Showing papers in "Publications of The Research Institute for Mathematical Sciences in 1995"

The Geometry of the Compactification of the Hurwitz Scheme

Abstract: Table of

Minimal affinizations of representations of quantum groups: the rank 2 case

Abstract: If Uq (g) is a finite-dimensional complex simple Lie algebra, an affinization of a finite-dimensional irreducible representation V of Uq (g) is a finite-dimensional irreducible representation V of Uq (g) which contains V with multiplicity one, and is such that all other Uq (g) -types in V have highest weights strictly smaller than that of V. We define a natural partial ordering ^ on the set of affinizations of V. If g is of rank 2, we show that there is a unique minimal element with respect to this order and give its Uq (g) -module structure when g is of type AT. or €2Introduction If g is a finite-dimensional complex simple Lie algebra, the associated 'untwisted' affine Lie algebra § is a central extension, with one-dimensional centre, of the space of Laurent polynomial maps C —* g (on which a Lie bracket is defined using pointwise operations). Since the cocycle of the extension vanishes on the constant maps, we can regard g as a subalgebra of g. If V is any representation of g, it is easy to extend the action of g on V to an action of g on the same space. If a^C , evaluation at a gives a homomorphism eva'. Q~^ Q (under which the centre maps to zero) which is the identity on g, so pulling back V by eva gives the desired extension. It follows from the results of [2] that, if V is finite-dimensional and irreducible, these are, up to isomorphism, the only possible extensions. Quantum deformations Uq (g) and Uq (g) of the universal enveloping algebras of g and g were introduced in 1985 by V. G. Drinfel'd and M. Jimbo. These Communicated by T. Miwa, January 18, 1995. 1991 Mathematics Subject Classification (s): 17B. * Dept. of Math., Univ. of California, Riverside, CA92521, U.S.A. Partially supported by the NSF, DMS-9207701. 874 VYJAYANTHI CHARI algebras depend on a parameter #^C X ; we assume throughout this paper that q is transcendental. It is well-known (see [5] or [10] , for example) that, up to twisting by certain simple automorphisms, there is a natural one-to-one correspondence between the finite-dimensional representations of Uq (g) and those of g. Corresponding representations have the same character, and hence the same dimension. However, the structure of the finite-dimensional representations of Uq (g) is not well-understood. A parametrization of these representations in the spirit of Cartan's highest weight classification of the finite-dimensional irreducible representations of g is proved in the case Q = sl2 in [3], and in [6] in general. As in the classical situation, we may regard Uq (g) as a subalgebra of Uq (§) . If g is of type sln, the action of Uq (g) on any representation V extends to a representation of Uq (g) . However, if g is not of type sln, it is not usually possible to extend the action of Uq (g) on an irreducible finite-dimensional represention V to an action of Uq (g) on V. Thus, it is natural to ask how V can be 'enlarged' so as to obtain a representation of Uq (g) . To make this question precise, we define in this paper a natural partial ordering on the set of isomorphism classes of representations of Uq (g). By an affinization of a finitedimensional irreducible representation V of Uq (g) , we mean an irreducible representation V of Uq (g) which contains V as a Uq (g) -subrepresentation with multiplicity one, and such that all other irreducible Uq (g) -subrepresentations of V are strictly smaller than V. (There is a clear analogy with the classical Harish Chandra theory of (g, K) -modules here.) We prove that any given representation V has only finitely many affinizations (at least one) up to Uq (g) -isomorphism, and one may ask if any of them is 'canonical'. A reasonable interpretation of this question is to look for the minimal affinization (s) of V, with respect to our partial order. If Q = sln, we show in [4] that every finite-dimensional irreducible representation of £/«(g) has, up to Uq(o) -isomorphism, a unique minimal affinization. In this paper, we prove that, if g is of type C2 or G^ there is again a unique minimal affinization, and we describe it precisely in terms of the highest weight classification of representations of Uq (g) mentioned above. In contrast to the sln case, the minimal affinization in these cases is not, in general, irreducible as a representation of Uq (g) . In fact, in the C% case we describe the structure of all minimal affinizations as representations of C/«(g); a consequence of this result is that the minimal affinization of V is irreducible under Uq (g) if and only if the value of its highest weight on the short simple root of g is 0 or 1. Subsequent papers will deal with the case when g has rank greater that 2. The problem of constructing aff inizations of representations of Uq (g) is important in several areas of mathematics and physics, as has been emphasized by I. B. Frenkel and N. Yu. Reshetikhin, among others (see Remark 4,2 in [9]). MINIMAL AFFINIZATIONS 875 As one example, recall that, to any finite-dimensional irreducible representation V of Uq(o) one can associate an R-matrix, i.e. an element R^End (V®V) which satisfies the 'quantum Yang-Baxter equation' (QYBE). There are many situations, however, in which it is important to have a solution of the 'QYBE with spectral parameters'. This is so, for example, in the theory of lattice models in statistical mechanics, for only when the R-matrix constructed from the Boltzmann weights of the model satisfies the QYBE with spectral parameters can one prove the existence of commuting transfer matrices and deduce the integrability of the model. (See [5], for example, for an introduction to these ideas.) Thus, it is natural to ask when R can be 'embedded' in a parameter-dependent R-matrix R(u) ^End (V® V). A sufficient condition for this is that the action of UQ(Q) on V extends to an action of UQ(Q) on V, for then V itself can be embedded in a 1-parameter family of representations of Uq(o) by twisting with a certain 1-parameter family of automorphisms of Uq(o) (which correspond, in the classical case, to 'rescaling' the C parameter in g). A second example concerns the affine Toda field theory associated to g. This admits Uq (g*) as a 'quantum symmetry group', where g* is the dual affine Lie algebra (whose generalized Cartan matrix is the transpose of that of g). It is well known that the classical solitons of this theory correspond essentially to the finite-dimensional irreducible representations of g. The solitons (or particle states) of the quantum theory should therefore correspond to the finitedimensional irreducible representations of Uq (g *). Since not all representations of Uq(o) are affinizable on the same space, the quantum solitons come in 'multiplets', and there are generally 'more' quantum solitons than classical ones. §1. Quantum Affine Algebras Let g be a finite-dimensional complex simple Lie algebra with Cartan subalgebra t) and Cartan matrix A = (an) ,-fj6j. Fix coprime positive integers (di) ,e/ such that (diatj) is symmetric. Let R be the set of roots and R a set of positive roots. The roots can be regarded as functions /— >Z; in particular, the simple roots at^R are given by Let 0= © 0 for all i^I\ 876 VYJAYANTHI CHARI be the set of dominant weights. Define a partial order > on P by iff XLet 6 be the unique highest root with respect to ^. Define a non-degenerate symmetric bilinear form ( , ) on t)* by (at, aj) =diaij , and denote by ( , ) also the induced form on t). Set d0 — y (0, 6), a0o — 2, and, for all i^I, 2(0, at) 2(0, a,-) 01 (0, 0) ' *° (a,-, a,-) Let /=/ II )0[ and A = (a//) ue/. Then, A is the generalized Cartan matrix of the untwisted affine Lie algebra g associated to g. From Section 5 onwards, we shall be interested in the case when g is of type C2. Then, /=|1,2| , d0 2 -I "-'.-* 2 the rows of A being numbered 0, 1, 2. Let q^C* be transcendental, and, for r, n&N, n>r, define

Traces, unitary characters and crossed products by Z

Abstract: We determine the character group of the infinite unitary group of a unital exact C*-algebra in terms of K-theory and traces and obtain a description of the infinite unitary group modulo the closure of its commutator subgroup by the same means. The methods are then used to decide when the state space SKQ(A X fl Z) of the KQ group of a crossed product by Z is homeomorphic to SK0(A\t or TG4) a. We also consider the crossed product A X aG by a discrete countable abelian group G and give necessary and sufficient conditions for the equality T(A xaG) = T(A\ to hold.

General Existence Theorems for Orthonormal Wavelets, an Abstract Approach

Abstract: Methods from noncommutative harmonic analysis are used to develop an abstract theory of orthonormal wavelets. The relationship between the existence of an orthonormal wavelet and the existence of a multi-resolution is clarified, and four theorems guaranteeing the existence of wavelets are proved. As a special case of the fourth theorem, a generalization of known results on the existence of smooth wavelets having compact support is obtained.

Euler-Poincare Characteristic and Polynomial Representations of Iwahori-Hecke Algebras

Abstract: The Hecke algebras of type A „ admit faithful representations by symmetrization operators acting on polynomial rings. These operators are related to the geometry of flag manifolds and in particular to a generalized Euler-Poincare characteristic denned by Hirzebruch. They provide g-idempotents, togetherwith a simple way to describe the irreducible representations of the Hecke algebra. The link with Kazhdan-Lusztig representations is discussed. We specially detail the case of hook representations, and as an application, we investigate the hamiltonian of a quantum spin chain with C/g(su(l/l)) symmetry.

Classification of paragroup actions in subfactors

Abstract: We define “a crossed product by a paragroup action on a subfactor” as a certain commuting square of type II1 factors and give their complete classification in a strongly amenable case (in the sense of S. Popa) in terms of a new combinatorial object which generalizes Ocneanu’s paragroup. In addition to the standard axioms of paragroups, we have the intertwining Yang-Baxter equation as the new additional axiom. We will show that Rational Conformal Field Theory in the sense of Moore-Seiberg and orbifold construction in the sense of D. E. Evans, the author, and F. Xu produce paragroup actions on

Perturbation Formulas for Traces on $C^*$-algebras

Abstract: We introduce the Frechet differential of operator functions on C*-aIgebras obtained via spectral theory from ordinary differentiate functions. In the finite-dimensional case this differential is expressed in terms of Hadamard products of matrices. A perturbation formula with applications to traces is given. §1. The Frechet Differentia! Definition 1.1. // & and ty are Banach spaces, and Q) is an open subset of &, we say that a function F: & -+& is Frechet differentiate, if for each x in 2 there is a bounded linear operator F^. in B(2£9 <&) such that lim Ufeir^FOc + h) F(x) Fx (h)) = 0 . If the differential map x -» F^ is continuous from & to B(3E9 $/\\ we say that F is continuously Frechet differ entiable. Straightforward computations give the following result, which we list for easy reference. Proposition 1.2. // F: X -» %/ and G: ty -> 3£ are continuously Frechet differentiable maps between Banach spaces 3£9 exp (A) is continuously Frechet differentiable with f 1 exp^ (B) = exp (sA)B exp ((1 s)A)ds Jo for all A, B in <$/. Proof. By elementary calculus we have 11 k s (l s)ds = Jo v ' (fc + m+1)! and we can prove either by direct calculation or by induction that (A + B) -A = ^(A + B)BA~v . k=0 Combining these two expressions we establish the Dyson formula (*) exp (A + B)exp (A) = £ V —(A + n=i fc=o nl f 1 = exp (s(A + B))B exp ((1 s)A)ds , Jo where we rearranged the sums by setting m = n — k— 1. It is clear that the proposed expression for exp^ is a bounded linear operator that depends continuously on A, and by subtraction we get from (*) that Hexp (A + B) exp (A) exp^ (B)|| 1 (exp (s(A + B)) exp (sA))B exp ((1 s)A)ds \\ J ||exp (s(A + B) exp (sA)\\\\ds . o PERTURBATION FORMULAS FOR TRACES ON C*-ALGEBRAS 171 From Lebesgues theorem of dominated convergence we see that the last integral converges to zero as B -» 0. We can thus conclude that exp is continuously Frechet differentiable with the desired differential. QED Definition 1.4. We denote by C£(R) the set of real ^-functions f of the form

The Eisenstein Quotient of the Jacobian Variety of a Drinfeld Modular Curve

Abstract: Let K = ¥q(T\ the rational function field over the finite field ¥q (T: indeterminate), and A=¥q[T]. For a non-zero ideal n of A, we can define a smooth proper geometrically connected curve X0(n) over K, called the Drinfeld modular curve of Hecke type with conductor n. In this article, we define the "Eisenstein quotient" J of the Jacobian variety / of Jf0(n) for n maximal and investigate its arithmetic properties. One of the main results is as follows :

Automorphisms of the Extended Affine Root System and Modular Property for the Flat Theta Invariants

Abstract: We define the central extension Aut+(jR) of the automorphism group Aut+(R) of the extended affine root system. We give the action of Aut +(.R) on the flat theta invariants (theta functions). This describes the modular property for the flat theta invariants.

Generalized Quantum Stochastic Processes on Fock Space

Abstract: As is highlighted in the excellent books by Meyer [21] and by Parthasarathy [26] quantum stochastic calculus on (Boson) Fock space has developed into a new field of mathematics keeping a profound contact with physical applications. Since Hudson and Parthasarathy [12] first formulated quantum stochastic integrals of Ito type in 1984 a crucial role has been played by three basic quantum stochastic processes:

Representations of Unitary Groups and Free Convolution

Abstract: To each finite dimensional representation of a unitary group U(n) is associated a probability measure on the set of integers, depending on the highest weights which occur in this representation. We show that asymptotically for large n and large irreducible representations of U(n) the measure associated to the tensor product of two representations, or to the restriction of a representation to a subgroup U(m) with m comparable to n, can be expressed in terms of the measures associated to the first representations by means of the notion of free convolution (namely additive free convolution for the tensor product problem and multiplicative free convolution for the restriction problem).

High-energy behavior of the scattering amplitude for a Dirac operator

Abstract: We study the high-energy behavior of the scattering amplitude and the total scattering cross section for a Dirac operator with a 4 X 4 matrix-valued potential. Moreover, in the electro-magnetic case, it is shown that the electric potential and the magnetic field can be reconstructed from the high-energy behavior of the scattering amplitude. The study of the high-energy behavior of the resolvent estimates is crucial for our proof.

The topological structure of Polish groups and groupoids of measure space transformations

Abstract: It is proved that the groupoid of nonsingular partial isomorphisms of a Lebesgue space (X,μ) is weakly contractible in a “strong” sense: we present a contraction path which preserves invariant the subgroupoid of μ-preserving partial isomorphisms as well as the group of nonsingular transformations of X. Moreover, let R be an ergodic measured discrete equivalence relation on X. The full group [R] endowed with the uniform topology is shown to be contractible. For an approximately finite R of type II or IIIλ, 0 ≤ λ < 1, the normalizer N [R] of R furnished with the natural Polish topology is established to be homotopically equivalent to the centralizer of the associated Poincare flow. These are the measure theoretical analogues of the resent results of S. Popa and M. Takesaki on the topological structure of the unitary and the automorphism group of a factor. The topological properties of automorphism groups of a Lebesgue space (X,μ) have been studied since 1944, when P. Halmos [Ha] introduced two metrizable topologies on the group Aut0(X,μ) of μ-preserving transformations: the weak dw and the uniform du. He proved that (Aut0(X,μ), dw) is a Polish group. S. Harada [Har] showed that it is simply connected and arcwise connected. His result was later considerably refined by M. Keane [K] who proved the contractibility of Aut0(X,μ) both in du and dw (see also [D, N]). A. Ionescu Tulsea [IT] and R. V. Chacon and N. A. Friedman [ChF] extended the weak and the uniform topology to the group Aut (X,μ) of nonsingular transformations of (X,μ) and generalized the results obtained in [Ha]. However, the homotopical properties of this group have not been studied so far. For a detailed exposition of the productive interplay between ergodic theory and operator algebras we refer to [M, C, S2]. The present work also was stimulated by a paper [PT] being pertained to operator algebras. In particular, given a countable group Γ ⊂ Aut (X,μ), then one can consider the full group [Γ] and its normalizer N [Γ] in Aut (X,μ) which are the measure theoretical analogues of the unitary group U(M) and the automorphism group Aut (M) of a von Neumann algebra M . Both groups, [Γ] andN [Γ], are Polish: the first with respect to du, the second with respect to some metric d defined by T. Hamachi and M. Osikawa [HO]. Further topological 1991 Mathematics Subject Classification. Primary 55P10, 22A05, 22A22, 28D15; Secondary 46L55.

Distributions with exponential growth and Bochner-Schwartz theorem for Fourier hyperfunctions

Abstract: Every positive definite Fourier hyperfunction is a Fourier transform of a positive and infra-exponentially tempered measure, which is the generalized Boehner-Schwartz theorem for the Fourier hyperfunctions. To prove this we characterize the distributions with exponential growth via the heat kernel method.

On Types of Quasifree Representations of Clifford Algebras

Abstract: The types of von Neumann algebras generated by quasifree representations of infinite dimensional Clifford algebras are studied in terms of spectral properties of positive operators parametrizing quasifree states. §

WKB Analysis to Global Solvability and Hypoellipticity

Abstract: This paper studies the global regularity and solvability of operators which could change their type at every point of the domain. Our object is to understand such operators from the viewpoints of a WKB analysis. To be more precise, let X be a compact manifold or an open domain in R. We denote by C°°(X) and C£°(X) the set of smooth functions on X and the set of smooth functions with compact supports respectively. We also denote the set of distributions on X by 9\"'(X). We say that a differential operator P is globally solvable (resp. globally hypoelliptic) in X if for every /eC£°(X) there exists ue3i'(X) satisfying Pu = f. (resp. ueC°°(X) when PueC°°(X) and we£^ ' (X)) . The operator P is said to be locally solvable (resp. locally hypoelliptic) at a point p e X if there exists a neighborhood U of p such that for every /£C~(£7) , there exists ue9\"'(U) satisfying Pu = f in U (resp. p £singsupp(Pu) implies p £singsupp(u)). By definition local hypoellipticity at each point p e X implies the global hypoellipticity in X, and the global solvability implies the local solvability at each point p e. X, while the corresponding inverse implications are not true. ([7]). Because the operators which we want to study are in general of mixed type the structures of local solutions may change drastically in every part of the domain. Therefore most of the methods such as those for degenerate elliptic operators, and for weakly hyperbolic operators are not applicable to such operators, (cf. [9], [17]). Moreover, because the structure of the characteristics is so complicated that the usual characteristic geometry does not seem adequate to

Whittaker functions for the large discrete series representations of SU (2,1) and related zeta integrals

Abstract: In terms of classical Bessel function, we represent explicitly the radial part of Whittaker functions on S£7(2,1) belonging to the large discrete series representations. Moreover we compute archimedean local L-factors corresponding to a construction of L-function by Gelbart and Piatetski-Shapiro.

Global existence of small solutions to semilinear Schro¨dinger equations with gauge invariance

Abstract: We present the global existence theorem for semilinear Schrodinger equations satisfying gauge invariance. Combining the local existence results and a priori estimates, we construct global solutions with small initial data. §

Estimation of the Number of the Critical Values at Infinity of a Polynomial Function f: C2 → C

Abstract: is not necessarily a locally trivial fibration. In general, we have to exclude a finite values Z^ c C from the base space so that / : C f~ (L u Z^ ) — > C (Z u Z^ ) is a locally trivial fibration. We say that r e C is a regular value at infinity of the function / : C — > C if there exist positive numbers R and 8 so that the restriction of/, f:f-(D£(T))-B*^>D£(T),is a trivial fibration over the disc D£(r) where De(T) = {r]£C\\ri-i\<£} and B*=[(x,y);x\ 2 =\y\

Quotients of Abelian Surfaces

Abstract: Let L be an abelian function field of two variables over C, and K be a Galois subfield of L, i.e., L is a finite algebraic Galois extension of K. We classify such K by a suitable complex representation of the Galois group G = Gal (L/K). Let A be the abelian surface with the function field L. Since g e G induces an automorphism of A, we have a complex representation gz = M(g)z + t(g\ where M(g) e GL2(C), z e C , and t(g) e C. Fixing the representation, we put G0 = {g eG\M(g) is the unit matrix}, H = {M(g)\g e G} and H1 = {M(g)e H\det M(g) = 1}. Then we have the following exact sequences of groups:

A generalization of Bochner's tube theorem in elliptic boundary value problems

Abstract: It is classically well known as Bochner's tube theorem that any holomorphic function defined on a tube domain T in a complex affine space has analytic continuation on the convex full of T [H, Theorem 2.5.10]. Now let M be a real analytic manifold, X its complexification. Let TM X be the normal bundle of M in X, VM the functor of specialization along M, and let J?M denote the sheaf H°vM(&x} on TMX. In [SKK, Chap.I], in connection with the theory of microfunctions, a proof is given to the following local version of Bochner's tube theorem :

Equivariant K-Theory and Maps between Representation Spheres

Abstract: The equivariant ^-theory has been successfully employed in the study of equivariant maps by Marzantowicz [5], Liulevicious [7] and Bartsch [3]. In the present paper, using the equivariant K- theory, we will obtain a necessary condition for the existence of G-maps SU — > SW , where SU and S W are the unit spheres of unitary representations U and W, respectively, of a compact Lie group G.

Equivariant Surgery Theory: Construction of Equivariant Normal Maps

Abstract: This procedure is one of the important ideas of equivariant surgery theory, and has enabled us to construct various exotic actions (see e.g. [BMol-2], [LaMo], [LaMoPa], [Mo 1-3], [MoU], [Pel-3], [PeR]). A method for (Step I) was presented by T. Petrie in [Pel-3], which we call the equivariant transversality construction. Roughly speaking, it is as follows: Let G be a finite group, and let Y be a compact, smooth G-manifold. If F is a real G-module and a : Y x F > 7 x F i s a proper G-map then a is properly G-homotopic to /?: Y x V -> Y x V such that j8 is transversal to Y x {0}. Then we obtain a G-normal map f:X-+Y, where X = p~(Yx {0}) and /= 0\x: X-+ Y. (Step II) is to convert f\X-*Y to a G-map f':X'^>Y belonging to a prescribed class of maps, e.g. of G-homotopy equivalences, of homotopy equivalences, of Zp-homology equivalences, etc. If some properties of X' are specified before the construction then it is a key to find an adequate real G-module V and an appropriate G-map a. Modified equivariant transversality construction has been employed in [BMol-2], [LaMoPa], [LaMo], [Mol-3], [MoU]. For

Symmetric Flat Connections, Triviality of Loi's Invariant and Orbifold Subfactors

Abstract: We define a notion of symmetric connections on subfactors and get a sufficient condition for a subfactor to have a symmetric connection. We also give a necessary and sufficient condition for Loi's invariant of a non-strongly outer automorphism of a subfactor to be trivial in the case with a symmetric connection. We apply this result to non-AFD SU(n)k subfactors and construct orbifold subfactors of non-AFD SU(n)k subfactors as well as the AFD case, as conjectured in our previous work. This generalizes constructions of Evans-Kawahig ashi and Xu.

Small Resolutions of Schubert Varieties and Kazhdan-Lusztig Polynomials

Abstract: Let G be a complex semisimple algebraic group. Let B, F, W, and S denote respectively a Borel subgroup, a fixed maximal torus contained in B, the Weyl group of G with respect to F, and the set of fundamental reflections in W. Let P denote a parabolic subgroup which contains B. We denote by Wp c W the Weyl group of P. For A,EW/Wp let e^ denote the corresponding F-fixed point of G/ P. Let X(A)cG/P denote the Schubert subvariety determined by k£W/Wp. Thus,X(A) is the Zariski closure of the 5-orbit of e^ e G/ P. The point X of an irreducible complex projective variety X is said to be small if, for each / > 0 one has codimx {x E X\ dim p~ l (x) > i} > 21. If p : X ^ > X is a small resolution, then for the intersection cohomology sheaf #'(X) (with respect to the middle perversity), the stalk 7/'(X)K is isomorphic to the singular cohomology group H'(p~(x);C) for any i > 0 . Suppose p: X(A) -> X(A) is a small resolution of Schubert variety X(A)czG/£ , A e W , and let X(r) e X(A) be a Schubert subvariety. It is well-known that the Poincare polynomial in q l / 2 of the fibre p~(eT} of p is equal to the Kazhdan-Lusztig polynomial P rA(g), where T < A, A, r e W [4]. Suppose that, for some g e G , gX(A) = X(A). Then g induces an isomorphism of the intersection cohomology sheaf #(X) such that # (X\ maps onto #(X)^. In particular / 8 A(^) = P r A(^) if PXX(9) = X(r) c X(A) where FA denotes the largest parabolic subgroup that stabilizes X(A). Suppose X(A) dGI P. Let A0 e W denote the representative of A of maximal length. Then PA = PA, the largest parabolic that stabilizes X(A). Also if X(T) e X(A) is /{-stable, then so is X(r0) c X(A0).

The Quantum weak coupling limit (II): Langevin equation and finite temperature case

Abstract: We complete the program started in [4] by proving that, in the weak coupling limit, the matrix elements, in the collective coherent vectors, of the Heisenberg evolved of an observable of a system coupled to a quasi-free reservoir through a laser type interaction, converge to the matrix elements of a quantum stochastic process satisfying a quantum Langevin equation driven by a quantum Brownian motion. Our results apply to an arbitrary quasi-free reservoir so, in particular, the finite temperature case is included.

Microlocal quasianalyticity for distributions and ultradistributions

Abstract: It was proved in [Bol] that (1.1) and (1.2) imply that/ must vanish in some neighborhood of S. The purpose of this note is to strengthen that result by replacing WFA(f) with WFM(f), the wave front set of/with respect to an arbitrary quasianalytic Denjoy-Carleman class C (Theorem 1), and by allowing/ to be an ultradistribution in the dual of a non-quasianalytic class (Theorem 2). By a counterexample of M. Sato ([Ka], Note 3.3) (1.1) and (1.2) do not imply that/ = 0 in a neighborhood of S, if/is only assumed to be a hyperf unction. In fact we have recently proved [Bo2] that Sato's example can be strengthened as follows: for an arbitrary quasianalytic class C there exists a hyperfunction in the dual of C such that (1.1) and (1.2) hold for some S but the support of/meets S. In the analytic case there is a well known closely related theorem proved by Hormander for distributions ([HI] ; [H2], Theorem 8.5.6) and independently by Kawai and Kashiwara for hyperf unctions ([Ka], Theorem 4.4.1), which reads as follows. If the distribution / vanishes on, one side of the C hypersurface S and