GL<Subscript>2</Subscript>(ℂ)-orbits of binary forms

In the present paper we study the orbits of action of the group GL₂(ℂ) on the space of binary forms. The main result of the paper is the proof of the criterion which divides the GL₂(ℂ)-orbits of binary forms.     

Weighted Bergman spaces¶associated with causal symmetric spaces

Let H\G be a causal symmetric space sitting inside its complexification H _ℂ\G _ℂ. Then there exist certain G-invariant Stein subdomains Ξ of H _ℂ\G _ℂ. The Haar measure on H _ℂ\G _ℂ gives rise to a G-invariant measure on Ξ. With respect to this measure one can define the Bergman space B ²(Ξ) of square integrable holomorphic functions on Ξ. The group G acts unitarily on the Hilbert space B ²(Ξ) by left translations in the arguments. The main result of this paper is the Plancherel Theorem for B ²(Ξ), i.e., the disintegration formula for the left regular representation into irreducibles. Received: Received: 23 November 1998     

The centralizer algebra of the diagonal action of the group <Emphasis Type="Italic">GL</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(ℂ) in a mixed tensor space

We consider the walled Brauer algebra Br _{k, l}(n) introduced by V. Turaev and K. Koike. We prove that it is a subalgebra of the Brauer algebra and that it is isomorphic, for sufficiently large n ∈ ℕ, to the centralizer algebra of the diagonal action of the group GL_n(ℂ) in a mixed tensor space. We also give the presentation of the algebra Br _{k, l}(n) by generators and relations. For a generic value of the parameter, the algebra is semisimple, and in this case we describe the Bratteli diagram for this family of algebras and give realizations for the irreducible representations. We also give a new, more natural proof of the formulas for the characters of the walled Brauer algebras. Bibliography: 29 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 331, 2006, pp. 170–198.     

Linear bounds for levels of stable rationality

Let G be one of the groups SL _n (ℂ), Sp_2n (ℂ), SO _m (ℂ), O _m (ℂ), or G ₂. For a generically free G-representation V, we say that N is a level of stable rationality for V/G if V/G × ℙ ^N is rational. In this paper we improve known bounds for the levels of stable rationality for the quotients V/G. In particular, their growth as functions of the rank of the group is linear for G being one of the classical groups.     

On the Equicontinuity Region of Discrete Subgroups of <Emphasis Type="Italic">PU</Emphasis>(1,<Emphasis Type="Italic">n</Emphasis>)

Let G be a discrete subgroup of PU(1,n). Then G acts on ℙ_ℂ ⁿ preserving the unit ball ℍ_ℂ ⁿ , where it acts by isometries with respect to the Bergman metric. In this work we look at its action on all of ℙ_ℂ ⁿ and determine its equicontinuity region Eq(G). This turns out to be the complement of the union of all complex projective hyperplanes in ℙ_ℂ ⁿ which are tangent to ∂ℍ_ℂ ⁿ at points in the Chen-Greenberg limit set Λ(G), a closed G-invariant subset of ∂ℍ_ℂ ⁿ which is minimal for non-elementary groups. We also prove that the action on Eq(G) is discontinuous. Also , if the limit set is “sufficiently general” (i.e. it is not contained in any proper k -chain), then each connected component of Eq(G) is a holomorphy domain and it is a complete Kobayashi hyperbolic space.     

Representation Results for Equivariant Rigid Analytic Functions

Given a prime number p and the Galois orbit O(x) of an element x of ℂ _p , the topological completion of the algebraic closure of the field of p-adic numbers, we are interested in the representation results for equivariant rigid analytic functions defined on ℙ¹(ℂ _p ) \ O(x) with values in ℂ _p that vanishes at ∞.     

Comparison of germ expansion¶on inner forms of GL(n)

We prove that the germ expansion of a discrete series representation π^′ on GL _n (D) where D is a division algebra over k of index m and the germ expansion of the representation π of GL _mn (k) associated to π^′ by the Deligne–Kazhdan–Vigneras correspondence are closely related, and therefore certain coefficients in the germ expansion of a discrete series representation of GL _mn (k) can be interpreted (and therefore sometimes calculated) in terms of the dimension of a certain space of (degenerate) Whittaker models on GL _n (D). Received: 30 September 1999 / Revised version: 11 February 2000     

On the iteration of holomorphic self-maps of ℂ*

Letf be a holomorphic self-map of the punctured plane ℂ^*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI ₀(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞} andI _∞(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞}. We try to find the relation betweenI ₀(f),I _∞(t) andJ(f). It is proved that both the boundary ofI ₀(f) and the boundary ofI _∞)f) equal toJ(f),I ₀(f) ∩J(f) ≠ θ andI _∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI ₀(f) andI _∞(f) are not doubly-bounded. Supported by the National Natural Science Foundation of China     

Homology of GL<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>: injectivity conjecture for GL<Subscript>4</Subscript>

The homology of GL _n (R) and SL _n (R) is studied, where R is a commutative ‘ring with many units’. Our main theorem states that the natural map H ₄(GL₃(R), k) → H ₄(GL₄(R), k) is injective, where k is a field with char(k) ≠ 2, 3. For an algebraically closed field F, we prove a better result, namely, is injective. We will prove a similar result replacing GL by SL. This is used to investigate the indecomposable part of the K-group K ₄(R).     

A larger GL2 large sieve in the level aspect

In this paper we study the orthogonality of Fourier coefficients of holomorphic cusp forms in the sense of large sieve inequality. We investigate the family of GL ₂ cusp forms modular with respect to the congruence subgroups Γ₁(q), with additional averaging over the levels q ∼ Q. We obtain the orthogonality in the range N ≪ Q ^2−δ for any δ > 0, where N is the length of linear forms in the large sieve.     

Coinvariants for a coadjoint action of quantum matrices

Let K be a (algebraically closed ) field. A morphism A ⟼ g ⁻¹ Ag, where A ∈ M(n) and g ∈ GL(n), defines an action of a general linear group GL(n) on an n × n-matrix space M(n), referred to as an adjoint action. In correspondence with the adjoint action is the coaction α: K[M(n)] → K[M(n)] ⊗ K[GL(n)] of a Hopf algebra K[GL(n)] on a coordinate algebra K[M(n)] of an n × n-matrix space, dual to the conjugation morphism. Such is called an adjoint coaction. We give coinvariants of an adjoint coaction for the case where K is a field of arbitrary characteristic and one of the following conditions is satisfied: (1) q is not a root of unity; (2) char K = 0 and q = ±1; (3) q is a primitive root of unity of odd degree. Also it is shown that under the conditions specified, the category of rational GL _q × GL _q -modules is a highest weight category.     

Bergman kernel and complex singularity exponent

We give a precise estimate of the Bergman kernel for the model domain defined by Ω _F = “(z,w) ∈ ℂ ⁿ⁺¹: Im w − |F(z)|² > 0”, where F = (f ₁, ..., f _m ) is a holomorphic map from ℂ ⁿ to ℂ ^m , in terms of the complex singularity exponent of F.     

Degenerate principal series representations of Sp(<Emphasis Type="Italic">p,q</Emphasis>)

Letp>q and letG=Sp(p, q). LetP=LN be the maximal parabolic subgroup ofG with Levi subgroupL≅GL _q (ℍ)×Sp(p−q). Forsεℂ andμ a highest weight of Sp(p−q), let п_s,μ be the representation ofP such that its restriction toN is trivial and ⊠T _p-q ^μ , where det _q is the determinant character of GL _q (ℍ) andT _p-q ^μ is the irreducible representation of Sp(p−q) with highest weightμ. LetI _p,q(s, μ) be the Harish-Chandra module of the induced representation Ind _P ^G . In this paper, we shall determine the module structure and unitarity ofI _{p, q}(s, μ). Partially supported by NUS grant R-146-000-026-112.     

Quotient singularities which are complete intersections

Let G be a finite subgroup of GL_n() acting naturally on an affine space ⁿ. In this note we will determine G such that the quotient variety ⁿ/G is a complete intersection. For n=2 and 3, such a group G was classified in [13, 24, 32].     

On cuspidal representations of general linear groups over discrete valuation rings

We define a new notion of cuspidality for representations of GL _n over a finite quotient o _k of the ring of integers o of a non-Archimedean local field F using geometric and infinitesimal induction functors, which involve automorphism groups G _λ of torsion o-modules. When n is a prime, we show that this notion of cuspidality is equivalent to strong cuspidality, which arises in the construction of supercuspidal representations of GL _n (F). We show that strongly cuspidal representations share many features of cuspidal representations of finite general linear groups. In the function field case, we show that the construction of the representations of GL _n (o _k ) for k ≥ 2 for all n is equivalent to the construction of the representations of all the groups G _λ . A functional equation for zeta functions for representations of GL _n (o _k ) is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for GL₄(o₂) are constructed. Not all these representations are strongly cuspidal.     

On the modularity of the GL<Subscript>2</Subscript>-twisted spinor L-function

In modern number theory there are famous theorems on the modularity of Dirichlet series attached to geometric or arithmetic objects. There is Hecke’s converse theorem, Wiles proof of the Taniyama-Shimura conjecture, and Fermat’s Last Theorem to name a few. In this article in the spirit of the Langlands philosophy we raise the question on the modularity of the GL₂-twisted spinor L-function Z _{G ⊗ h} (s) related to automorphic forms G,h on the symplectic group GSp₂ and GL₂. This leads to several promising results and finally culminates into a precise very general conjecture. This gives new insights into the Miyawaki conjecture on spinor L-functions of modular forms. We indicate how this topic is related to Ramakrishnan’s work on the modularity of the Rankin-Selberg L-series.     

A basis for the symplectic group branching algebra

The symplectic group branching algebra, B\mathcal {B}, is a graded algebra whose components encode the multiplicities of irreducible representations of Sp_2n−2(ℂ) in each finite-dimensional irreducible representation of Sp_2n (ℂ). By describing on B\mathcal {B} an ASL structure, we construct an explicit standard monomial basis of B\mathcal {B} consisting of Sp_2n−2(ℂ) highest weight vectors. Moreover, B\mathcal {B} is known to carry a canonical action of the n-fold product SL₂×⋯×SL₂, and we show that the standard monomial basis is the unique (up to scalar) weight basis associated to this representation. Finally, using the theory of Hibi algebras we describe a deformation of Spec(B)\mathrm{Spec}(\mathcal {B}) into an explicitly described toric variety.     

Discrete subgroups of <Emphasis Type="Italic">PU</Emphasis> (2, 1) acting on P<Stack><Subscript>ℂ</Subscript><Superscript>2</Superscript></Stack> and the Kobayashi metric

In this paper, we prove following: If G ⊂ PU (2, 1) is an infinite, discrete group, acting on P_ℂ² without complex invariant lines, then the component containing ℍP_ℂ² of the domain of discontinuity Ω(G) = PP_ℂ²∖ Λ (G), according to Kulkarni, is G-invariant complete Kobayashi hyperbolic. The authors were supported by the Universidad Autónoma de Yucatán and the Universidad Nacional Autónoma de México.     

Quelques théorèmes sur la structure complexe

We prove the following result: If the function Max (log|ω -f ₁(z)|, ..., log|ω -f _k(z)|) is plurisubharmonic in the open setD×ℂ (D open of ℂ ⁿ ), thenf ₁,...,f _k are analytic functions iff ₁,...,f _k are continuous functions onD(k≥2). We prove also some other results.     

Estimates of the number of singular points of a complex hypersurface and related questions

It is well known that the number of isolated singular points of a hypersurface of degree d in ℂP^m does not exceed the Arnol’d number A_m(d), which is defined in combinatorial terms. In the paper it is proved that if b _m−1 ^± (d) are the inertia indices of the intersection form of a nonsingular hypersurface of degree d in ℂP^m, then the inequality A_m(d)<min{b _m−1 ⁺ (d), b _m−1 ⁻ (d)} holds if and only if (m−5)(d−2)≥18 and (m,d)≠(7,12). The table of the Arnol’d numbers for 3≤m≤14, 3≤d≤17 and for 3≤m≤14, d=18, 19 is given. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 180–190. Translated by O. A. Ivanov and N. Yu. Netsvetev.