Ellipticity and discrete series

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.

Flat deformations of type IIB S-folds

Type IIB S-folds of the form AdS4× S1× S5 have been shown to contain axion-like deformations parameterising flat directions in the 4D scalar potential and corresponding to marginal deformations of the dual S-fold CFT’s. In this note we present a group-theoretical characterisation of such flat deformations and provide a 5D interpretation thereof in terms of $$ \mathfrak{s}(6) $$ s 6 -valued duality twists inducing a class of Cremmer-Scherk-Schwarz flat gaugings in a reduction from 5D to 4D. In this manner we establish the existence of two flat deformations for the $$ \mathcal{N} $$ N = 4 and SO(4) symmetric S-fold causing a symmetry breaking down to its U(1)2 Cartan subgroup. The result is a new two-parameter family of non-supersymmetric S-folds which are perturbatively stable at the lower-dimensional supergravity level, thus providing the first examples of such type IIB backgrounds.

A prime geodesic theorem for SL3(ℤ)

We 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.

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

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 $\text{GL}_{2}(\mathbb{F}_{p})$.

On the action of the Hua subgroups in special Moufang sets

We 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

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.

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

In 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.

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

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.