Local integrability results in harmonic analysis on reductive groups in large positive characteristic

Let $G$ be a connected reductive algebraic group over a non-Archimedean local field $K$, and let $g$ be its Lie algebra. By a theorem of Harish-Chandra, if $K$ has characteristic zero, the Fourier transforms of orbital integrals are represented on the set of regular elements in $g(K)$ by locally constant functions, which, extended by zero to all of $g(K)$, are locally integrable. In this paper, we prove that these functions are in fact specializations of constructible motivic exponential functions. Combining this with the Transfer Principle for integrability [R. Cluckers, J. Gordon, I. Halupczok, "Transfer principles for integrability and boundedness conditions for motivic exponential functions", preprint arXiv:1111.4405], we obtain that Harish-Chandra's theorem holds also when $K$ is a non-Archimedean local field of sufficiently large positive characteristic. Under the hypothesis on the existence of the mock exponential map, this also implies local integrability of Harish-Chandra characters of admissible representations of $G(K)$, where $K$ is an equicharacteristic field of sufficiently large (depending on the root datum of $G$) characteristic.Comment: Compared to v2/v3: some proofs simplified, the main statement generalized; slightly reorganized. Regarding the automatically generated text overlap note: it overlaps with the Appendix B (which is part of arXiv:1208.1945) written by us; the appendix and this article cross-reference each other, and since the set-up is very similar, some overlap is unavoidabl

Integrable Systems and Discrete Geometry

This is an expository article for Elsevier's Encyclopedia of Mathematical Physics on the subject in the title. Comments/corrections welcome.Comment: 22 pages, 7 figure

Discrete asymptotic nets and W-congruences in Plucker line geometry

The asymptotic lattices and their transformations are studied within the line geometry approach. It is shown that the discrete asymptotic nets are represented by isotropic congruences in the Plucker quadric. On the basis of the Lelieuvre-type representation of asymptotic lattices and of the discrete analog of the Moutard transformation, it is constructed the discrete analog of the W-congruences, which provide the Darboux-Backlund type transformation of asymptotic lattices.The permutability theorems for the discrete Moutard transformation and for the corresponding transformation of asymptotic lattices are established as well. Moreover, it is proven that the discrete W-congruences are represented by quadrilateral lattices in the quadric of Plucker. These results generalize to a discrete level the classical line-geometric approach to asymptotic nets and W-congruences, and incorporate the theory of asymptotic lattices into more general theory of quadrilateral lattices and their reductions.Comment: 28 pages, 4 figures; expanded Introduction, new Section, added reference

Deforming Lie algebras to Frobenius integrable non-autonomous Hamiltonian systems

Motivated by the theory of Painlev\'e equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra of Hamiltonian vector fields can be deformed to a time-dependent Lie algebra of Frobenius integrable vector fields spanning the same distribution as the original algebra. The results are applied to quasi-St\"ackel systems.Comment: 14 pages, no figures. We repaired Example 5 and also made some minor amendments in the tex

Intertwining operator for AG2AG_2 Calogero-Moser-Sutherland system

We consider generalised Calogero-Moser-Sutherland quantum Hamiltonian $H$ associated with a configuration of vectors $AG_2$ on the plane which is a union of $A_2$ and $G_2$ root systems. The Hamiltonian $H$ depends on one parameter. We find an intertwining operator between $H$ and the Calogero-Moser-Sutherland Hamiltonian for the root system $G_2$. This gives a quantum integral for $H$ of order 6 in an explicit form thus establishing integrability of $H$.Comment: 24 page

Classification of the Killing Vectors in Nonexpanding HH-Spaces with Lambda

Conformal Killing equations and their integrability conditions for nonexpanding hyperheavenly spaces with Lambda are studied. Reduction of ten Killing equations to one master equation is presented. Classification of homothetic and isometric Killing vectors in nonexpanding hyperheavenly spaces with Lambda and homothetic Killing vectors in heavenly spaces is given. Some nonexpanding complex metrics of types [III,N]x[N] are found. A simple example of Lorentzian real slice of the type [N]x[N] is explicitly given

Reproducing formulas for generalized translation invariant systems on locally compact abelian groups

In this paper we connect the well established discrete frame theory of generalized shift invariant systems to a continuous frame theory. To do so, we let $\Gamma_j$, $j \in J$, be a countable family of closed, co-compact subgroups of a second countable locally compact abelian group $G$ and study systems of the form $\cup_{j \in J}\{g_{j,p}(\cdot - \gamma)\}_{\gamma \in \Gamma_j, p \in P_j}$ with generators $g_{j,p}$ in $L^2(G)$ and with each $P_j$ being a countable or an uncountable index set. We refer to systems of this form as generalized translation invariant (GTI) systems. Many of the familiar transforms, e.g., the wavelet, shearlet and Gabor transform, both their discrete and continuous variants, are GTI systems. Under a technical $\alpha$ local integrability condition ($\alpha$-LIC) we characterize when GTI systems constitute tight and dual frames that yield reproducing formulas for $L^2(G)$. This generalizes results on generalized shift invariant systems, where each $P_j$ is assumed to be countable and each $\Gamma_j$ is a uniform lattice in $G$, to the case of uncountably many generators and (not necessarily discrete) closed, co-compact subgroups. Furthermore, even in the case of uniform lattices $\Gamma_j$, our characterizations improve known results since the class of GTI systems satisfying the $\alpha$-LIC is strictly larger than the class of GTI systems satisfying the previously used local integrability condition. As an application of our characterization results, we obtain new characterizations of translation invariant continuous frames and Gabor frames for $L^2(G)$. In addition, we will see that the admissibility conditions for the continuous and discrete wavelet and Gabor transform in $L^2(\mathbb{R}^n)$ are special cases of the same general characterizing equations.Comment: Minor changes (v2). To appear in Trans. Amer. Math. So