Discrete Series Characters as Lifts from Two-structure Groups
Rebecca
A.
Herb
2557-2599
Abstract: Let $G$ be a connected reductive Lie group with a relatively compact Cartan subgroup. Then it has relative discrete series representations. The main result of this paper is a formula expressing relative discrete series characters on $G$ as ``lifts'' of relative discrete series characters on smaller groups called two-structure groups for $G$. The two-structure groups are connected reductive Lie groups which are locally isomorphic to the direct product of an abelian group and simple groups which are real forms of $SL(2, \mathbf{C})$ or $SO(5, { {\bf C} })$. They are not necessarily subgroups of $G$, but they ``share'' the relatively compact Cartan subgroup and certain other Cartan subgroups with $G$. The character identity is similar to formulas coming from endoscopic lifting, but the two-structure groups are not necessarily endoscopic groups, and the characters lifted are not stable. Finally, the formulas are valid for non-linear as well as linear groups.
On Bessel distributions for quasi-split groups
Ehud
Moshe
Baruch
2601-2614
Abstract: We show that the Bessel distribution attached by Gelfand and Kazhdan and by Shalika to a generic representation of a quasi-split reductive group over a local field is given by a function when it is restricted to the open Bruhat cell. As in the case of the character distribution, this function is real analytic for archimedean fields and locally constant for non-archimidean fields.
Linear functionals of eigenvalues of random matrices
Persi
Diaconis;
Steven
N.
Evans
2615-2633
Abstract: Let $M_n$ be a random $n \times n$ unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of $M_n$ to converge to a Gaussian limit as $n \rightarrow \infty$. By Fourier analysis, this result leads to central limit theorems for the measure on the circle that places a unit mass at each of the eigenvalues of $M_n$. For example, the integral of this measure against a function with suitably decaying Fourier coefficients converges to a Gaussian limit without any normalisation. Known central limit theorems for the number of eigenvalues in a circular arc and the logarithm of the characteristic polynomial of $M_n$ are also derived from the criterion. Similar results are sketched for Haar distributed orthogonal and symplectic matrices.
Higher type adjunction inequalities for Donaldson invariants
Vicente
Muñoz
2635-2654
Abstract: We prove new adjunction inequalities for embedded surfaces in four-manifolds with non-negative self-intersection number using the Donaldson invariants. These formulas are completely analogous to the ones obtained by Ozsváth and Szabó using the Seiberg-Witten invariants. To prove these relations, we give a fairly explicit description of the structure of the Fukaya-Floer homology of a surface times a circle. As an aside, we also relate the Floer homology of a surface times a circle with the cohomology of some symmetric products of the surface.
Bi-Lipschitz homogeneous curves in $\mathbb{R}^2$ are quasicircles
Christopher
J.
Bishop
2655-2663
Abstract: We show that a bi-Lipschitz homogeneous curve in the plane must satisfy the bounded turning condition, and that this is false in higher dimensions. Combined with results of Herron and Mayer this gives several characterizations of such curves in the plane.
Multiplier ideals of monomial ideals
J.
A.
Howald
2665-2671
Abstract: In this note we discuss a simple algebraic calculation of the multiplier ideal associated to a monomial ideal in affine $n$-space. We indicate how this result allows one to compute not only the multiplier ideal but also the log canonical threshold of an ideal in terms of its Newton polygon.
Invariants and projections of six lines in projective space
Dana
R.
Vazzana
2673-2688
Abstract: Given six lines in $\mathbf{P}^3$, quartics through the six lines define a map from $\mathbf{P}^3$ to $\mathbf{P}^4$, and the image of this map is described in terms of invariants of the six lines. The map can be interpreted as projection of the six lines, and this permits a description of the canonical model of the octic surface which is given by points which project the lines so that they are tangent to a conic. We also define polarity for sets of six lines, and discuss the above map in the case of a self-polar set of lines and in the case of six lines which form a ``double-sixer'' on a cubic surface.
On the shellability of the order complex of the subgroup lattice of a finite group
John
Shareshian
2689-2703
Abstract: We show that the order complex of the subgroup lattice of a finite group $G$ is nonpure shellable if and only if $G$ is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups.
Primes in short arithmetic progressions with rapidly increasing differences
P.
D. T. A.
Elliott
2705-2724
Abstract: Primes are, on average, well distributed in short segments of arithmetic progressions, even if the associated moduli grow rapidly.
Centralizers of Iwahori-Hecke algebras
Andrew
Francis
2725-2739
Abstract: To date, integral bases for the centre of the Iwahori-Hecke algebra of a finite Coxeter group have relied on character theoretical results and the isomorphism between the Iwahori-Hecke algebra when semisimple and the group algebra of the finite Coxeter group. In this paper, we generalize the minimal basis approach of an earlier paper, to provide a way of describing and calculating elements of the minimal basis for the centre of an Iwahori-Hecke algebra which is entirely combinatorial in nature, and independent of both the above mentioned theories. This opens the door to further generalization of the minimal basis approach to other cases. In particular, we show that generalizing it to centralizers of parabolic subalgebras requires only certain properties in the Coxeter group. We show here that these properties hold for groups of type $A$ and $B$, giving us the minimal basis theory for centralizers of any parabolic subalgebra in these types of Iwahori-Hecke algebra.
Sur le rang du $2$-groupe de classes de $Q({\sqrt{m}},{\sqrt{d}})$ où $m=2$ ou un premier $p\equiv 1(mod4)$
Abdelmalek
Azizi;
Ali
Mouhib
2741-2752
Abstract: On the rank of the $2$-class group of $Q({\sqrt{m}},{\sqrt{d}})$. Let $d$ be a square-free positive integer and $p$ be a prime such that $p\equiv 1\,(mod\, 4)$. We set $K = Q({\sqrt{m}},{\sqrt{d}})$, where $m=2$ or $m=p$. In this paper, we determine the rank of the $2$-class group of $K$. RÉSUMÉ. Soit $K = Q({\sqrt{m}},{\sqrt{d}})$, un corps biquadratique où $m=2$ ou bien un premier $p\equiv 1\,(mod\,4)$ et $d$ étant un entier positif sans facteurs carrés. Dans ce papier, on détermine le rang du $2$-groupe de classes de $K$.
Degree-one maps between hyperbolic 3-manifolds with the same volume limit
Teruhiko
Soma
2753-2772
Abstract: Suppose that $f_n:M_n\longrightarrow N_n$ $(n\in {\mathbf N})$ are degree-one maps between closed hyperbolic 3-manifolds with \begin{displaymath}\lim_{n\rightarrow \infty} \operatorname{Vol} (M_n)=\lim_{n\rightarrow \infty}{\operatorname{Vol}}(N_n) <\infty. \end{displaymath} Then, our main theorem, Theorem 2, shows that, for all but finitely many $n\in {\mathbf N}$, $f_n$ is homotopic to an isometry. A special case of our argument gives a new proof of Gromov-Thurston's rigidity theorem for hyperbolic 3-manifolds without invoking any ergodic theory. An example in §3 implies that, if the degree of these maps is greater than 1, the assertion corresponding to our theorem does not hold.
$SL_n$-character varieties as spaces of graphs
Adam
S.
Sikora
2773 - 2804
Abstract: An $SL_n$-character of a group $G$ is the trace of an $SL_n$-representation of $G.$ We show that all algebraic relations between $SL_n$-characters of $G$ can be visualized as relations between graphs (resembling Feynman diagrams) in any topological space $X,$ with $\pi_1(X)=G.$ We also show that all such relations are implied by a single local relation between graphs. In this way, we provide a topological approach to the study of $SL_n$-representations of groups. The motivation for this paper was our work with J. Przytycki on invariants of links in 3-manifolds which are based on the Kauffman bracket skein relation. These invariants lead to a notion of a skein module of $M$ which, by a theorem of Bullock, Przytycki, and the author, is a deformation of the $SL_2$-character variety of $\pi_1(M).$This paper provides a generalization of this result to all $SL_n$-character varieties.
A model structure on the category of pro-simplicial sets
Daniel
C.
Isaksen
2805-2841
Abstract: We study the category ${pro-}\mathcal{SS}$ of pro-simplicial sets, which arises in étale homotopy theory, shape theory, and pro-finite completion. We establish a model structure on ${pro-}\mathcal{SS}$ so that it is possible to do homotopy theory in this category. This model structure is closely related to the strict structure of Edwards and Hastings. In order to understand the notion of homotopy groups for pro-spaces we use local systems on pro-spaces. We also give several alternative descriptions of weak equivalences, including a cohomological characterization. We outline dual constructions for ind-spaces.
Spectral lifting in Banach algebras and interpolation in several variables
Gelu
Popescu
2843-2857
Abstract: Let ${\mathcal{A}}$ be a unital Banach algebra and let $J$ be a closed two-sided ideal of ${\mathcal{A}}$. We prove that if any invertible element of ${\mathcal{A}}/J$ has an invertible lifting in ${\mathcal{A}}$, then the quotient homomorphism $\Phi :{\mathcal{A}}\to {\mathcal{A}}/J$ is a spectral interpolant. This result is used to obtain a noncommutative multivariable analogue of the spectral commutant lifting theorem of Bercovici, Foias, and Tannenbaum. This yields spectral versions of Sarason, Nevanlinna-Pick, and Carathéodory type interpolation for $F_{n}^{\infty }\bar \otimes B({\mathcal{K}})$, the WOT-closed algebra generated by the spatial tensor product of the noncommutative analytic Toeplitz algebra $F_{n}^{\infty }$ and $B({\mathcal{K}})$, the algebra of bounded operators on a finite dimensional Hilbert space ${\mathcal{K}}$. A spectral tangential commutant lifting theorem in several variables is considered and used to obtain a spectral tangential version of the Nevanlinna-Pick interpolation for $F_{n}^{\infty }\bar \otimes B({\mathcal{K}})$. In particular, we obtain interpolation theorems for matrix-valued bounded analytic functions on the open unit ball of $\mathbb{C} ^{n}$, in which one bounds the spectral radius of the interpolant and not the norm.
Hyperbolic automorphisms and Anosov diffeomorphisms on nilmanifolds
Karel
Dekimpe
2859-2877
Abstract: We translate the problem of finding Anosov diffeomorphisms on a nilmanifold which is covered by a free nilpotent Lie group into a problem of constructing matrices in $\mathrm{GL}(n,\mathbb{Z})$ whose eigenvalues satisfy certain conditions. Afterwards, we show how this translation can then be solved in some specific situations. The paper starts with a section on polynomial permutations of $\mathbb{Q}^K$, a subject which is of interest on its own.
Livsic theorems for connected Lie groups
M.
Pollicott;
C.
P.
Walkden
2879-2895
Abstract: Let $\phi$ be a hyperbolic diffeomorphism on a basic set $\Lambda$ and let $G$ be a connected Lie group. Let $f : \Lambda \rightarrow G$ be Hölder. Assuming that $f$ satisfies a natural partial hyperbolicity assumption, we show that if $u : \Lambda \rightarrow G$ is a measurable solution to $f=u\phi \cdot u^{-1}$ a.e., then $u$ must in fact be Hölder. Under an additional centre bunching condition on $f$, we show that if $f$ assigns `weight' equal to the identity to each periodic orbit of $\phi$, then $f = u\phi \cdot u^{-1}$ for some Hölder $u$. These results extend well-known theorems due to Livsic when $G$ is compact or abelian.
Derived equivalence in $SL_2(p^2)$
Joseph
Chuang
2897-2913
Abstract: We present a proof that Broué's Abelian Defect Group Conjecture is true for the principal $p$-block of the group $SL_2(p^2)$. Okuyama has independently obtained the same result using a different approach.
Minimal projective resolutions
E.
L.
Green;
Ø.
Solberg;
D.
Zacharia
2915-2939
Abstract: In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this resolution to the study of the $\operatorname{Ext}$-algebra of the algebra; namely, we present a new method for computing Yoneda products using the constructions of the resolutions. We also use our resolution to prove a case of the ``no loop'' conjecture.
Galois groups of some vectorial polynomials
Shreeram
S.
Abhyankar;
Nicholas
F. J.
Inglis
2941-2969
Abstract: Previously nice vectorial equations were constructed having various finite classical groups as Galois groups. Here such equations are constructed for the remaining classical groups. The previous equations were genus zero equations. The present equations are strong genus zero.
Invariant ideals of abelian group algebras and representations of groups of Lie type
D.
S.
Passman;
A.
E.
Zalesskii
2971-2982
Abstract: This paper contributes to the general study of ideal lattices in group algebras of infinite groups. In recent years, the second author has extensively studied this problem for $G$ an infinite locally finite simple group. It now appears that the next stage in the general problem is the case of abelian-by-simple groups. Some basic results reduce this problem to that of characterizing the ideals of abelian group algebras stable under certain (simple) automorphism groups. Here we begin the analysis in the case where the abelian group $A$ is the additive group of a finite-dimensional vector space $V$ over a locally finite field $F$ of prime characteristic $p$, and the automorphism group $G$ is a simple infinite absolutely irreducible subgroup of $GL(V)$. Thus $G$ is isomorphic to an infinite simple periodic group of Lie type, and $G$ is realized in $GL(V)$ via a twisted tensor product $\phi$ of infinitesimally irreducible representations. If $S$ is a Sylow $p$-subgroup of $G$ and if $\langle v\rangle$ is the unique line in $V$ stabilized by $S$, then the approach here requires a precise understanding of the linear character associated with the action of a maximal torus $T_G$ on $\langle v\rangle$. At present, we are able to handle the case where $\phi$ is a rational representation with character field equal to $F$.