it has implications to the study of entanglement of mixed 2 qubit states in quantum computation.

A prime geodesic theorem for SL3(ℤ)

2019 ◽  
Vol 31 (5) ◽  
pp. 1179-1201
Author(s):  
Anton Deitmar ◽  
Polyxeni Spilioti ◽  
Yasuro Gon
Keyword(s):  
Cartan Subgroup ◽  
Prime Geodesic Theorem ◽  
Higher Rank

AbstractWe show a prime geodesic theorem for the group {\mathrm{SL}_{3}(\mathbb{Z})} counting those geodesics whose lifts lie in the split Cartan subgroup. This is the first arithmetic prime geodesic theorem of higher rank for a non-cocompact group.

scholarly journals ON THE AUTOMORPHISMS OF THE NONSPLIT CARTAN MODULAR CURVES OF PRIME LEVEL

2016 ◽  
Vol 224 (1) ◽  
pp. 74-92 ◽  
Author(s):  
VALERIO DOSE
Keyword(s):  
Automorphism Group ◽  
Rational Point ◽  
Cartan Subgroup ◽  
Modular Curves ◽  
Modular Curve

We study the automorphisms of the nonsplit Cartan modular curves $X_{\text{ns}}(p)$ of prime level $p$. We prove that if $p\geqslant 29$ all the automorphisms preserve the cusps. Furthermore, if $p\equiv 1~\text{mod}~12$ and $p\neq 13$, the automorphism group is generated by the modular involution given by the normalizer of a nonsplit Cartan subgroup of $\text{GL}_{2}(\mathbb{F}_{p})$. We also prove that for every $p\geqslant 29$ the existence of an exceptional rational automorphism would give rise to an exceptional rational point on the modular curve $X_{\text{ns}}^{+}(p)$ associated to the normalizer of a nonsplit Cartan subgroup of $\text{GL}_{2}(\mathbb{F}_{p})$.

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