Showing papers on "Reductive group published in 2000"

The quantization conjecture revisited

Abstract: A strong version of the quantization conjecture of Guillemin and Sternberg is proved. For a reductive group action on a smooth, compact, polarized variety (X, L), the cohomologies of L over the GIT quotient X//G equal the invariant part of the cohomologies over X. This generalizes the theorem of [GS] on global sections, and strengthens its subsequent extensions ([JK], [li]) to RiemannRoch numbers. Remarkable by-products are the invariance of cohomology of vector bundles over X//G under a small change in the defining polarization or under shift desingularization, as well as a new proof of Boutot's theorem. Also studied are equivariant holomorphic forms and the equivariant Hodgeto-de Rham spectral sequences for X and its strata, whose collapse is shown. One application is a new proof of the Borel-Weil-Bott theorem of [Ti] for the moduli stack of C-bundles over a curve, and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures. Collapse of the Hodge-to-de Rham sequences for these stacks is also shown.

Perverse Sheaves on affine Grassmannians and Langlands Duality

Abstract: This is an expanded version of the text ``Perverse Sheaves on Loop Grassmannians and Langlands Duality'', AG/9703010. The main new result is a topological realization of algebraic representations of reductive groups over arbitrary rings. We outline a proof of a geometric version of the Satake isomorphism. Given a connected, complex algebraic reductive group G we show that the tensor category of representations of the Langlands dual group is naturally equivalent to a certain category of perverse sheaves on the loop Grassmannian of G. The above result has been announced by Ginsburg in and some of the arguments are borrowed from his approach. However, we use a more "natural" commutativity constraint for the convolution product, due to Drinfeld. Secondly, we give a direct proof that the global cohomology functor is exact and decompose this cohomology functor into a direct sum of weights. The geometry underlying our arguments leads to a construction of a canonical basis of Weyl modules given by algebraic cycles and an explicit construction of the group algebra of the dual group in terms of the affine Grassmannian. We completely avoid the use of the decomposition theorem which makes our techniques applicable to perverse sheaves with coefficients over an arbitrary commutative ring. We deduce the classical Satake isomorphism using the affine Grassmannian defined over a finite field. This note contains indications of proofs of some of the results. The details will appear elsewhere.

Essential Dimensions of Algebraic Groups and a Resolution Theorem for G-Varieties

Abstract: Let G be an algebraic group and let X be a generically free G-variety. We show that X can be trans- formed, by a sequence of blowups with smooth G-equivariant centers, into a G-variety Xwith the following property: the stabilizer of every point of Xis isomorphic to a semidirect product U⋊ A of a unipotent group U and a diagonalizable group A. As an application of this result, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus trans- formation.

Unipotent elements, tilting modules, and saturation

Abstract: The results in this paper were motivated by work of Serre [19] where semisimplicity results for representations of arbitrary groups in positive characteristic were established. A key ingredient in the arguments was the notion of saturation where one embeds a unipotent element of prime order of a simple algebraic group in a 1-dimensional unipotent subgroup. One of our goals here is to show that saturation can be achieved rather generally and to establish a uniqueness result for the resulting unipotent subgroups. As a by product we show for a simple algebraic group in good characteristic a unipotent element of prime order is contained in a particularly nice subgroup of type A1 and that the Lie algebra of the ambient algebraic group usually affords a tilting module for this subgroup. The tilting decompositions are applied to show that the centralizers of the unipotent element, a corresponding 1-dimensional unipotent group, and its Lie algebra all coincide and are closely related to the centralizer of the A1 subgroup. We also establish a convenient factorization of the centralizer of the unipotent element. Results for finite groups of Lie type are also obtained.

Résolutions de Demazure affines et formule de Casselman-Shalika géométrique

Abstract: We prove a conjecture of Frenkel, Gaitsgory, Kazhdan and Vilonen, related to Fourier coefficients of spherical perverse sheaves on the affine Grassmannian associated to a a split reductive group Our proof is an extension of the proof given by the first author in the case of GL(n) (see math/9801109); it relies on the study of certain resolutions of Schubert varieties in the affine Grassmannian, built from the so-called minuscule or quasi-minuscule cases

The GIT-Equivalence for G-Line Bundles

Abstract: Let X be a projective variety with an action of a reductive group G. Each ample G-line bundle L on X defines an open subset Xss(L) of semi-stable points. Following Dolgachev and Hu, define a GIT-class as the set of algebraic equivalence classes of L's with fixed XssL. We show that the GIT-classes are the relative interiors of rational polyhedral convex cones, which form a fan in the G-ample cone. We also study the corresponding variations of quotients Xss(L)//G. This sharpens results of Thaddeus and Dolgachev-Hu.

A positivity property of the Satake isomorphism

Abstract: Let G be an unramified reductive group over a local field. We consider the matrix describing the Satake isomorphism in terms of the natural bases of the source and the target. We prove that all coefficients of this matrix which are not obviously zero are in fact positive numbers. The result is then applied to an existence problem of F-crystals which is a partial converse to Mazur's theorem relating the Hodge polygon and the Newton polygon.

Simple algebraic and semialgebraic groups over real closed fields

Abstract: We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In Definably simple groups in o-minimal structures, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a compositionisomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.

On Orbit Closures of Symmetric Subgroups in Flag Varieties

Abstract: We study K-orbits in G/P where G is a complex connected reductive group, P ⊆ G is a parabolic subgroup, and K ⊆ G is the fixed point subgroup of an involutive automorphism �. Generalizing work of Springer, we parametrize the (finite) orbit set K\ G/P and we determine the isotropy groups. As a conse- quence, we describe the closed (resp. affine) orbits in terms of �-stable (resp. �-split) parabolic subgroups. We also describe the decomposition of any (K, P)-double coset in G into (K, B)-double cosets, where B⊆ P is a Borel subgroup. Finally, for certain K-orbit closures X ⊆ G/B, and for any homogeneous line bundle L on G/B having nonzero global sections, we show that the restriction map resX : H 0 (G/B,L)→ H 0 (X,L) is surjective and that H i (X,L) = 0 for i ≥ 1. Moreover, we describe the K-module H 0 (X,L). This gives information on the restriction to K of the simple G-module H 0 (G/B,L). Our construction is a geometric analogue of Vogan and Sepanski's approach to extremal K-types.

Exceptional groups and del Pezzo surfaces

Abstract: Given a del Pezzo surface of degree d between 1 and 6, possibly with rational double points, we construct a "tautological" holomorphic G-bundle over X, where G is a reductive group which is an appropriate conformal form of the simply connected complex linear group whose coroot lattice is isomorphic to the primitive cohomology of the minimal resolution of X For example, in case d=3 and X is a smooth cubic surface, the rank 27 vector bundle over X associated to the G-bundle constructed above and the standard 27-dimensional representation of E_6 is a direct sum of the line bundles associated to the 27 lines on X We also discuss the restriction of the G-bundle to smooth hyperplane sections

Finite quotients of the algebraic fundamental group of projective curves in positive characteristic

Abstract: Let X be a smooth connected projective curve defined over an algebraically closed field k of characteristic p >0 Let G be a finite group whose order is divisible by p Suppose that G has a normal p-Sylow subgroup We give a necessary and sufficient condition for G to be a quotient of the algebraic fundamental group pi(1)(X) of X

Abelian Unipotent Subgroups of Reductive Groups

Abstract: Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the ``order formula'' of D. Testerman for unipotent elements in G; moreover, we show that the same formula determines the p-nilpotence degree of the corresponding nilpotent elements in the Lie algebra of G. Second, if G is semisimple and p is sufficiently large, we show that G always has a faithful representation (r,V) with the property that the exponential of dr(X) lies in r(G) for each p-nilpotent X in Lie(G). This property permits a simplification of the description given by Suslin, Friedlander, and Bendel of the (even) cohomology ring for the Frobenius kernels G_d, d > 1. The previous authors already observed that the natural representation of a classical group has the above property (with no restriction on p). Our methods apply to any Chevalley group and hence give the result also for quasisimple groups with ``exceptional type'' root systems. The methods give explicit sufficient conditions on p; for an adjoint semisimple G with Coxeter number h, the condition p > 2h -2 is always good enough.

A characterization of linearly reductive groups by their invariants

Abstract: The theorem of Hochster and Roberts says that, for every moduleV of a linearly reductive groupG over a fieldK, the invariant ringK[V] G is Cohen-Macaulay. We prove the following converse: ifG is a reductive group andK[V] G is Cohen-Macaulay for every moduleV, thenG is linearly reductive.

On the Auslander-Reiten quiver of an infinitesimal group

Abstract: Let ${\cal G}$ be an infinitesimal group scheme, defined over an algebraically closed field of characteristic $p$. We employ rank varieties of ${\cal G}$-modules to study the stable Auslander-Reiten quiver of the distribution algebra of ${\cal G}$. As in case of finite groups, the tree classes of the AR-components are finite or infinite Dynkin diagrams, or Euclidean diagrams. We classify the components of finite and Euclidean type in case ${\cal G}$ is supersolvable or a Frobenius kernel of a smooth, reductive group.

The characters of the generalized Steinberg representations of finite general linear groups on the regular elliptic set

Abstract: Let k be a finite field, k Ik the degree n extension of k, and G = GLn(k) the general linear group with entries in k. This paper studies the "generalized Steinberg" (GS) representationis of G and proves the equivalence of several different characterizations for this class of representations. As our main result we show that the union of the class of cuspidal and GS representations of G is in natural one-one correspondence with the set of Galois orbits of characters of kx, the regular orbits of course corresponding to the cuspidal representations. Besides using Green's character formulas to define GS representations, we characterize GS representations by associating to them idempotents in certain commuting algebras corresponding to parabolic inductions and by showing that GS representations are the sole components of these induced representations which are "generic" (have Whittaker vectors). Let k = Fq be a finite field of cardinality q, let kIkm Ik be,_ respectively, an algebraic closure of k and the degree rm extension of k contained in k. Let q x _ Xq denote the Frobenius automorphism of klk and of any subextension km1k. Let G = GL (k) be the group of non-singular n x n matrices with entries in k. Also write Gm = GLrn(k) for any m > 1. In a signally important paper published in 1955 [GR] J. A. Green showed how to calculate formulas for the irreducible characters of G. In Green's work appeared for the first time general character formulas for the cuspidal representations and for a family of "generalized Steinberg" (GS) representations. For their study of the "level zero" discrete series characters of unit groups of simple algebras over a p-adic field the authors need diverse characterizations of cuspidal and GS characters of finite general linear groups and to be able to pass between these different characterizations. In this paper we give these characterizar tions and prove their equivalence. It is fruitful to view the set of GS representatio:ns as a class of representations of G which contains the class of cuspidal representations as a subelass. Of course the cusp form property, the propert;y of not being a component of Ind u 1 for any unipotent radical U :& (I) of a parabolic subgroup of G, clearly distinguishes the class of cuspidal representations from all other representations of G. However, other important properties which are usually associated to the class of cuspidal representations generalize to the union of the two classes of representations of C. Received by the editors May 26, 1997 and, in revised form, April 18, 1998 and June 26, 1998. 1991 Mathematics Subject Classification. Primary 22E50, 11T24.

Pro-algebraic and differential algebraic group structures on affine spaces

Abstract: We prove the following result, solving a problem raised in an article by A. Buium and Ph. J. Cassidy, to appear in the collected works of E. Kolchin. If G is a differential algebraic group whose underlying set is some affine n -space, then G is unipotent. A key result, possibly of independent interest, concerns infinite-dimensional group schemes: a group scheme whose underlying scheme is Spec ( K [ X 1 , X 2 ,...]) ( K an algebraically closed field of characteristic 0) is a projective limit of unipotent algebraic groups.

A Langlands classification for unitary representations

Abstract: The Langlands classification theorem describes all admissible representations of a reductive group $G$ in terms of the tempered representations of Levi subgroups of $G$. I will describe work with Susana Salamanca-Riba that provides (conjecturally) a similar description of the unitary representations of $G$ in terms of certain very special unitary representations of Levi subgroups.

Norm principle for reductive algebraic groups

Abstract: We begin by recalling the norm principles of Knebusch and Scharlau from the algebraic theory of quadratic forms. Let (V, q) be a non-degenerate quadratic form over a field F and let L/F be a finite field extension. Let D(q) denote the subgroup in F× generated by the non-zero values of q. Knebusch norm principle asserts that NL/F ( D(q⊗F L) ) ⊂ D(q), where NL/F : L× → F× is the norm map. Let G(q) be the group of multipliers of the quadratic form q, i.e. the group of all x ∈ F× such that (V, xq) ≃ (V, q). Scharlau norm principle states that NL/F ( G(q ⊗F L) ) ⊂ G(q). More generally, let φ : G→ T be an algebraic group homomorphism defined over a field F . Assume that the group T is commutative, so that the norm homomorphism NL/F : T (L) → T (F ) is defined for any finite separable field extension L/F . The norm principle for φ and L/F claims that the norm homomorphism NL/F maps the image of the induced homomorphism φL : G(L) → T (L) to the image of φF : G(F ) → T (F ). Knebusch and Scharlau norm principles are the special cases of the norm principle for certain group homomorphisms ( cf. Examples (3.2) and (3.3) ) . In the general setting the validity of the norm principle is an open problem. The main result of the paper is the following

Weak approximation, Brauer and R-equivalence in algebraic groups over arithmetical fields, II

Abstract: We prove some new relations between weak approximation and some rational equivalence relations (Brauer and R-equivalence) in algebraic groups over arithmetical fields. By using weak approximation and local - global approach, we compute completely the group of Brauer equivalence classes of connected linear algebraic groups over number fields, and also completely compute the group of R-equivalence classes of connected linear algebraic groups $G$, which either are defined over a totally imaginary number field, or contains no anisotropic almost simple factors of exceptional type $^{3,6}\\D_4$, nor $\\E_6$. We discuss some consequences derived from these, e.g., by giving some new criteria for weak approximation in algebraic groups over number fields, by indicating a new way to give examples of non stably rational algebraic groups over local fields and application to norm principle.

On the action of the dual group on the cohomology of perverse sheaves on the affine grassmannian

Abstract: It was proved by Ginzburg and Mirkovic-Vilonen that the $G(O)$-equivariant perverse sheaves on the affine grassmannian of a connected reductive group $G$ form a tensor category equivalent to the tensor category of finite dimensional representations of the dual group $G^\vee$. The proof use the Tannakian formalism. The purpose of this paper is to construct explicitely the action of $G^\vee$ on the global cohomology of a perverse sheaf.

On one class of unipotent subgroups of semisimple algebraic groups

Abstract: This paper contains a complete proof of a fundamental theorem on the normalizers of unipotent subgroups in semisimple algebraic groups.

On the converse to a theorem of Atiyah and Bott

Abstract: Let G be a complex reductive group and let C be a smooth curve of genus at least one. We prove a converse to a theorem of Atiyah-Bott concerning the stratification of the space of holomorphic G-bundles on C. In case the genus of C is one, we establish that one has a stratification in the strong sense. The paper concludes with a characterization of the minimally unstable strata in case G is simple.

Stable Groups and Algebraic Groups

Abstract: A certain property of some type-definable subgroups of superstable groups with finite U-rank is closely related to the Mordell–Lang conjecture. This property is discussed in the context of algebraic groups.

BOUNDARY COHOMOLOGY OF SHIMURA VARIETIES, III: The nightmare continues

Abstract: The present article continues the study of the boundary cohomology of Shimura varieties initiated in [HZ1, HZ2]. Let G be a reductive group over Q, X the symmetric space associated to G(R), and Γ an arithmetic subgroup (e.g., a congruence subgroup) of G(Q). We consider the cohomology of Γ\X with coefficients in the local system V constructed from a representation V of G, i.e., H•(Γ\X, V) ' H•(Γ, V ). It is standard that this cohomology can be decomposed as the direct sum of “interior” cohomology, defined as the image of the cohomology with compact supports H• c (Γ\X, V), and a complementary “boundary cohomology” that restricts nontrivially to the boundary of the Borel-Serre (manifold-with-corners) compactification of Γ\X. The designation of boundary cohomology is generally non-canonical, and much work has been devoted to constructing canonical decompositions using Eisenstein series. By an elaboration on the de Rham theorem, one knows that the cohomology group H•(Γ\X, V) can be expressed as the relative Lie algebra cohomology of the space of V -valued C∞ functions on Γ\G(R), or even the functions of moderate growth ([B2, §7]). Thanks to the work of Franke [Fr1], one can replace the functions of moderate growth by the subspace of automorphic forms, and this can provide the starting point for an approach to the boundary cohomology. However, in this series of articles we are concerned only tangentially with the relation between boundary cohomology and automorphic forms. We choose to work at a more intrinsic level, concentrating instead on the additional structures on H•(Γ\X, V) when X is a hermitian symmetric domain. In that case, Γ\X is an algebraic variety, and V underlies a natural variation of Hodge structure. Morihiko Saito’s theory of mixed Hodge modules [Sa3] then gives that H•(Γ\X, V) has a corresponding mixed Hodge

Algebraicity criteria for almost homogeneous complex spaces

Abstract: We give some algebraicity criteria on almost homogeneous spaces and thereby generalize the following result of D. Luna: every smooth equivariant Moishezon compactification of an algebraic torus is in fact algebraic. Introduction. D. Luna proved in (3) that every smooth Moishezon space that is almost homogeneous with respect to an algebraic torus action is in fact an algebraic variety. The purpose of this note is to give some generalizations of Lunas result. We consider the following situation: Let G denote a connected reductive complex Lie group and let X be a normal irreducible complex space. Assume that X is almost homogeneous with respect to a holomorphic action of G, i.e., there is a point x02 X such that the orbit G x0 is open (and hence dense) in X. The first algebraicity criterion we prove in this note is Theorem 1. Suppose that the isotropy group Gx0 is connected and that there are points x1; ... ; x r such that the isotropy groups Gx1 ; ... ; G x r are reductive and moreover for every x2 X the closure of the orbit G x contains at least one of the xi. Then X is a complex algebraic G-variety. This result implies in particular that an analytic normal equivariant compactification of an algebraic torus T is (equivariantly) algebraic if and only if every T-orbit contains a fixed point in its closure. The latter condition is always satisfied if T acts algebraically on a Moishezon space. Now assume that X is in addition a (compact) Moishezon space and the reductive group G acts algebraically, i.e., the action of G on X is given by a morphism G! AutOXU of algebraic groups. In this setting we prove:

Wilson's theorem in algebraic number fields

Abstract: In this paper a generalization of Wilson's theorem (p — 1)! = — 1 (mod p), pa prime, in algebraic number fields is proved. Gauss [DICKSON, L. E.: History of the Theory of Numbers, Vol. I, Carnegie Institute, Washington, 1919] generalized this proving that the product of positive integers less than n and prime to n is congruent modulo n to — 1 if n = 4 ,p ,2p , where p is an odd prime, and to -f 1 if n is not of one of these three forms. Further extensions of this result to products n>> ri> aeP(e) a£G(e) where P(e), G(e) are respectively a maximal semigroup and a maximal group in Z n belonging to the idempotent e, are given in [SCHWARZ, S.: The role of semigroups in the elementary theory of numbers, Math. Slovaca 31 (1981), 369-395]. Extending this method based on investigation of idempotents and the structure of the maximal (semi)groups, we prove analogous theorems for the residue class ring S/3 of the ring of integers of an algebraic number field and give specialization to some special cases of algebraic number fields. 1. P r imit ive idempotents Let R be a finite commutative ring with unit element 1 and let E be the set of its idempotents. The set E is non empty (0,1 G E) and finite. Endowed 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n : Primary 13G05; Secondary 11A05; 11R04.