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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).

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.

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.

Let FFn be a family of subgroups of EnX which is closed under taking subgroups and conjugates. Such a family has a classifying space, BYFn, and we showed in an earlier paper that a cormpatible choice of TFn for each n gives a simplicial monoid JJn BTF,, which group completes to an infinite loop space. In this paper we define a filtration of the associated spectrum whose filtration quotients, given an extra condition on the families, can be identified in terms of the classifying spaces of the families of subgroups that were chosen. This gives a way to go from group theoretic data about the families to homotopy theoretic information about the associated spectrum. We calculate two examples. The first is related to elementary abelian p-groups, and the second gives a new expression for the desuspension of Sp""' (SO)/Spm--I (SO) as a suspension spectrum.

LetGGbe a finitely generated residually finite group and letan(G)a_n(G)denote the number of indexnnsubgroups ofGG. Ifan(G)≤nαa_n(G) \le n^{\alpha }for someα\alphaand for allnn, thenGGis said to have polynomial subgroup growth (PSG, for short). The degree ofGGis then defined bydeg⁡(G)=lim suplog⁡an(G)log⁡n\operatorname {deg}(G) = \limsup {{\log a_n(G)} \over {\log n}}. Very little seems to be known about the relation betweendeg⁡(G)\operatorname {deg}(G)and the algebraic structure ofGG. We derive a formula for computing the degree of certain metabelian groups, which serves as a main tool in this paper. Addressing a problem posed by Lubotzky, we also show that ifH≤GH \le Gis a finite index subgroup, thendeg⁡(G)≤deg⁡(H)+1\operatorname {deg}(G) \le \operatorname {deg}(H)+1. A large part of the paper is devoted to the structure of groups of small degree. We show thatan(G)a_n(G)is bounded above by a linear function ofnnif and only ifGGis virtually cyclic. We then determine all groups of degree less than3/23/2, and reveal some connections with plane crystallographic groups. It follows from our results that the degree of a finitely generated group cannot lie in the open interval(1,3/2)(1, 3/2). Our methods are largely number-theoretic, and density theorems à la Chebotarev play essential role in the proofs. Most of the results also rely implicitly on the Classification of Finite Simple Groups.

Let S be an F -finite regular local ring and I an ideal contained in S. Let R = S/I. Fedder proved that R is F -pure if and only if (I [p] : I) * m[p]. We have noted a new proof for his criterion, along with showing that (I [q] : I) ⊆ (τ [q] : τ), where τ is the pullback of the test ideal for R. Combining the the F -purity criterion and the above result we see that if R = S/I is F pure then R/τ is also F -pure. In fact, we can form a filtration of R, I ⊆ τ = τ0 ⊆ τ1 ⊆ . . . ⊆ τi ⊆ . . . that stabilizes such that each R/τi is F -pure and its test ideal is τi+1. To find examples of these filtrations we have made explicit calculations of test ideals in the following setting: Let R = T/I, where T is either a polynomial or a power series ring and I = P1 ∩ . . . ∩ Pn is generated by monomials and the R/Pi are regular. Set J = Σ(P1 ∩ . . . ∩ Pi ∩ . . . ∩ Pn). Then J = τ = τpar. This paper concerns the study of the test ideal in F -finite quotients of regular local rings. Test elements play a key role in tight closure theory. Once known, they make computing tight closures of ideals and modules easier. In fact, in an excellent Gorenstein local ring with an isolated singularity, R/τ ∼= Hom(I∗/I, E) where I is generated by a system of parameters that are test elements and E is the injective hull of R (see [Hu1] and [S1]). We also know for parameter ideals I that I : τ = I∗. Thus knowing τ is basically equivalent to knowing the tight closure of a system of parameters which is contained in the test ideal. A recent paper of Huneke and Smith [HS] links tight closure to Kodaira vanishing for graded rings R with characteristic either 0 or p where p 0. Recall that the a-invariant for a graded ring R, denoted a, is equal to −min{i|[ωR]i 6= 0}, where ωR is the canonical module for R. If R is Gorenstein, then ωR = R(a). Huneke and Smith prove that the test ideal is exactly the ideal generated by elements of degree greater than the a-invariant of R if and only if a strong Kodaira vanishing holds on R. A recent paper of Hara [Ha] confirms that this strong Kodaira vanishing holds in finitely generated algebras over a field of characteristic zero. In this paper we study test ideals of F -finite rings which are reduced quotients of a regular local ring. Reduced quotients of F -finite regular local rings have been studied by both Fedder [Fe] and Glassbrenner [Gl]. Fedder’s work concerns F purity aspects of these rings, and Glassbrenner’s results use Fedder’s techniques to examine strong F -regularity. The object that plays a key role in their work is Received by the editors November 4, 1996. 1991 Mathematics Subject Classification. Primary 13A35.

If ¢ is a generic cubic rnetaplectic form on GSp(4), that is also an eigenfunction for all the Hecke operators, then corresponding to X is an Euler product of degree 4 that has a functional equation and rnerornorphic continu.ation to the whole cornplex plane. This correspondence is obtained by convolving (b with the cubic 0-function on GL(3) in a Shirnura type RankinSelberg integral. 0. INTRODUCT[ON Suppose 0 is a metaplectic automorphic form of minimal level on the 3-fold cover of GSp(4) that is an eigenfunction of all the Hecke operators. If 0 has any non-zero Whittaker coefficients, then 0 is called generic. In this case, this paper will show tha1; there is a Dirichlet series in the Whittaker coefficients of 0 that has a formula1;ion as a degree 4 Euler product. Moreover, this Euler product has a meromorphic continuastion to the whole complex plane. This association of the Euler product with 0 will be obtained via a Shimura type JRankin-Selberg integral involving 0 and a H-function on the 3-fold cover of GL(3). Historically) the problem of associating an Euler product which has meromorphic continuation and functional equation with a metaplectic automorphic form originated with the work of Shimura [Shi]. More specifically, suppose f (z). = E a(n) qn is a holomorphic modular form of half-integral weight k/2, which is an eigenform of the Hecke operators Tp2, i.e. Tp2 f = )pf. Then via a JRankin-Selberg integral of the form f __ (0.1) | f (z) H(z) E(z, s) dz, where H(z) is a classical theta function and E(z, s) is an integral weight Eisenstein series, Shimura obtains an Euler product of the form (0.2) t| (1 _ Ap ps + pk-2-2s)-1 p The analytic continuation and functional equation of this Euler product follow from the siirlilar properties of E(z, s) in (0. 1) . Bump and Hoffstein [BH2] have subsequently extended these techniques of [Shig to GL(3) by finding a Rankin-Selberg integral of a metaplect1c automorphic form on the 3-fold cover of GL(3) which produces an Euler product of degree 3. Just as in [ShiX, this Euler product is shown to have meromorphic continuation to the whole Received by the editors November 14, 1995 and, in revised forrn, May 21, 1996. 1991 Mathematics S?lbject Classification. Prirnary llF55, llF30. @1998 Ameri( an Mathematical Society 975 This content downloaded from 157.55.39.45 on Wed, 05 Oct 2016 05:22:35 UTC All use subject to http://about.jstor.org/terms THOMAS GOETZE 976 complex plane and to have a functional equation under s 1-s. The integral which represents this Euler product involves the 0-function on the 3-fold cover of GL(3) over the field Q (e21rt/3) ? wllich has been studied independently by Proskurin [Pr] and Bump and HofEstein [BH12. In addition, Bump and HofEstein [BH2] have conjectured that Euler products with meromorphic continuation and functional equation may be obtained by convolving metaplectic automorphic forms on the n-fold cover of GL(r) against H-functions on the n-fold cover of GL(n). This was carried out in [BH3] in the case r-2 and n > 2. Friedberg and Wong [FrW] have also used Shimura's method to associate an Euler product to a generic metaplectic automorphic form on the double cover of the symplectic group GSp(4). They have found an integral (inspired by Novo dvorsky's GSp(4)xGL(2) convolution) involving ametaplectic automorphic form on the double cover of GSp(4), the H-function on the double cover of GL(2), and a (non-metaplectic) Eisenstein series on GL(2), that yields a degree 5 Euler product. This Euler product is shown to have meromorphic continuation and a functional equation, and furthermore it has the same local Euler factors as the L-function of an automorphic form on GSp(4). As in [Shi, BH2], the Euler product found by Friedberg and Wong is explicitly constructed from the Whittaker coeEcients of the metaplectic automorphic form. Alternatively, Flicker [Fli], Kazhdan and Patterson [KaP2], and Flicker and Kazhdan [FliKa] have used the trace formula to generalize [Shi] by showing that (in many situations) there exists a correspondence between metaplectic automorphic forms and (non-metaplectic) automorphic forms. Indeed: Shimura actually proves in [Shi] that the Euler product (0.2) is the L-function of a holomorphic integral weight modular form. In using the trace formula, however: explicit information about the interplay between the metaplectic Fourier coefficients and the corresponding L-functions (see (0.2)) is not obtained. If a generalized Shimura correspondence does exist between generic metaplectic and non-metaplectic automorphic forms, then the associated Euler products obtained by Bump-Hofistein, Friedberg-Wong, and this paper will be the L-functions of the corresponding non-metaplectic forms. There is evidence that the degree 4 Euler product obtained in this paper is the L-function of an automorphic form on GSp(4) Savin [Sa] has shown that there is an algebra isomorphism between the local Iwahori Hecke algebra of GSp(4) and the local Iwahori Hecke algebra on the 3-fold cover of GSp(4) This suggests that; if a Shimura correspondent exists in this situation, it should be an automorphic form on GSp(4)e Since there is a representation of degree 4 on the L-group of GSp(4): automorphic forms on GSp(4) will have natural L-functions with Euler products of degree 4 In this sense, having a degree 4 Euler product is consistent with Savin's results. The main results of this paper will be found in Theorems 3.1 and 5.1, which are summarized as follows: Mairl Theorem. Suppose Q is a generic metaplectic C?lsp form of minimal level on the S-fold cover of G8p(4) that is an eigenf?lnction of all the lTecke operators. Then there is a degree J E?ller prod?lct, with a meromorphic contin?ltation, which cctn be explicitly constr?lcted from the Whittaker coefficients of f0Je This association is realized as a Shim?lra type Rankin-Selberg integral of 0 against an Eisenstein series ind?lced from a f)-f?lnction on the S-fold cover of GL(3). This content downloaded from 157.55.39.45 on Wed, 05 Oct 2016 05:22:35 UTC All use subject to http://about.jstor.org/terms

In this paper, we construct explicit eigenfunctions of the local Weil representation on unitary groups of one variable in the p-adic case when p is odd. The idea is to use the lattice model, and the results will be used to compute special values of certain Hecke L-functions in separate papers. We also recover Moen’s results on when a local theta lifting from U(1) to itself does not vanish. 0. Introduction and Notation Let E/F be a quadratic extension of local fields. If (V, ( , )) is an Hermitian space over E, and (W, 〈 , 〉) is a skew-Hermitian space over E, the unitary groups G = G(W ) and G′ = G(V ) form a reductive dual pair in Sp(W), where W = V ⊗W has the symplectic form 12 trE/F ( , ) ⊗ 〈 , 〉 over F . According to a well-known result, this dual pair splits in the metaplectic cover of Sp(W), and thus has a Weil representation ω. We consider the very special case in which dimE V = dimEW = 1. In this case, G ∼= G′ = U(1) = E is compact and abelian, where E is the kernel of the norm map N : E∗ −→ F ∗. So irreducible representations of G are just characters, and the Weil representation has a direct sum decomposition

In the study of integrable systems of ODE’s arising from a Lax pair with a parameter, the constants of the motion occur as spectral curves. Many of these systems are algebraically completely integrable in that they linearize on the Jacobian of a spectral curve. In an earlier paper the authors gave a classification of the spectral curves in terms of the Weyl group and arranged the spectral curves in a hierarchy. This paper examines the Jacobians of the spectral curves, again exploiting the Weyl group action. A hierarchy of Jacobians will give a basis of comparison for flows from various representations. A construction of V. Kanev is generalized and the Jacobians of the spectral curves are analyzed for abelian subvarieties. Prym-Tjurin varieties are studied using the group ring of the Weyl group W W and the Hecke algebra of double cosets of a parabolic subgroup of W . W. For each algebra a subtorus is identified that agrees with Kanev’s Prym-Tjurin variety when his is defined. The example of the periodic Toda lattice is pursued.

The main new results of this paper concern the formulation of algebraic conditions for the Fredholm property of elliptic systems of P.D.E.’s with boundary values, which are equivalent to the Lopatinskii condition. The Lopatinskii condition is reformulated in a new algebraic form (based on matrix polynomials) which is then used to study the existence of homotopies of elliptic boundary value problems. The paper also contains an exposition of the relevant parts of the theory of matrix polynomials and the theory of elliptic systems of P.D.E.’s. Introduction Let A denote an elliptic operator in Ω and let B be a boundary operator, where Ω is a bounded domain in R. In this paper several versions of the algebraic condition for the Fredholm property of (A,B) are formulated, equivalent to the Lopatinskii condition of [Lo]. The main new result is the following. A square matrix function ∆+B defined on the unit cotangent bundle of ∂Ω is constructed from the principal symbols of the coefficients of the boundary operator and a spectral pair for the family of matrix polynomials associated with the principal symbol of the elliptic operator. The Lopatinskii condition is equivalent to the following condition: the function ∆+B must have invertible values. The proofs use the theory of matrix polynomials due to Gohberg, Lancaster, Rodman, and others; for instance, see [GLR], Chapter 14 in [LT], and Chapter 6 in [R]. There is a natural map defined by (A,B) 7→ A, going from the space of elliptic boundary value problems to the space of elliptic operators. Now let Aτ , 0 ≤ τ ≤ 1, be a homotopy of elliptic operators and let B0 be a boundary operator such that (A0,B0) satisfies the Lopatinksii condition. We will show that the given homotopy of elliptic operators can be lifted to a homotopy (Aτ ,Bτ ) in the space of elliptic boundary value problems satisfying the Lopatinskii condition. This result is proved in Theorem 7.3 and was motivated by the article [Ge]. The key element in the proof, i.e. the construction of the boundary operators Bτ , requires that we use pseudodifferential operators on ∂Ω and the theory of matrix polynomials mentioned above. Received by the editors August 16, 1994 and, in revised form, February 12, 1996. 1991 Mathematics Subject Classification. Primary 35J45, 35J55, 15A22.

Let K K be a compact connected semi-simple Lie group, let G = K C G = K_{\mathbf C} , and let G = K A N G = KAN be an Iwasawa decomposition. To a given K K -invariant Kaehler structure ω \omega on G / N G/N , there corresponds a pre-quantum line bundle L {\mathbf L} on G / N G/N . Following a suggestion of A.S. Schwarz, in a joint paper with V. Guillemin, we studied its holomorphic sections O ( L ) {\mathcal O}({\mathbf L}) as a K K -representation space. We defined a K K -invariant L 2 L^2 -structure on O ( L ) {\mathcal O}({\mathbf L}) , and let H ω ⊂ O ( L ) H_\omega \subset {\mathcal O}({\mathbf L}) denote the space of square-integrable holomorphic sections. Then H ω H_\omega is a unitary K K -representation space, but not all unitary irreducible K K -representations occur as subrepresentations of H ω H_\omega . This paper serves as a continuation of that work, by generalizing the space considered. Let B B be a Borel subgroup containing N N , with commutator subgroup ( B , B ) = N (B,B)=N . Instead of working with G / N = G / ( B , B ) G/N = G/(B,B) , we consider G / ( P , P ) G/(P,P) , for all parabolic subgroups P P containing B B . We carry out a similar construction, and recover in H ω H_\omega the unitary irreducible K K -representations previously missing. As a result, we use these holomorphic sections to construct a model for K K : a unitary K K -representation in which every irreducible K K -representation occurs with multiplicity one.

This paper deals with a reducible s ℓ ( 2 , C ) s\ell (2, \mathbf {C}) action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau Conjecture: suppose that s ℓ ( 2 , C ) s\ell (2, \mathbf {C}) acts on the formal power series ring via ( 0.1 ) (0.1) . Then I ( f ) = ( ℓ i 1 ) ⊕ ( ℓ i 2 ) ⊕ ⋯ ⊕ ( ℓ i s ) I(f)=(\ell _{i_{1}})\oplus (\ell _{i_{2}})\oplus \cdots \oplus (\ell _{i_{s}}) modulo some one dimensional s ℓ ( 2 , C ) s\ell (2, \mathbf {C}) representations where ( ℓ i ) (\ell _{i}) is an irreducible s ℓ ( 2 , C ) s\ell (2, \mathbf {C}) representation of dimension ℓ i \ell _{i} or empty set and { ℓ i 1 , ℓ i 2 , … , ℓ i s } ⊆ { ℓ 1 , ℓ 2 , … , ℓ r } \{\ell _{i_{1}},\ell _{i_{2}},\ldots ,\ell _{i_{s}}\}\subseteq \{\ell _{1},\ell _{2},\ldots ,\ell _{r}\} . Unlike classical invariant theory which deals only with irreducible action and 1–dimensional representations, we treat the reducible action and higher dimensional representations succesively.

This paper deals with the problem of describing the set M M of all solutions of an equation over a free semigroup S S . The standard way to do this involves the introduction of auxiliary equations containing polynomials in natural number parameters of arbitrarily high degree. Since S S has a solvable word problem, M M must be computable. However, M M cannot necessarily be computed from the standard description of M M . The present paper shows that the only polynomials needed to describe M M are just products of one parameter by a linear combination of some other parameters. The resulting simplification of the standard description of M M clearly can be used to compute M M .

Let Ω \Omega be a non-compact complex manifold of dimension n n , ω = ∂ ∂ ¯ Ψ \omega =\partial \overline \partial \Psi a Kähler form on Ω \Omega , and K α ( x , y ¯ ) K_\alpha ( x,\overline y) the reproducing kernel for the Bergman space A α 2 A^2_\alpha of all analytic functions on Ω \Omega square-integrable against the measure e − α Ψ | ω n | e^{-\alpha \Psi } |\omega ^n| . Under the condition \[ K α ( x , x ¯ ) = λ α e α Ψ ( x ) K_\alpha ( x,\overline x)= \lambda _\alpha e^{\alpha \Psi (x)} \] F. A. Berezin [Math. USSR Izvestiya 8 (1974), 1109–1163] was able to establish a quantization procedure on ( Ω , ω ) (\Omega ,\omega ) which has recently attracted some interest. The only known instances when the above condition is satisfied, however, are just Ω = C n \Omega = \mathbf {C} ^n and Ω \Omega a bounded symmetric domain (with the euclidean and the Bergman metric, respectively). In this paper, we extend the quantization procedure to the case when the above condition is satisfied only asymptotically, in an appropriate sense, as α → + ∞ \alpha \to +\infty . This makes the procedure applicable to a wide class of complex Kähler manifolds, including all planar domains with the Poincaré metric (if the domain is of hyperbolic type) or the euclidean metric (in the remaining cases) and some pseudoconvex domains in C n \mathbf {C}^n . Along the way, we also fix two gaps in Berezin’s original paper, and discuss, for Ω \Omega a domain in C n \mathbf {C}^n , a variant of the quantization which uses weighted Bergman spaces with respect to the Lebesgue measure instead of the Kähler-Liouville measure | ω n | |\omega ^n| .

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

In this paper we prove the existence of the symmetric-square lifting of admissible and of automorphic representations from the group SL ( 2 ) {\text {SL}}(2) to the group PGL ( 3 ) {\text {PGL}}(3) . Complete local results are obtained, relating the character of an SL ( 2 ) {\text {SL}}(2) -packet with the twisted character of self-contragredient PGL ( 3 ) {\text {PGL}}(3) -modules. Our global results relate packets of cuspidal representations of SL ( 2 ) {\text {SL}}(2) with a square-integrable component, and self-contragredient automorphic PGL ( 3 ) {\text {PGL}}(3) -modules with a component coming from a square-integrable one. The sharp results, which concern SL ( 2 ) {\text {SL}}(2) rather than GL ( 2 ) {\text {GL}}(2) , are afforded by the usage of the trace formula. The surjectivity and injectivity of the correspondence implies that any self-contragredient automorphic PGL ( 3 ) {\text {PGL}}(3) -module as above is a lift, and that the space of cuspidal SL ( 2 ) {\text {SL}}(2) -modules with a square-integrable component admits multiplicity one theorem and rigidity ("strong multiplicity one") theorem for packets (and not for individual representations). The techniques of this paper, based on the usage of regular functions to simplify the trace formula, are pursued in the sequel [ VI \text {VI} ] to extend our results to all cuspidal SL ( 2 ) {\text {SL}}(2) -modules and self-contragredient PGL ( 3 ) {\text {PGL}}(3) -modules

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.

Here we give a classification (in characteristic zero) of pairs (V, E) with V being a smooth, connected, complete 3-fold and E a rank-2 spanned ample vector bundle on V with sectional genus 1. The proof uses the partial classification of Fano 3-folds and Mori theory. Let X be an integral complete manifold, dim(X) = n, and E a rank(n 1) vector bundle on X. We define the sectional genus g(E) using the following formula: 2g(E) 2 := (Kx + c 1(E))Cn_ -(E) . It is easy to check (see Remark 4.1) that g(E) is an integer. If E has a section with zero-locus C of codimension n 1, then g(E) = pa(C). In this paper we give a reasonable classification in characteristic zero (see Theorem 0) of the pairs (V, E) with V being a smooth, complete 3-fold, and E a rank-2 ample spanned vector bundle on E with g(E) = 1 . Note that if g(E) = 0, then Kv + cl (E) is not nef; under this assumption (but a far weaker assumption on V, E) Wisniewski in [W] gave a classification, using Mori theory (see also 4.2 for a similar result). The case g(E) = 1 seems to be of a different order of difficulty. But we will use very much of Wisniewski's work [W], the partial classification of Fano 3folds, and classifications (e.g., [Io]) based on Mori theory. In an interesting paper [Fl] Fujita gave two definitions of sectional genus for higher rank vector bundles. The second one is the sectional genus of the tautological line bundle on P(E). The first one (and more interesting) is called the cl-sectional genus of E; it is the sectional genus of the line bundle det(E), i.e., g is defined by: 2g 2 := (Kx + (n 1)c1 (E))(c1 (E))n-1 . For instance, if E is the direct sum of n 1 line bundles all isomorphic to a line bundle A, with our definition 2g(E) 2 := (Kx + (n 1)A)An-I while the cl-sectional genus g of E is defined by 2g 2 := (n 1)(n -)(Kx + (n 1)2A)An-I , which is much bigger if A is ample. For instance, the cl -sectional genus of the bundle in (b) in the statement of Theorem 0 is 33. The first three sections of this paper are devoted to the proof of the following result (over an algebraically closed base field with characteristic zero). Received by the editors February 20, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 1 4J30; Secondary 14F05.

This paper is a study of the structure of the space R ( K ) R(K) of representations of classical knot groups into SU ( 2 ) {\text {SU}}(2) . Let R ^ ( K ) \hat R(K) equal the set of conjugacy classes of irreducible representations. In § I \S I , we interpret the relations in a presentation of the knot group in terms of the geometry of SU ( 2 ) {\text {SU}}(2) ; using this technique we calculate R ^ ( K ) \hat R(K) for K K equal to the torus knots, twist knots, and the Whitehead link. We also determine a formula for the number of binary dihedral representations of an arbitrary knot group. We prove, using techniques introduced by Culler and Shalen, that if the dimension of R ^ ( K ) \hat R(K) is greater than 1 1 , then the complement in S 3 {S^3} of a tubular neighborhood of K K contains closed, nonboundary parallel, incompressible surfaces. We also show how, for certain nonprime and doubled knots, R ^ ( K ) \hat R(K) has dimension greater than one. In § I I \S II , we calculate the Zariski tangent space, T ρ ( R ( K ) ) {T_\rho }(R(K)) , for an arbitrary knot K K , at a reducible representation ρ \rho , using a technique due to Weil. We prove that for all but a finite number of the reducible representations, dim ⁡ T ρ ( R ( K ) ) = 3 \dim {T_\rho }(R(K))= 3 . These nonexceptional representations possess neighborhoods in R ( K ) R(K) containing only reducible representations. At the exceptional representations, which correspond to real roots of the Alexander polynomial, dim ⁡ T ρ ( R ( K ) ) = 3 + 2 k \dim {T_\rho }(R(K)) = 3 + 2k for a positive integer k k . In those examples analyzed in this paper, these exceptional representations can be expressed as limits of arcs of irreducible representations. We also give an interpretation of these "extra" tangent vectors as representations in the group of Euclidean isometries of the plane.

Let X be the flag variety of a compact Lie group and let h* be a complex-oriented generalized cohomology theory. We introduce operators on h*(X) which generalize operators introduced by Bernstein, Gel'fand, and Gel'fand for rational cohomology and by Demazure for K-theory. Using the Becker-Gottlieb transfer, we give a formula for these operators, which enables us to prove that they satisfy braid relations only for the two classical cases, thereby giving a topological interpretation of a theorem proved by the authors and extended by Gutkin. One of the central issues in Lie theory is the geometry of the flag variety associated to a compact Lie group G. An important problem concerning the flag variety is the Schubert calculus, which studies the ring structure of the cohomology of the flag variety. Work initiated by Borel, Bott and Kostant, which culminated in a paper by Bernstein, Gel'fand and Gel'fand [BGG], gave a complete solution to the problem. Demazure studied the same problem for K-theory. Moreover, these techniques have been generalized to the Kac-Moody situation by Kac-Peterson, Kostant-Kumar, and others. This work has focussed on algebro-geometric properties of the flag variety. Here, on the other hand we study the flag variety from the point of view of algebraic topology. As a consequence, not only do we recover the classical results described above, but we extend these results to a certain class of cohomology theories-those which are complex-oriented. Examples of complex-oriented theories include ordinary cohomology, K-theory, complex cobordism, and elliptic cohomology. Since the context we have chosen in very general, the proofs are universal and are often simpler than the classical arguments. In the work of BGG, a crucial role is played by operators Ai associated to each simple reflection si of the Weyl group of G (defined by Demazure in K-theory). These operators Ai satisfy the braid relations, which are the relations between pairs of simple reflections. In this paper, we generalize the A, to give operators D, acting on h*(G/T) for any complex-oriented theory h*. We prove that braid relations are satisfied only for cohomology theories with the formal group law of rational cohomology or of K-theory (Theorem Received by the editors June 21, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 55N20, 57T15. The second author was supported by an NSF graduate fellowship. (3 1990 American Mathematical Society 0002-9947/90 $1.00 + $.25 perpage

In this paper, a type of algebra is introduced and studied from a rational homotopy point of view, using differential graded Lie algebras. The main aim of the paper is to establish whether or not such an algebra is the rational cohomology algebra of a unique rational homotopy type of spaces. That is, in the language of rational homotopy, whether or not such an algebra is intrinsically formal. Examples are given which show that, in general, this is not so-7.8 and 7.9. However, whilst it is true that not all such algebras are intrinsically formal, some of them are. The main results of this paper show a certain class of these algebras to be intrinsically formal-Theorem 2 (6.1); and a second, different type of algebra also to be intrinsically formal-Theorem 1 (5.2), which type of algebra overlaps with the first type in many examples of interest. Examples are given in ?7. 1. PRELIMINARIES, NOTATION AND AN ACKNOWLEDGMENT In [D-Gr-Mo-Su] it is proven that compact Kahler manifolds are formal spaces. From the rational homotopy point of view, this means that compact Kahler manifolds are particularly interesting as examples. This paper is concerned with a more algebraic approach than that of [D-Gr-Mo-Su]. The cohomological properties of compact Kahler manifolds are abstracted out-see 3.1-and algebras satisfying these properties are studied, with a particular view to deciding whether or not such algebras are intrinsically formal-see 2.1. The main results of this paper are Theorems 1 and 2-5.2 and 6.1; many examples are also given in ?7. Also in ?7, examples are given of algebras that are not intrinsically formal, and yet satisfy the properties referred to above-see 7.8, 7.9. Rather than give a step by step breakdown of the paper here, I have put one or two sentences heading each section. These section headings can be read in order, at this stage, so as to give an overview of the contents of the paper. For the fundamentals of rational homotopy theory, the basic references are, in alphabetical order, [Bou-Gu, Gr-Mo, HI, Q, Su,, and Su2]. This paper will, throughout, only be concerned with the "simply connected with rational homology of finite type" case. For references which give all the results needed for this Received by the editors July 29, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 55P62, 55P15; Secondary 14M10, 53C55.

First cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts, including Wiles's famous paper (1995). Internal to group theory, 1-cohomology plays a role in the general theory of maximal subgroups of finite groups, as developed by Aschbacher and Scott (1985).One can pass to the case where the group acts faithfully and the underlying module is absolutely irreducible. In this case, R. Guralnick (1986) conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups. This paper provides the first general positive results on this conjecture, proving that the generic 1-cohomology H 1 gen (G,L) := lim H 1 (G(q), L) (see Cline, Parshall, Scott, and van der Kallen) (1977) of a finite group G(q) of Lie type, with absolutely irreducible coefficients L (in the defining characteristic of G), is bounded by a constant depending only on the root system. In all cases, we are able to improve this result to a bound on H 1 (G(q),L) itself, still depending only on the root system. The generic H 1 result, and related results for Ext 1 , emerge here as a consequence of a general study, of interest in its own right, of the homological properties of certain rational modules Δ red (λ),∇ red (λ), indexed by dominant weights λ, for a reductive group G. The modules Δ red (λ) and ∇ red (λ) arise naturally from irreducible representations of the quantum enveloping algebra U ζ (of the same type as G) at a pth root of unity, where p > 0 is the characteristic of the defining field for G. Finally, we apply our Ext 1 -bounds, results of Bendel, Nakano, and Pillen (2006), as well as results of Sin (1993), (1992), (1994) on the Ree and Suzuki groups to obtain the (non-generic) bounds on H 1 (G(q),L).

In this paper, we consider the problem of solving the Cauchy–Riemann equation with prescribed support in a domain of a complex manifold for forms or currents. We are especially interested in the case when the domain is a Hartogs triangle in C 2 \mathbb {C}^2 or C P 2 \mathbb {C}\mathbb {P}^2 . In particular, we show that the strong L 2 L^2 Dolbeault cohomology group on the Hartogs triangle in C P 2 \mathbb {C}\mathbb {P}^2 is infinitely dimensional.