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Papers by Andrei Minchenko

Communications in Algebra

Transformation Groups

Anétale module for a linear algebraic group G is a complex vector space V with a rational G-actio... more Anétale module for a linear algebraic group G is a complex vector space V with a rational G-action on V that has a Zariski-open orbit and dim G = dim V. Such a module is called super-étale if the stabilizer of a point in the open orbit is trivial. Popov (2013) proved that reductive algebraic groups admitting super-étale modules are special algebraic groups. He further conjectured that a reductive group admitting a super-étale module is always isomorphic to a product of general linear groups. Our main result is the construction of counterexamples to this conjecture, namely a family of super-étale modules for groups with a factor Sp n for arbitrary n ≥ 1. A similar construction provides a family ofétale modules for groups with a factor SO n , which shows that groups withétale modules with non-trivial stabilizer are not necessarily special. Both families of examples are somewhat surprising in light of the previously known examples ofétale and super-étale modules for reductive groups. Finally, we show that the exceptional groups F 4 and E 8 cannot appear as simple factors in the maximal semisimple subgroup of an arbitrary Lie group with a lineaŕ etale representation.

Communications in Contemporary Mathematics

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of t... more Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation.

Selecta Mathematica

We prove that any relative character (a.k.a. spherical character) of any admissible representatio... more We prove that any relative character (a.k.a. spherical character) of any admissible representation of a real reductive group with respect to any pair of spherical subgroups is a holonomic distribution on the group. This implies that the restriction of the relative character to an open dense subset is given by an analytic function. The proof is based on an argument from algebraic geometry and thus implies also analogous results in the p-adic case. As an application, we give a short proof of some results of Kobayashi-Oshima and Kroetz-Schlichtkrull on boundedness and finiteness of multiplicities of irreducible representations in the space of functions on a spherical space. To deduce this application we prove the relative and quantitative analogs of the Bernstein-Kashiwara theorem, which states that the space of solutions of a holonomic system of differential equations in the space of distributions is finite-dimensional. We also deduce that, for every algebraic group G defined over R, the space of G(R)equivariant distributions on the manifold of real points of any algebraic G-manifold X is finite-dimensional if G has finitely many orbits on X .

Mathematische Annalen, 2016

The main motivation of our work is to create an efficient algorithm that decides hypertranscenden... more The main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations, via the parameterized differential and Galois theories. To achieve this, we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential Galois groups for differential equations whose coefficients are rational functions. P. Berman and M.F. Singer presented an algorithm calculating the differential Galois group for differential equations without parameters whose differential operator is a composition of two completely reducible differential operators. We use their algorithm as a part of our algorithm. As a result, we find an effective criterion for the algebraic independence of the solutions of parameterized differential equations and all of their derivatives with respect to the parameter.

Transformation Groups, 2010

The root system Σ of a complex semisimple Lie algebra is uniquely determined by its basis (also c... more The root system Σ of a complex semisimple Lie algebra is uniquely determined by its basis (also called a simple root system). It is natural to ask whether all homomorphisms of root systems come from homomorphisms of their bases. Since the Dynkin diagram of Σ is, in general, not large enough to contain the diagrams of all subsystems of Σ, the answer to this question is negative. In this paper we introduce a canonical enlargement of a basis (called an enhanced basis) for which the stated question has a positive answer. We use the name an enhanced Dynkin diagram for a diagram representing an enhanced basis. These diagrams in combination with other new tools (mosets, core groups) allow to obtain a transparent picture of the natural partial order between Weyl orbits of subsystems in Σ. In this paper we consider only ADE root systems (that is, systems represented by simply laced Dynkin diagrams). The general case will be the subject of the next publication.

Успехи математических наук, 2007

Transformation Groups, Jun 22, 2010

The root system R of a complex semisimple Lie algebra is uniquely determined by its basis (also c... more The root system R of a complex semisimple Lie algebra is uniquely determined by its basis (also called a simple root system). It is natural to ask whether all homomorphisms of root systems come from homomorphisms of their bases. Since the Dynkin diagram of R is, in general, not large enough to contain the diagrams of all subsystems of R, the answer to this question is negative. In this paper we introduce a canonical enlargement of a basis (called an enhanced basis) for which the stated question has a positive answer. We use the name an enhanced Dynkin diagram for a diagram representing an enhanced basis. These diagrams in combination with other new tools (mosets, core groups) allow to obtain a transparent picture of the natural partial order between Weyl orbits of subsystems in R. In this paper we consider only ADE root systems (that is, systems represented by simply laced Dynkin diagrams). The general case will be the subject of the next publication.

Communications in Algebra

Transformation Groups

Anétale module for a linear algebraic group G is a complex vector space V with a rational G-actio... more Anétale module for a linear algebraic group G is a complex vector space V with a rational G-action on V that has a Zariski-open orbit and dim G = dim V. Such a module is called super-étale if the stabilizer of a point in the open orbit is trivial. Popov (2013) proved that reductive algebraic groups admitting super-étale modules are special algebraic groups. He further conjectured that a reductive group admitting a super-étale module is always isomorphic to a product of general linear groups. Our main result is the construction of counterexamples to this conjecture, namely a family of super-étale modules for groups with a factor Sp n for arbitrary n ≥ 1. A similar construction provides a family ofétale modules for groups with a factor SO n , which shows that groups withétale modules with non-trivial stabilizer are not necessarily special. Both families of examples are somewhat surprising in light of the previously known examples ofétale and super-étale modules for reductive groups. Finally, we show that the exceptional groups F 4 and E 8 cannot appear as simple factors in the maximal semisimple subgroup of an arbitrary Lie group with a lineaŕ etale representation.

Communications in Contemporary Mathematics

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of t... more Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation.

Selecta Mathematica

We prove that any relative character (a.k.a. spherical character) of any admissible representatio... more We prove that any relative character (a.k.a. spherical character) of any admissible representation of a real reductive group with respect to any pair of spherical subgroups is a holonomic distribution on the group. This implies that the restriction of the relative character to an open dense subset is given by an analytic function. The proof is based on an argument from algebraic geometry and thus implies also analogous results in the p-adic case. As an application, we give a short proof of some results of Kobayashi-Oshima and Kroetz-Schlichtkrull on boundedness and finiteness of multiplicities of irreducible representations in the space of functions on a spherical space. To deduce this application we prove the relative and quantitative analogs of the Bernstein-Kashiwara theorem, which states that the space of solutions of a holonomic system of differential equations in the space of distributions is finite-dimensional. We also deduce that, for every algebraic group G defined over R, the space of G(R)equivariant distributions on the manifold of real points of any algebraic G-manifold X is finite-dimensional if G has finitely many orbits on X .

Mathematische Annalen, 2016

The main motivation of our work is to create an efficient algorithm that decides hypertranscenden... more The main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations, via the parameterized differential and Galois theories. To achieve this, we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential Galois groups for differential equations whose coefficients are rational functions. P. Berman and M.F. Singer presented an algorithm calculating the differential Galois group for differential equations without parameters whose differential operator is a composition of two completely reducible differential operators. We use their algorithm as a part of our algorithm. As a result, we find an effective criterion for the algebraic independence of the solutions of parameterized differential equations and all of their derivatives with respect to the parameter.

Transformation Groups, 2010

The root system Σ of a complex semisimple Lie algebra is uniquely determined by its basis (also c... more The root system Σ of a complex semisimple Lie algebra is uniquely determined by its basis (also called a simple root system). It is natural to ask whether all homomorphisms of root systems come from homomorphisms of their bases. Since the Dynkin diagram of Σ is, in general, not large enough to contain the diagrams of all subsystems of Σ, the answer to this question is negative. In this paper we introduce a canonical enlargement of a basis (called an enhanced basis) for which the stated question has a positive answer. We use the name an enhanced Dynkin diagram for a diagram representing an enhanced basis. These diagrams in combination with other new tools (mosets, core groups) allow to obtain a transparent picture of the natural partial order between Weyl orbits of subsystems in Σ. In this paper we consider only ADE root systems (that is, systems represented by simply laced Dynkin diagrams). The general case will be the subject of the next publication.

Успехи математических наук, 2007

Transformation Groups, Jun 22, 2010

The root system R of a complex semisimple Lie algebra is uniquely determined by its basis (also c... more The root system R of a complex semisimple Lie algebra is uniquely determined by its basis (also called a simple root system). It is natural to ask whether all homomorphisms of root systems come from homomorphisms of their bases. Since the Dynkin diagram of R is, in general, not large enough to contain the diagrams of all subsystems of R, the answer to this question is negative. In this paper we introduce a canonical enlargement of a basis (called an enhanced basis) for which the stated question has a positive answer. We use the name an enhanced Dynkin diagram for a diagram representing an enhanced basis. These diagrams in combination with other new tools (mosets, core groups) allow to obtain a transparent picture of the natural partial order between Weyl orbits of subsystems in R. In this paper we consider only ADE root systems (that is, systems represented by simply laced Dynkin diagrams). The general case will be the subject of the next publication.