Lifting automorphisms of generalized adjoint quotients

Let G be a reductive algebraic group over an algebraically closed field K of characteristic zero. Let p:\mathfrakg^r ? X = \mathfrakg^r//G \pi :{\mathfrak{g}^r} \to X = {\mathfrak{g}^r}//G be the categorical quotient where \mathfrakg \mathfrak{g} is the adjoint representation of G and r is a suitably large integer (in general r ≥ 5, but for many cases r ≥ 3 or even r ≥ 2 suffices). We show that every automorphism φ of X lifts to a map F:\mathfrakg^r ? \mathfrakg^r \Phi :{\mathfrak{g}^r} \to {\mathfrak{g}^r} commuting with π. As an application we consider the action of φ on the Luna stratification of X.     

A Noncommutative Version of <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and Characterizations of Subdiagonal Algebras

Let M{\mathcal M} be a σ-finite von Neumann algebra and \mathfrak A{\mathfrak A} a maximal subdiagonal algebra of M{\mathcal M} with respect to a faithful normal conditional expectation F{\Phi} . Based on Haagerup’s noncommutative L ^p space L^p(M){L^p(\mathcal M)} associated with M{\mathcal M} , we give a noncommutative version of H ^p space relative to \mathfrak A{\mathfrak A} . If h ₀ is the image of a faithful normal state j{\varphi} in L¹(M){L^1(\mathcal M)} such that j°F = j{\varphi\circ \Phi=\varphi} , then it is shown that the closure of {\mathfrak Ah₀^\frac1p}{\{\mathfrak Ah_0^{\frac1p}\}} in L^p(M){L^p(\mathcal M)} for 1 ≤ p < ∞ is independent of the choice of the state preserving F{\Phi} . Moreover, several characterizations for a subalgebra of the von Neumann algebra M{\mathcal M} to be a maximal subdiagonal algebra are given.     

Dual partially harmonic tensors and Brauer-Schur-Weyl duality

Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let $ \mathfrak{B}_n^{(f)} Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let \mathfrakB_n^(f) \mathfrak{B}_n^{(f)} be the two-sided ideal of the Brauer algebra \mathfrakB_n( - 2m ) {\mathfrak{B}_n}\left( { - 2m} \right) over K generated by e ₁ e _3⋯ e _2f-1 where 0 ≤ f ≤ [n/2]. Let HT_f ^?n \mathcal{H}\mathcal{T}_f^{ \otimes n} be the subspace of partial-harmonic tensors of valence f in V ^⊗n . In this paper we prove that dimHT_f ^?n \mathcal{H}\mathcal{T}_f^{ \otimes n} and dim \textEn\textd_K\textSp(V)( V ^?n \mathord

Bounds on smooth matrix coefficients on L 2-spaces

Let G be a semisimple Lie group with a finite number of connected components and a finite center. Let K be a maximal compact subgroup. Let X be a smooth G-space equipped with a G-invariant measure. In this paper, we give upper bounds for K-finite and ${\mathfrak k}Let G be a semisimple Lie group with a finite number of connected components and a finite center. Let K be a maximal compact subgroup. Let X be a smooth G-space equipped with a G-invariant measure. In this paper, we give upper bounds for K-finite and \mathfrak k{\mathfrak k}-smooth matrix coefficients of the regular representation L ²(X) under an assumption about supp(L²(X)) ?[^(G)]_K{{\rm supp}(L^2(X)) \cap \hat G_K}. Furthermore, we show that this bound holds for unitary representations that are weakly contained in L ²(X). Our result generalizes a result of Cowling–Haagerup–Howe (J Reine Angew Math 387:97–110, 1988). As an example, we discuss the matrix coefficients of the O(p, q) representation L²(\mathbbR^p+q){L^2(\mathbb{R}^{p+q})}.     

An abstract setting for hamiltonian actions

In this paper we develop an abstract setup for hamiltonian group actions as follows: Starting with a continuous 2-cochain ω on a Lie algebra \mathfrak h{\mathfrak h} with values in an \mathfrak h{\mathfrak h}-module V, we associate subalgebras \mathfrak sp(\mathfrak h,w) ê \mathfrak ham(\mathfrak h,w){\mathfrak {sp}(\mathfrak h,\omega) \supseteq \mathfrak {ham}(\mathfrak h,\omega)} of symplectic, resp., hamiltonian elements. Then \mathfrak ham(\mathfrak h,w){\mathfrak {ham}(\mathfrak h,\omega)} has a natural central extension which in turn is contained in a larger abelian extension of \mathfrak sp(\mathfrak h,w){\mathfrak {sp}(\mathfrak h,\omega)}. In this setting, we study linear actions of a Lie group G on V which are compatible with a homomorphism \mathfrak g ? \mathfrak ham(\mathfrak h,w){\mathfrak g \to \mathfrak {ham}(\mathfrak h,\omega)}, i.e., abstract hamiltonian actions, corresponding central and abelian extensions of G and momentum maps J : \mathfrak g ? V{J : \mathfrak g \to V}.     

The upper bound of the $${{\rm L}_{\mu}^2}$$ spectrum

Let M be an n-dimensional complete non-compact Riemannian manifold, dμ = e ^h (x)dV(x) be the weighted measure and \triangle_m{\triangle_{\mu}} be the weighted Laplacian. In this article, we prove that when the m-dimensional Bakry–émery curvature is bounded from below by Ric _m ≥ −(m − 1)K, K ≥ 0, then the bottom of the L_m²{{\rm L}_{\mu}^2} spectrum λ₁(M) is bounded by

On Representations of Integers in Thin Subgroups of {{\rm SL}_2({\mathbb {Z}})}

Let ${\Gamma < {\rm SL}(2, {\mathbb Z})}Let G < SL(2, \mathbb Z){\Gamma < {\rm SL}(2, {\mathbb Z})} be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors v₀, w₀ ? \mathbb Z²  \  {0}{v_{0}, w_{0} \in \mathbb {Z}^{2} \, {\backslash} \, \{0\}}. We consider the set S{\mathcal {S}} of all integers occurring in áv₀g, w₀?{\langle v_{0}\gamma, w_{0}\rangle}, for g ? G{\gamma \in \Gamma} and the usual inner product on \mathbb R²{\mathbb {R}^2}. Assume that the critical exponent δ of Γ exceeds 0.99995, so that Γ is thin but not too thin. Using a variant of the circle method, new bilinear forms estimates and Gamburd’s 5/6-th spectral gap in infinite-volume, we show that S{\mathcal {S}} contains almost all of its admissible primes, that is, those not excluded by local (congruence) obstructions. Moreover, we show that the exceptional set \mathfrak E(N){\mathfrak {E}(N)} of integers |n| < N which are locally admissible (n ? S   (mod  q)   for all   q 3 1){(n \in \mathcal {S} \, \, ({\rm mod} \, q) \, \, {\rm for\,all} \,\, q \geq 1)} but fail to be globally represented, n ? S{n \notin \mathcal {S}}, has a power savings, |\mathfrak E(N)| << N^1-e₀{|\mathfrak {E}(N)| \ll N^{1-\varepsilon_{0}}} for some ${\varepsilon_{0} > 0}${\varepsilon_{0} > 0}, as N → ∞.     

Cusp forms

LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L ²(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP _f(g)=Σ_H∩ΓΓ f(γ{}itg) is an automorphic form on Γ\G. We show thatP _f is cuspidal in some cases, whenH ∩Γ\H is compact. Partially supported by NSF Grant # DMS 9103608.     

Regelflächenscharen

Let K be any commutative field and V:=K ⁴. A collection of ruled quadrics in V is called a flock of ruled quadrics if the following holds true. (1) ⋃_{ℱ∈ G} ℱ = V; (2) There is a line S⊂V such that ℱ₁⋂ℱ₂= S for all distinct ℱ₁, ℱ₂∈ . The group ΓL(V) decomposes the set of all those flocks into equivalence classes. Besides that, we consider any cone R in V, say R:= {x∈V|x ₁ x ₃ - x ₂ ² = 0}. Let R denote the set of all regular points of R. Plane sections of R which do not contain the singular point of ℜ are called regular sections. We consider decompositions of R ^* by regular sections and their equivalence classes with respect to the symmetry group ΓL(V)R of the cone ℜ. The main result is as follows. There is a (natural) bijection between the classes of equivalent flocks of rules quadrics and the classes of equivalent decompositions of R ^* by regular sections. A brief discussion of those flocks of ruled quadrics on which the construction of the so-called Betten-Walker planes is based ends the paper. Provided that char K≠3, these planes exist if and only if x∈K→x ³∈K is bijective.      

Web bases for the general linear groups

Let V be the representation of the quantized enveloping algebra of \mathfrakgl(n)\mathfrak{gl}(n) which is the q-analogue of the vector representation and let V ^∗ be the dual representation. We construct a basis for ?^r(V ?V^*)\bigotimes^{r}(V \oplus V^{*}) with favorable properties similar to those of Lusztig’s dual canonical basis. In particular our basis is invariant under the bar involution and contains a basis for the subspace of invariant tensors.     

Alternating and Sporadic Simple Groups are Determined by Their Character Degrees

Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let cd(G) be the set of all irreducible complex character degrees of G forgetting multiplicities, that is, cd(G) = {χ(1) : χ ∈ Irr(G)} and let cd ^*(G) be the set of all irreducible complex character degrees of G counting multiplicities. Let H be an alternating group of degree at least 5, a sporadic simple group or the Tits group. In this paper, we will show that if G is a non-abelian simple group and cd(G) í cd(H)cd(G)\subseteq cd(H) then G must be isomorphic to H. As a consequence, we show that if G is a finite group with cd^*(G) í cd^*(H)cd^*(G)\subseteq cd^*(H) then G is isomorphic to H. This gives a positive answer to Question 11.8 (a) in (Unsolved problems in group theory: the Kourovka notebook, 16th edn) for alternating groups, sporadic simple groups or the Tits group.     

Slices for biparabolics of index 1

Let \mathfraka \mathfrak{a} be an algebraic Lie subalgebra of a simple Lie algebra \mathfrakg \mathfrak{g} with index \mathfraka \mathfrak{a}  ≤ rank \mathfrakg \mathfrak{g} . Let Y( \mathfraka ) Y\left( \mathfrak{a} \right) denote the algebra of \mathfraka \mathfrak{a} invariant polynomial functions on \mathfraka^* {\mathfrak{a}^*} . An algebraic slice for \mathfraka \mathfrak{a} is an affine subspace η + V with h ? \mathfraka^* \eta \in {\mathfrak{a}^*} and V ì \mathfraka^* V \subset {\mathfrak{a}^*} subspace of dimension index \mathfraka \mathfrak{a} such that restriction of function induces an isomorphism of Y( \mathfraka ) Y\left( \mathfrak{a} \right) onto the algebra R[η + V] of regular functions on η + V. Slices have been obtained in a number of cases through the construction of an adapted pair (h, η) in which h ? \mathfraka h \in \mathfrak{a} is ad-semisimple, η is a regular element of \mathfraka^* {\mathfrak{a}^*} which is an eigenvector for h of eigenvalue minus one and V is an h stable complement to ( \textad  \mathfraka )h \left( {{\text{ad}}\;\mathfrak{a}} \right)\eta in \mathfraka^* {\mathfrak{a}^*} . The classical case is for \mathfrakg \mathfrak{g} semisimple [16], [17]. Yet rather recently many other cases have been provided; for example, if \mathfrakg \mathfrak{g} is of type A and \mathfraka \mathfrak{a} is a “truncated biparabolic” [12] or a centralizer [13]. In some of these cases (in particular when the biparabolic is a Borel subalgebra) it was found [13], [14], that η could be taken to be the restriction of a regular nilpotent element in \mathfrakg \mathfrak{g} . Moreover, this calculation suggested [13] how to construct slices outside type A when no adapted pair exists. This article makes a first step in taking these ideas further. Specifically, let \mathfraka \mathfrak{a} be a truncated biparabolic of index one. (This only arises if \mathfrakg \mathfrak{g} is of type A and \mathfraka \mathfrak{a} is the derived algebra of a parabolic subalgebra whose Levi factor has just two blocks whose sizes are coprime.) In this case it is shown that the second member of an adapted pair (h, η) for \mathfraka \mathfrak{a} is the restriction of a particularly carefully chosen regular nilpotent element of \mathfrakg \mathfrak{g} . A by-product of our analysis is the construction of a map from the set of pairs of coprime integers to the set of all finite ordered sequences of ±1.     

Reductions of points on elliptic curves

Let E be an elliptic curve defined over \mathbbQ{\mathbb{Q}}. Let Γ be a subgroup of rank r of the group of rational points E(\mathbbQ){E(\mathbb{Q})} of E. For any prime p of good reduction, let [`(G)]{\bar{\Gamma}} be the reduction of Γ modulo p. Under certain standard assumptions, we prove that for almost all primes p (i.e. for a set of primes of density one), we have

Ergodic Subequivalence Relations Induced by a Bernoulli Action

Let Γ be a countable group and denote by S{\mathcal{S}} the equivalence relation induced by the Bernoulli action G\curvearrowright [0, 1]^G{\Gamma\curvearrowright [0, 1]^{\Gamma}}, where [0, 1]^Γ is endowed with the product Lebesgue measure. We prove that, for any subequivalence relation R{\mathcal{R}} of S{\mathcal{S}}, there exists a partition {X _i } _i≥0 of [0, 1]^Γ into R{\mathcal{R}}-invariant measurable sets such that R_|X0{\mathcal{R}_{\vert X_{0}}} is hyperfinite and R_|Xi{\mathcal{R}_{\vert X_{i}}} is strongly ergodic (hence ergodic and non-hyperfinite), for every i ≥ 1.     

Calculus of functors,operad formality,and rational homology of embedding spaces

Let M be a smooth manifold and V a Euclidean space. Let [`(\textEmb)] \overline{{{\text{Emb}}}} (M,V) be the homotopy fiber of the map Emb(M,V) → Imm(M,V). This paper is about the rational homology of [`(\textEmb)] \overline{{{\text{Emb}}}} (M,V). We study it by applying embedding calculus and orthogonal calculus to the bifunctor (M,V)↦ HQ ∧ [`(\textEmb)] \overline{{{\text{Emb}}}} (M,V)₊. Our main theorem states that if

On the outer automorphism group of a hyperbolic group

LetG be a one-ended, word-hyperbolic group. Let Γ be an irreducible lattice in a connected semi-simple Lie group of rank at least 2. Ifh: Γ→Out(G) is a homomorphism, then Im(h) is finite. Dedicated to Professor Takushiro Ochiai for his sixtieth birthday     

A Modification of the Lyons-Sullivan Discretization of Positive Harmonic Functions

A modification of the Lyons-Sullivan discretization of positive harmonic functions on a Riemannian manifold M is proposed. This modification, depending on a choice of constants C = {C _n :n = 1,2,..}, allows for constructing measures n_x^C, x ? M\nu_x^\mathbf{C},\ x\in M, supported on a discrete subset Γ of M such that for every positive harmonic function f on M

Exotic smooth structures on small 4-manifolds with odd signatures

Let M be (2n-1)\mathbbCP²#2n[`(\mathbbCP)]<sup>2</sup>(2n-1)\mathbb{CP}^{2}\#2n\overline{\mathbb{CP}}{}^{2} for any integer n≥1. We construct an irreducible symplectic 4-manifold homeomorphic to M and also an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to M. We also construct such exotic smooth structures when M is \mathbbCP<sup>2</sup>#4[`(\mathbbCP)]²\mathbb{CP}{}^{2}\#4\overline {\mathbb{CP}}{}^{2} or 3\mathbbCP²#k[`(\mathbbCP)]²3\mathbb{CP}{}^{2}\#k\overline{\mathbb{CP}}{}^{2} for k=6,8,10.     

Principal bundles on compact complex manifolds with trivial tangent bundle

Let G be a connected complex Lie group and G ì G{\Gamma \subset G} a cocompact lattice. Let H be a complex Lie group. We prove that a holomorphic principal H-bundle E _H over G/Γ admits a holomorphic connection if and only if E _H is invariant. If G is simply connected, we show that a holomorphic principal H-bundle E _H over G/Γ admits a flat holomorphic connection if and only if E _H is homogeneous.