Holomorphic factorization of mappings into SL_n(C)

Journal of Mathematical Sciences, 2009

We suggest a generalization of Pontryagin duality from the category of commutative, complex Lie groups to the category of (not necessarily commutative) Stein groups with algebraic connected component of identity. In contrast to the other similar generalizations, in our approach the enveloping category consists of Hopf algebras (in a proper symmetrical monoidal category).

Journal of Mathematical Analysis and Applications, 2013

We consider matrix-valued functions that are holomorphic in the unit disk that are Cauchy transforms of finite matrix-valued measures. For non-zero roots z j of the determinants of such functions, we provide estimates for

The Journal of Geometric Analysis, 1996

Given a real-analytic hypersurface invariant under a finite unitary group, we construct an invariant holomorphic mapping to a hyperquadric, and prove the basic properties of this mapping. When the hypersurface is the unit sphere, the groups are cyclic, and the quotient is a Lens space, we prove that the coefficients of this mapping must be square roots of integers. For the Lens spaces L(p, p-1) we evaluate these integers by some combinatorial reasoning. We indicate how these calculations bear on a conjecture about the multiplicity of proper mappings between balls in different dimensions.

Transactions of the American Mathematical Society, 1986

In response to one often problems posed by G. Warner, we assign (to the extent that it is possible) a geometric or cohomological interpretationin the sense of Langlands-to the multiplicty in L2(T\G) of an irreducible unitary representation it of a semisimple Lie group G, where T is a discrete subgroup of G, in the case when 7r has a highest weight.

manuscripta mathematica, 2009

We study and classify actions of the complex multiplicative group on a nonsingular Stein surface with an isolated nondicritical singularity. We prove that the corresponding foliation exhibits a holomorphic first integral of a type F = f n g m where f and g are global holomorphic functions and n, m ∈ N. Under some additional conditions on the functions f and g we prove analytic linearization for the action. Our results can be viewed as extension of the original work of Masakazu Suzuki.

manuscripta mathematica, 2004

Let X be an irreducible reduced complex space on which a connected compact Lie group K acts by holomorphic automorphisms. Let G be the complexification of K and g the Lie algebra of G. Following the theory of algebraic transformation groups, we call the complex space X spherical if X is normal and its tangent space at some point is generated by the vector fields from a Borel subalgebra b ⊂ g. We give several characterizations of spherical Stein spaces. In particular, we prove that a connected Stein manifold X is spherical if and only if the algebra of K-invariant differential operators on X is commutative.

Advances in Mathematics, 2017

Let X, Y be smooth algebraic varieties of the same dimension. Let f, g : X −→ Y be finite regular mappings. We say that f, g are equivalent if there exists a regular automorphism Φ ∈ Aut(X) such that f = g • Φ. Of course if f, g are equivalent, then they have the same discriminants (i.e., the same set of critical values) and the same geometric degree. We show that conversely, for every hypersurface V ⊂ Y and every k ∈ N, there are only a finite number of non-equivalent finite regular mappings f : X → Y such that the discriminant D(f) equals V and µ(f) = k. As one of applications we show that if f : X → Y is a finite mapping of topological degree two, then there exists a regular automorphism Φ : X → X which acts transitively on the fibers of f and Y = X/G, where G = {id, Φ} and f is equivalent to the canonical projection X → X/G. We prove the same statement in the local (and sometimes global) holomorphic situation. In particular we show that if f : (C n , 0) → (C n , 0) is a proper and holomorphic mapping of topological degree two, then there exist biholomorphisms Ψ, Φ : (C n , 0) → (C n , 0) such that Ψ • f • Φ(x1, x2,. .. , xn) = (x 2 1 , x2,. .. , xn). Moreover, for every proper holomorphic mapping f : (C n , 0) → (C n , 0) with smooth discriminant there exist biholomorphisms Ψ, Φ : (C n , 0) → (C n , 0) such that Ψ • f • Φ(x1, x2,. .. , xn) = (x k 1 , x2,. .. , xn), where k = µ(f).

2015

We prove that any two algebraic embeddings of $\mathbb{C}$ into $\textrm{SL}_n(\mathbb{C})$ are the same up to an algebraic automorphism of $\textrm{SL}_n(\mathbb{C})$, provided that $n$ is at least $3$. Moreover, we prove that two algebraic embeddings of $\mathbb{C}$ into $\textrm{SL}_2(\mathbb{C})$ are the same up to a holomorphic automorphism of $\textrm{SL}_2(\mathbb{C})$.

Advances in Mathematics, 1981

In complex analysis of several variables it has turned out that Stein spaces are the equivalent to non-compact Riemann surfaces in classic function theory. On a Stein space there exist "enough" holomorphic functions, i.e., the global holomorphic functions describe completely the local function theory of this space. More precisely: let be given the algebra r(X, P) of all holomorphic functions on a Stein space (X, 8), X the basic topological space and P the sheaf of germs of holomorphic functions on X, then both, X as well as P, can be rediscovered by r (X, Fp) uniquely up to biholomorphic maps. Namely, X turns out to be homeomorphic to the spectrum uT(X, @) and P can be obtained by a certain power series technique due to Forster [S]. We consider only reduced complex analytic spaces (X, 8). It is well known that T(X, P) becomes a uniform Frechet-nuclear algebra when endowed with the topology of compact convergence on X. A topological Calgebra XZ' is called a Stein algebra if there exists a Stein space (X, 8) such that & is topologically isomorphic to r(X, 6). In this paper we wish to characterize Stein algebras by "intrinsic properties," that is, by purely topoiogically algebraic properties which do not depend on the function theory of the corresponding Stein space. Thereby we achieve a reconstruction of holomorphy by functionalanalytic principles. (0.2) The simplest class of non-trivial Stein algebras is the one of onedimensional regular Stein algebras. Since they correspond with the class of non-compact Riemann surfaces they are called Riemann algebras in [ 111. A first and rather difficult characterization of Riemann algebras is due to Richards [ 141. A simple characterization has been gicen in [ 111 by using Gleason's famous theorem [7] and a theorem of Carpenter [3]. Unfortunately, these methods do not seem to be transferable to dimension >2. In order to introduce analytic structure into spectra we apply a theorem

Colloquium Mathematicum, 2009

We describe a part of the recent developments in the theory of separately holomorphic mappings between complex analytic spaces. Our description focuses on works using the technique of holomorphic discs.