Minimal generating subspaces of ''weak PBW type'' for vertex operator algebras are studied and a procedure is developed for finding such subspaces. As applications, some results on generalized modules are obtained for vertex operator algebras that satisfy a certain condition, and a minimal generating space of weak PBW type is produced for V with L any positive-definite even lattice. ᮊ 1999 L
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.Theorem 3.2, we derive functional equations for Whittaker functions arising from spherical functions, assuming convergence of interwining integrals.In Section 4 we discuss the case of local singular series. Here we are able to show that the relevant characters on N' are generic, there is no problem with convergence of intertwining integrals, and hence, there are functional equations.It is a pleasure to acknowledge my debt to R. Langlands for suggesting that the functional equations might follow from the uniqueness of certain Whittaker models and to W. Casselman for explaining his ideas on Whittaker models. In particular, Section 1 is just a recasting of Casselman's argument in the Chevalley group case. During the preparation of this paper the author was supported in part by National Science Foundation Grant MPS75-07055. Notations and Conventions.0.1. The field of real (resp. complex, rational) numbers is denoted R (resp. C, Q), and if k is a field of finite degree over Q, then k, will denote the completion of k at a place v; Q will be a locally compact, totally disconnected field. 0.2. All manifolds X will be analytic over the field Q and countable at infinity. By C, (X) we denote the space of locally constant functions of compact support on X. Let G be a group. If H is a subgroup of G, then Z(H) = ZG(H) = {g E Gg*x .g-1 = x(x EH)} isthecentralizerofHinG. IfNis a subgroup of G, then G = H oc N means that N is normal in G and G isthe semi-direct product of H and N. If g, x E G, then Int(g)(x) = g*x.g-1. Algebraic groups will be affine and defined over fields of characteristic zero. If G is an 9-group, then G(Q) is an analytic group over Q. If dg is left Haar measure on a locally compact group G, then the moduluscharacter bG is defined by d(gh) = &G(h) -1 dg. 0.5. A smooth representation of a locally compact group G on a complex vector space V is a homomorphism p: G -GL ( V) such that the isotropy group of each vector is an open subgroup of G. 1.3.3. We will also need the exactness of the functor V -( ON' V. This follows easily from the identity 4 ON' V V/V(+), where V(Q) = {J E V: u w(x) . * (x) dx = 0 for all sufficiently large compact subgroups U of N' }. The Filtration. Given w E G let V(w) = Ind, (x1P, PwP'). Suppose Pw,P' is open in G. Then there is an inclusion of V(w,) in V = PS(x). We shall show in Section 2 that whenever 4 is generic we have an isomorphism of i ON' V(w1) with i ON' V induced by the inclusion of V(w,) in V.Given w E G, let l(w) = dim P \PwP'. In case P = P' is a minimal 9-parabolic and w represents an element w in the r...
This is a continuation of a previous study of quantum vertex algebras of Zamolodchikov-Faddeev type. In this paper, we focus our attention on the special case associated to diagonal unitary rational quantum Yang-Baxter operators. We prove that the associated weak quantum vertex algebras, if not zero, are irreducible quantum vertex algebras with a normal basis in a certain sense.
Let K be a field of characteristic 7*2,3 and let 3* be the exceptional Jordan algebra of dimension 27 consisting of hermitian 3x3 matrices with entries in the Cayley-Dickson algebra & K . The product X o Y in 3 is j{XY + YX\ where XY is the matrix product. In [3], there are defined a norm (det) and a trace (tr) on 3. Let ( , , ) be the symmetric trilinear form on 3 x 3 x 3 such that {A, A, A) = fe\(A\ and define a bilinear map 3 x 3 -» 3? which takes {A, B) to Ax J5, by requiring that (A x B 9 Q = 3(A 9 B 9 Q for each C G 3, where (X, 7) = tr(Z o Y). Then Ax A plays the role of the matrix adjoint of A, and the notions just introduced can be used to define the rank of each element A e 3-We denote this by rk(A). where!/ = {Yety. Y = X 2 for some Xe 3 K }. The group of holomorphic automorphisms of % is isogenous to a certain algebraic ô-group which is of type E 7 . Baily [1] has defined an arithmetic subgroup T of G Q which is a unicuspidal subgroup of G and a maximal discrete subgroup of G R . Let J(Z, y) be the functional determinant of y at Z, Z G 2. Let T 0 be the subgroup of T which stabilizes a certain zero-dimensional rational boundary component %$ of % 9 as in [1, §7]. We let
T e A + /IMS 1970 Mityerf classifications. Primary 10D20; Secondary 20G30. Key words and phrases. Fourier coefficients, Eisenstein series, algebraic g-group of Type E 7 , exceptional Jordan algebra, arithmetic subgroup. f This paper describes a portion of the author's doctoral thesis written under the direction of Professor Walter L. Baily, Jr. at the University of Chicago.
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