Showing papers on "Reductive group published in 1978"

Instability in invariant theory

Abstract: Let V be a representation of a reductive group G. A fundamental theorem in geometric invariant theory states that there are enough polynomial functions on V, which are invariant under G, to distinguish disjoint closed G-invariant subsets of V. This result was given by D. Mumford in [11] for linearly reductive groups and conjectured by him to hold in general. Subsequently, it has been established for arbitrary reductive groups due to the efforts of C.S. Seshadri, M. Nagata and W. Haboush among others (see [15] and [12]). To use geometry to study invariants, for any closed G-invariant subset S of V, one may define a vector v in V to be S-unstable if the closure of the orbit G v meets S. In the classical case, S contains just the zero point O of V. The other fundamental theorem in geometric invariant theory states that v is S-unstable if and only if there is a one-parameter subgroup X of G such that v is S-unstable for the induced Gm-action on V. This result appears in Chapter 2 of Mumford in the classical case over an algebraically closed field. The above theorems, together with the numerical study of the action of the one-parameter subgroups of G, form the Hilbert-Mumford criterion for instability, which gives an effective means for finding all vectors v for which all invariants vanish (without actually finding any invariants!). In this paper, I will prove the second fundamental theorem for arbitrary S over a perfect ground field (Theorem 4-2). This solves a rationality question mentioned on page 64 of Mumford's book. For special ground fields, this has been done by D. Birkes in [1] for the real numbers and by M. Raghunathan in [13] for the algebraic number fields. In fact, M. Raghunathan uses Birkes' results and A. Borel's reduction theory to deduce his results. My proof of this theorem rests on the solution of another problem

Some applications of the Frobenius in characteristic 0

Abstract: Several applications are given of the technique of proving theorems in char 0 (as well as char/?) by, in some sense, "reducing" to char/? and then applying the Frobenius. A "metatheorem" for reduction to char/? is discussed and the proof is sketched. This result is used later to give the idea of the proof of the existence of big Cohen-Macaulay modules in the equicharacteristic case. Homological problems related to the existence of big Cohen-Macaulay modules are discussed. A different application of the same circle of ideas is the proof that rings of invariants of reductive linear algebraic groups over fields of char 0 acting on regular rings are CohenMacaulay. Despite the fact that this result is false in char/?, the proof depends on reduction to char/?. A substantial number of examples of rings of invariants is considered, and a good deal of time is spent on the question, what does it really mean for a ring to be Cohen-Macaulay? The paper is intended for nonspecialists.

Coherent states and square integrable representations

Abstract: The purpose of this paper -is to put in evidence the close connection between the existence of coherent states and the square integrability of an irreducible unitary representation of a Lie group. It is shown that, in the case of a reductive group, an irreducible unitary representation with discrete kernel admits a system of coherent states if and only if it belongs to the relative discrete series, and that, in such a situation, it is in fact a coherent state representation (in a sense to be precised in the body of the paper). Results of the same nature are proved in the solvable case, but only for coherent state representations, and under the additional . assumption that the group involved is an extension of a torus by an exponential Lie group.

Algebraic curves with an infinite automorphism group

Abstract: Irreducible algebraic curves X (not necessarily smooth and complete) for which the group Aut X of biregular automorphisms is infinite are classified. Applications of the results obtained to the theory of algebraic transformation groups are given.

Conjugations of arithmetic automorphic function fields

Abstract: In [1], Doi and Naganuma showed that the conjugation ofa Shimura curve is again a Shimura curve. The present paper deals with the generalization of their result. Consider in general a reductive algebraic group G defined over Q. Let G u be the semi-simple part of G. Assume that G~t modulo a maximal compact subgroup defines a bounded symmetric domain ~ , and that a system of canonical models (in the sense of Shimura [11, 2.13]) for the quotients of ~ by the arithmetic subgroups Yx of G exists. Let {Vx, (gx, Jxw(U)} be such a system. Take an arbitrary automorphism z of the complex number field C, and conjugate all the Vx's and Jxw(U)'S by z. Then one expects that {V:~, C~x, Jxw(u) ~} with suitable ~x's forms a system of canonical models for some reductive group G1. In this paper we show that this is the case if G is the type of groups investigated by Shimura in [11]. The corresponding group G1 is defined explicitly in 1.2, and the precise statement of the result is given as Theorem 1.3. In Shimura's construction, for special Yx the model Vx is first realized as a s ubvariety of a moduli-variety Vo for some PEL-type f2. Consider V~ as embedded in V~. It is known that the conjugate variety V~is isomorphic to the moduli-variety V w of another PEL-type f2'. The relations between f2 and f2' are provided by Shimura's work [6]. One of our main task then is to prove that the isomorphism of V~ to V u, induces an isomorphism ofV~to Vxl for some arithmetic subgroup Fx~ of G1. This can be achieved by studying the isolated fixed points of G o and G1 o" We carry out these considerations in Sections 2 and 3. The same functorial property also holds for the models constructed by Miyake in [3]. In Section 4 we deal with this case briefly. We shall not give the proof, because the argument is similar to, and actually simpler than, the one presented in this paper. We assume the reader is familiar with Shimura's work [10] and [11], which will be quoted respectively as [A] and [C] hereafter.

On the group of automorphisms of affine algebraic groups

Abstract: We study the conservativeness property of affine algebraic groups over an algebraically closed field of characteristic O and of their group of automorphisms. We obtain a certain decomposition of affine algebraic groups, and this, together with the result of Hochschild and Mostow, becomes a major tool in our study of the conservativeness property of the group of automorphisms. 1. Introduction. Let G be an affine algebraic group over a field F, with Hopf algebra 6g(G) of polynomial functions on G, in the sense of (2) and let W(G) denote the group of all affine algebraic group automorphisms of G. Then 6D(G) may be viewed as a right W(G)-module, with W(G) acting by compositionff o ol on 6D(G). We recall, from (3), that G is said to be conservative if (G) is locally finite as a W(G)-module. As is shown in (3), the conservativeness of G character- izes the existence of a suitable affine algebraic group structure on W(G) and the obstruction to the conservativeness of a connected G is realized as the presence of certain central tori in G, when the base field F is algebraically closed and of characteristic 0. In the present study of W(G), we exploit the above results and technique of (3) and, accordingly, we refer to (2) and (3) for standard facts concerning affine algebraic groups and their automorphism group. The following are brief descriptions of the contents appearing in each section: In §2, we examine reductive affine algebraic groups and their conservativeness and, in §3, we establish a certain W(G)-invariant decompo- sition of G when G is conservative. Finally, in §4, we use the result obtained in §3 to study the structure of W(G). The following notation is standard throughout: Let G be an affine algebraic group. Then Gl denotes the connected component of the identity element of G and Z(G) the center of G. If x E G, we use lx to denote the inner automorphism of G that is induced by x, and, for a subset S of G,