On Pairs of Additive Subgroups Associated with Intermediate Subgroups of Groups of Lie Type over Nonperfect Fields

The author has previously (Trudy IMM UrO RAN, 19(2013), no. 3) described the groups lying between twisted Chevalley groups G(K) and G(F) of type 2Al, 2Dl, 2E6, 3D4 in the case when the larger field F is an algebraic extension of the smaller nonperfect field K of exceptional characteristic for the group G(F) (characteristics 2 and 3 for the type 3D4 and only 2 for other types). It turned out that apart from, perhaps, the type 2Dl, such intermediate subgroups are standard, that is, they are exhausted by the groups G(P)H for some intermediate subfield P, K ⊆ P ⊆ F, and of the diagonal subgroup H normalizing the group G(P). In this note, it is established that intermediate subgroups are also standard for the type 2Dl

New integrable coset sigma models

By using the general framework of affine Gaudin models, we construct a new class of integrable sigma models. They are defined on a coset of the direct product of N copies of a Lie group over some diagonal subgroup and they depend on 3N − 2 free parameters. For N = 1 the corresponding model coincides with the well-known symmetric space sigma model. Starting from the Hamiltonian formulation, we derive the Lagrangian for the N = 2 case and show that it admits a remarkably simple form in terms of the classical ℛ-matrix underlying the integrability of these models. We conjecture that a similar form of the Lagrangian holds for arbitrary N. Specifying our general construction to the case of SU(2) and N = 2, and eliminating one of the parameters, we find a new three-parametric integrable model with the manifold T1,1 as its target space. We further comment on the connection of our results with those existing in the literature.

Periods of automorphic forms: The case of (U n+1×U n ,U n )

Following Jacquet, Lapid and Rogawski, we define regularized periods of automorphic forms on \mathrm{U}_{n+1} \times \mathrm{U}_{n} along the diagonal subgroup {\mathrm{U}_{n}} and compute the regularized periods of cuspidal Eisenstein series and their residues. The formula for the periods of residues has an application to the Gan–Gross–Prasad conjecture.

Integration over the quantum diagonal subgroup and associated Fourier-like algebras

By analogy with the classical construction due to Forrest, Samei and Spronk, we associate to every compact quantum group [Formula: see text], a completely contractive Banach algebra [Formula: see text], which can be viewed as a deformed Fourier algebra of [Formula: see text]. To motivate the construction, we first analyze in detail the quantum version of the integration over the diagonal subgroup, showing that although the quantum diagonal subgroups in fact never exist, as noted earlier by Kasprzak and Sołtan, the corresponding integration represented by a certain idempotent state on [Formula: see text] makes sense as long as [Formula: see text] is of Kac type. Finally, we analyze as an explicit example the algebras [Formula: see text], [Formula: see text], associated to Wang’s free orthogonal groups, and show that they are not operator weakly amenable.

Periods of automorphic forms: the case of

Following Jacquet, Lapid and Rogawski, we define a regularized period of an automorphic form on $\text{GL}_{n+1}\times \text{GL}_{n}$ along the diagonal subgroup $\text{GL}_{n}$ and express it in terms of the Rankin–Selberg integral of Jacquet, Piatetski-Shapiro and Shalika. This extends the theory of Rankin–Selberg integrals to all automorphic forms on $\text{GL}_{n+1}\times \text{GL}_{n}$.

CONSTRUCTIVE WALL-CROSSING AND SEIBERG–WITTEN

We outline a comprehensive and first-principle solution to the wall-crossing problem in D = 4N = 2 Seiberg–Witten theories. We start with a brief review of the multi-centered nature of the typical BPS states and of how this allows them to disappear abruptly as parameters or vacuum moduli are continuously changed. This means that the wall-crossing problem is really a bound state formation/dissociation problem. A low energy dynamics for arbitrary collections of dyons is derived, with the proximity to the so-called marginal stability wall playing the role of the small expansion parameter. We discover that the low energy dynamics of such BPS dyons cannot be reduced to one on the classical moduli space, [Formula: see text], yet the index can be phrased in terms of [Formula: see text]. The so-called rational invariant, first seen in Kontsevich–Soibelman formalism of wall-crossing, is shown to incorporate Bose/Fermi statistics automatically. Furthermore, an equivariant version of the index is shown to compute the protected spin character of the underlying D = 4N = 2 theory, where [Formula: see text] isometry of [Formula: see text] is identified as a diagonal subgroup of rotation SU(2)L and R-symmetry SU(2)R.

Equidistribution of joinings under off-diagonal polynomial flows of nilpotent Lie groups

Let $G$ be a connected nilpotent Lie group. Given probability-preserving$G$-actions $(X_i,\Sigma _i,\mu _i,u_i)$, $i=0,1,\ldots ,k$, and also polynomial maps $\phi _i:\mathbb {R}\to G$, $i=1,\ldots ,k$, we consider the trajectory of a joining $\lambda $ of the systems $(X_i,\Sigma _i,\mu _i,u_i)$ under the ‘off-diagonal’ flow \[ (t,(x_0,x_1,x_2,\ldots ,x_k))\mapsto (x_0,u_1^{\phi _1(t)}x_1,u_2^{\phi _2(t)}x_2,\ldots ,u_k^{\phi _k(t)}x_k). \] It is proved that any joining $\lambda $ is equidistributed under this flow with respect to some limit joining $\lambda '$. This is deduced from the stronger fact of norm convergence for a system of multiple ergodic averages, related to those arising in Furstenberg’s approach to the study of multiple recurrence. It is also shown that the limit joining $\lambda '$ is invariant under the subgroup of $G^{k+1}$generated by the image of the off-diagonal flow, in addition to the diagonal subgroup.

On endoscopy and the refined Gross–Prasad conjecture for (SO5, SO4)

We prove an explicit formula for periods of certain automorphic forms on SO5 × SO4 along the diagonal subgroup SO4 in terms of L-values. Our formula also involves a quantity from the theory of endoscopy, as predicted by the refined Gross–Prasad conjecture.