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We continue our exploration of various approaches to integration of representations from a Lie algebra $\mbox{Lie} (G)$ to an algebraic group $G$ in positive characteristic. In the present paper we concentrate on an approach exploiting exponentials. This approach works well for over-restricted representations, introduced in this paper, and takes no note of $G$-stability., Accepted by Transactions of the AMS. This paper is split off the earlier versions (1, 2 and 3) of arXiv:1708.06620. Some of the statements in these versions of arXiv:1708.06620 contain mistakes corrected here. Version 2 of this paper: close to the accepted version by the journal, minor improvements, compared to Version 1

For a finite involutive non-degenerate solution $(X,r)$ of the Yang--Baxter equation it is known that the structure monoid $M(X,r)$ is a monoid of I-type, and the structure algebra $K[M(X,r)]$ over a field $K$ share many properties with commutative polynomial algebras, in particular, it is a Noetherian PI-domain that has finite Gelfand--Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid $M(X,r)$ and the algebra $K[M(X,r)]$ is much more complicated than in the involutive case, we provide some deep insights. In this general context, using a realization of Lebed and Vendramin of $M(X,r)$ as a regular submonoid in the semidirect product $A(X,r)\rtimes\mathrm{Sym}(X)$, where $A(X,r)$ is the structure monoid of the rack solution associated to $(X,r)$, we prove that $K[M(X,r)]$ is a module finite normal extension of a commutative affine subalgebra. In particular, $K[M(X,r)]$ is a Noetherian PI-algebra of finite Gelfand--Kirillov dimension bounded by $|X|$. We also characterize, in ring-theoretical terms of $K[M(X,r)]$, when $(X,r)$ is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of $M(X,r)$. These results allow us to control the prime spectrum of the algebra $K[M(X,r)]$ and to describe the Jacobson radical and prime radical of $K[M(X,r)]$. Finally, we give a matrix-type representation of the algebra $K[M(X,r)]/P$ for each prime ideal $P$ of $K[M(X,r)]$. As a consequence, we show that if $K[M(X,r)]$ is semiprime then there exist finitely many finitely generated abelian-by-finite groups, $G_1,\dotsc,G_m$, each being the group of quotients of a cancellative subsemigroup of $M(X,r)$ such that the algebra $K[M(X,r)]$ embeds into $\mathrm{M}_{v_1}(K[G_1])\times\dotsb\times \mathrm{M}_{v_m}(K[G_m])$., A subtle mistake in the proof of Theorem 4.4 has been corrected (will appear in a corrigendum et addendum, TAMS). In the latter paper we also strengthen some of the results by removing the "square free'' condition in Section 5 and in this paper we also prove new homological equivalences in Theorem 4.4

In this paper it is shown that the wave front set of a direct integral of singular, irreducible representations of a real, reductive algebraic group is contained in the singular set. Combining this result with the results of the first paper in this series, the author obtains asymptotic results on the occurrence of tempered representations in induction and restriction problems for real, reductive algebraic groups., Accepted to Transactions of the American Mathematical Society

Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues, known as affine Kac-Moody groups. We solve this problem and construct affine Kac-Moody symmetric spaces in a series of several papers. This paper focuses on the algebraic side; more precisely, we introduce OSAKAs, the algebraic structures used to describe the connection between affine Kac-Moody symmetric spaces and affine Kac-Moody algebras and describe their classification.

The primary theorem of this paper concerns the Poincaré (Hilbert) series for the cohomology ring of a finite group G G with coefficients in a prime field of characteristic p p . This theorem is proved using the ideas of equivariant cohomology whereby one considers more generally the cohomology ring of the Borel construction H ∗ ( E G × G X ) H^*(EG \times _G X) , where X X is a manifold on which G G acts. This work results in a formula that computes the “degree” of the Poincaré series in terms of corresponding degrees of certain subgroups of the group G G . In this paper, we discuss the theorem and the method of proof.

Iterated function systems have been most extensively studied when the functions are affine transformations of Euclidean space and, more recently, projective transformations on real projective space. This paper investigates iterated function systems consisting of Möbius transformations on the extended complex plane or, equivalently, on the Riemann sphere. The main result is a characterization, in terms of topological, geometric, and dynamical properties, of Möbius iterated function systems that possess an attractor. The paper also includes results on the duality between the attractor and repeller of a Möbius iterated function system.

We improve some of the results in our 1999 paper concerning weighted norm estimates for homogeneous singular integrals with rough kernels. Using a representation of such integrals in terms of averages over starlike sets, we prove a two-weight L p L^{p} inequality for 1 > p > 2 1 > p > 2 which we were previously able to obtain only for p ≥ 2 p \geq 2 . We also construct examples of weights that satisfy conditions which were shown in our earlier paper to be sufficient for one-weight inequalities when 1 > p > ∞ 1>p>\infty .

We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the mul- tiplicity one case and extends to higher multiplicity their dilation approach. We prove several results for operator-valued frames concerning duality, dis- jointeness, complementarity , and composition of operator valued frames and the relationship between the two types of similarity (left and right) of such frames. A key technical tool is the parametrization of Parseval operator val- ued frames in terms of a class of partial isometries in the Hilbert space of the analysis operator. We apply these notions to an analysis of multiframe gener- ators for the action of a discrete group G on a Hilbert space. One of the main results of the Han-Larson work was the parametrization of the Parseval frame generators in terms of the unitary operators in the von Neumann algebra gen- erated by the group representation, and the resulting norm path-connectedness of the set of frame generators due to the connectedness of the group of unitary operators of an arbitrary von Neumann algebra. In this paper we general- ize this multiplicity one result to operator-valued frames. However, both the parameterization and the proof of norm path-connectedness turn out to be necessarily more complicated, and this is at least in part the rationale for this paper. Our parameterization involves a class of partial isometries of a different von Neumann algebra. These partial isometries are not path-connected in the norm topology, but only in the strong operator topology. We prove that the set of operator frame generators is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. As in the multiplicity one theory there are analogous results for general (non-Parseval) frames.

We use the notion of rank defined in an earlier paper (2007) to introduce and study two correspondences between small irreducible unitary representations of the split real simple Lie groups of types E n , where n ∈ {6, 7}, and two reductive classical groups. We show that these correspondences classify all of the unitary representations of rank two (in the sense of our earlier paper) of these exceptional groups. We study our correspondences for a specific family of degenerate principal series representations in detail.

"For a connected linear semisimple Lie group G, this paper considers those nonzero limits of discrete series representations having infinitesimal character~0, calling them ""totally degenerate"". Such representations exist if and only if G has a compact Cartan subgroup, is quasisplit, and is acceptable in the sense of Harish-Chandra. Totally degenerate limits of discrete series are natural objects of study in the theory of automorphic forms: in fact, those automorphic representations of adelic groups that have totally degenerate limits of discrete series as archimedean components correspond conjecturally to complex continuous representations of Galois groups of number fields. The automorphic representations in question have important arithmetic significance, but very little has been proved up to now toward establishing this part of the Langlands conjectures. There is some hope of making progress in this area, and for that one needs to know in detail the representations of G under consideration. The aim of this paper is to determine the classification parameters of all totally degenerate limits of discrete series in the Knapp-Zuckerman classification of irreducible tempered representations, i.e., to express these representations as induced representations with ""nondegenerate data"". The paper uses a general argument, based on the finite abelian reducibility group R attached to a specific unitary principal series representation of G. First an easy result gives the aggregate of the classification parameters. Then a harder result uses the easy result to match the classification parameters with the representations of G under consideration in representation-by-representation fashion. The paper includes tables of the classification parameters for all such groups G."

This paper continues previous work in which we developed the Galois theory of strongly normal extensions using differential schemes. In the present paper we derive two main results. First, we show that an extension is strongly normal if and only if a certain differential scheme splits, i.e. is obtained by base extension of a scheme over constants. This gives a geometric characterization to the notion of strongly normal. Second, we show that Picard-Vessiot extensions are characterized by their Galois group being affine. Our proofs are elementary and do not use "group chunks" or cohomology. We end by recalling some important results about strongly normal extensions with the hope of spurring future research.

We build a model of ZFC+CH in which every first countable, countably compact space is either compact or contains a homeomorphic copy of ω 1 \omega _1 with the order topology. The majority of the paper consists of developing forcing technology that allows us to conclude that our iteration adds no reals. Our results generalize Saharon Shelah’s iteration theorems appearing in Chapters V and VIII of Proper and improper forcing (1998), as well as Eisworth and Roitman’s (1999) iteration theorem. We close the paper with a ZFC example (constructed using Shelah’s club–guessing sequences) that shows similar results do not hold for closed pre–images of ω 2 \omega _2 .

This paper examines the clean intersection hypothesis required for the expression of the wave invariants, computed from the asymptotic expansion of the classical wave trace by Duistermaat and Guillemin. The main result of this paper is the calculation of a necessary and sufficient condition for an arbitrary Riemannian two-step nilmanifold to satisfy the clean intersection hypothesis. This condition is stated in terms of metric Lie algebra data. We use the calculation to show that generic two-step nilmanifolds satisfy the clean intersection hypothesis. In contrast, we also show that the family of two-step nilmanifolds that fail the clean intersection hypothesis are dense in the family of two-step nilmanifolds. Finally, we give examples of nilmanifolds that fail the clean intersection hypothesis.

JOSE MANUEL MARCO AND JAVIER PARCET´Abstract. In this paper we present a unified approach to the spectral analysisof an hypergeometric type operator whose eigenfunctions include the classicalorthogonal polynomials. We write the eigenfunctions of this operator by meansof a new Taylor formula for operators of Askey-Wilson type. This gives rise tosome expressions for the eigenfunctions, which are unknown in such a generalsetting. Our methods also give a general Rodrigues formula from which severalwell known formulas of Rodrigues type can be obtained directly. Moreover,other new Rodrigues type formulas come out when seeking for regular solutionsof the associated functional equations. The main difference here is that, incontrast with the formulas appearing in the literature, we get non-ramifiedsolutions which are useful for applications in combinatorics. Another fact,that becomes clear in this paper, is the role played by the theory of ellipticfunctions in the connection between ramified and non-ramified solutions.

This paper concerns the homotopy type of hyperplane arrangements associated to infinite Coxeter groups acting as reflection groups on C n \mathbb C^n . A long-standing conjecture states that the complement of such an arrangement should be aspherical. Some partial results on this conjecture were previously obtained by the author and M. Davis. In this paper, we extend those results to another class of Coxeter groups. The key technical result is that the spherical Deligne complex for the 4-strand braid group is CAT(1).

Let $R$ be the power series ring or the polynomial ring over a field $k$ and let $I $ be an ideal of $R.$ Macaulay proved that the Artinian Gorenstein $k$-algebras $R/I$ are in one-to-one correspondence with the cyclic $R$-submodules of the divided power series ring $\Gamma. $ The result is effective in the sense that any polynomial of degree $s$ produces an Artinian Gorenstein $k$-algebra of socle degree $s.$ In a recent paper, the authors extended Macaulay's correspondence characterizing the $R$-submodules of $\Gamma $ in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras. However, these submodules in positive dimension are not finitely generated. Our goal is to give constructive and finite procedures for the construction of Gorenstein $k$-algebras of dimension one and any codimension. This has been achieved through a deep analysis of the $G$-admissible submodules of $\Gamma. $ Applications to the Gorenstein linkage of zero-dimensional schemes and to Gorenstein affine semigroup rings are discussed., Comment: To appear in Trans. Am. Math. Soc

Test ideals are an important concept in tight closure theory and their behavior via flat base change can be very difficult to understand. Our paper presents results regarding this behavior under flat maps with reasonably nice (but far from smooth) fibers. This involves analyzing, in depth, a special type of ideal of test elements, called the CS test ideal. Besides providing new results, the paper also contains extensions of a theorem by G. Lyubeznik and K. E. Smith on the completely stable test ideal and of theorems by F. Enescu and, independently, M. Hashimoto on the behavior of F F -rationality under flat base change.

This paper deals with generalizations of Hilbert's irreducibility theorem. The classical Hilbert irreducibility theorem states that for any cover f of the projective line defined over a number field k, there exist infinitely many k-rational points on the projective line such that the fiber of f over P is irreducible over k. In this paper, we consider similar statements about algebraic points of higher degree on curves of any genus. We prove that Hilbert's irreducibility theorem admits a natural generalization to rational points on an elliptic curve and thus, via a theorem of Abramovich and Harris, to points of degree 3 or less on any curve. We also present examples that show that this generalization does not hold for points of degree 4 or more. These examples come from an earlier geometric construction of Debarre and Fahlaoui; some additional necessary facts about this construction can be found in the appendix provided by Debarre. We exhibit a connection between these irreducibility questions and the sharpness of Vojta's inequality for algebraic points on curves. In particular, we show that Vojta's inequality is not sharp for the algebraic points arising in our examples.

Together with Cogdell, Piatetski-Shapiro and Shahidi, we proved earlier the existence of a weak functorial lift of a generic cuspidal representation of SO 2n+1 to GL 2n . Recently, Ginzburg, Rallis and Soudry obtained a more precise form of the lift using their integral representation technique, namely, the lift is an isobaric sum of cuspidal representations of GLn i (more precisely, cuspidal representations of GL2n i such that the exterior square L-functions have a pole at s = 1). One purpose of this paper is to give a simpler proof of this fact in the case that a cuspidal representation has one supercuspidal component. In a separate paper, we prove it without any condition using a result on spherical unitary dual due to Barbasch and Moy. We give several applications of the functorial lift: First, we parametrize square integrable representations with generic supercuspidal support, which have been classified by Moeglin and Tadic. Second, we give a criterion for cuspidal reducibility of supercuspidal representations of GL m x SO 2n+1 . Third, we obtain a functorial lift from generic cuspidal representations of SO 5 to automorphic representations of GL 5 , corresponding to the L-group homomorphism Sp 4 (C) → GL 5 (C), given by the second fundamental weight.

An S L n SL_n -character of a group G G is the trace of an S L n SL_n -representation of G . G. We show that all algebraic relations between S L n SL_n -characters of G G can be visualized as relations between graphs (resembling Feynman diagrams) in any topological space X , X, with π 1 ( X ) = G . \pi _1(X)=G. We also show that all such relations are implied by a single local relation between graphs. In this way, we provide a topological approach to the study of S L n SL_n -representations of groups. The motivation for this paper was our work with J. Przytycki on invariants of links in 3-manifolds which are based on the Kauffman bracket skein relation. These invariants lead to a notion of a skein module of M M which, by a theorem of Bullock, Przytycki, and the author, is a deformation of the S L 2 SL_2 -character variety of π 1 ( M ) . \pi _1(M). This paper provides a generalization of this result to all S L n SL_n -character varieties.

In this paper, we study some algebraic topology aspects of String c ^c structures, more precisely, from the perspective of Whitehead tower and the perspective of the loop group of S p i n c ( n ) Spin^c(n) . We also extend the generalized Witten genera constructed for the first time by Chen et al. [J. Differential Geom. 88 (2011), pp. 1–40] to correspond to String c ^c structures of various levels and give vanishing results for them.

We consider the problem of counting solutions to a trinomial Thue equation — that is, an equation ( ∗ ) | F ( x , y ) | = 1 , \begin{equation*} |F(x,y)| = 1,\tag {$*$} \end{equation*} where F F is an irreducible form in Z [ x , y ] Z[x,y] with degree at least three and with three non-zero coefficients. In a 1987 paper J. Mueller and W. Schmidt gave effective bounds for this problem. Their work was based on a series of papers by Bombieri, Bombieri-Mueller and Bombieri-Schmidt, all concerned with the “Thue-Siegel principle" and its relation to ( ∗ ) (*) . In this paper we give specific numerical bounds for the number of solutions to ( ∗ ) (*) by a somewhat different approach, the difference lying in the initial step — solving a certain diophantine approximation problem. We regard this as a real variable extremal problem, which we then solve by elementary calculus.