@article{benim_dometrius_helminck_wu_2016, title={Isomorphy classes of involutions of SO(n, k, beta), n > 2}, volume={26}, number={2}, journal={Journal of Lie Theory}, author={Benim, R. W. and Dometrius, C. E. and Helminck, A. G. and Wu, L.}, year={2016}, pages={383–438} }
 @article{benim_helminck_ward_2016, title={Isomorphy classes of involutions of SP(2n, k), n > 2 (vol 25, pg 903, 2015)}, volume={26}, number={1}, journal={Journal of Lie Theory}, author={Benim, R. W. and Helminck, A. G. and Ward, F. J.}, year={2016}, pages={293–295} }
 @article{helminck_helminck_panasenko_2014, title={Cauchy problems related to integrable deformations of pseudo differential operators}, volume={85}, ISSN={["1879-1662"]}, DOI={10.1016/j.geomphys.2014.05.004}, abstractNote={In this paper we discuss the solvability of two Cauchy problems in the pseudo differential operators. The first is associated with a set of pseudo differential operators of negative order, the prominent example being the set of strict integral operator parts of the different powers of a solution of the KP hierarchy. We show that it can be solved, provided the setting possesses a compatibility completeness. In such a setting all solutions of the KP hierarchy are obtained by dressing with the solution of the related Cauchy problem. The second Cauchy problem is slightly more general and links up with a set of pseudo differential operators of order zero or less. The key example here is the collection of integral operator parts of the different powers of a solution of the strict KP hierarchy. This system is solvable as soon as exponential and compatibility completeness holds. Also under these circumstances, all solutions of the strict KP hierarchy are obtained by dressing with the solution of the corresponding Cauchy problem.}, journal={JOURNAL OF GEOMETRY AND PHYSICS}, author={Helminck, G. F. and Helminck, A. G. and Panasenko, E. A.}, year={2014}, month={Nov}, pages={196–205} }
 @article{helminck_helminck_2014, title={Infinite dimensional symmetric spaces and Lax equations compatible with the infinite Toda chain}, volume={85}, ISSN={["1879-1662"]}, DOI={10.1016/j.geomphys.2014.05.023}, abstractNote={In this paper we present a natural embedding of the infinite Toda chain in a set of Lax equations in the algebra L T consisting of Z × Z -matrices that possess only a finite number of nonzero diagonals above the main central diagonal. This hierarchy of Lax equations describes the evolution of deformations of a set of commuting anti-symmetric matrices and corresponds to splitting this algebra into its anti-symmetric part and the subalgebra of matrices in L T that have no component above the main diagonal. We show that the projections of these deformations satisfy a set of zero curvature relations, which demonstrates the compatibility of the system. Further we introduce a suitable L T -module in which we can distinguish elements, the so-called wave matrices, that will lead you to solutions of the hierarchy. We conclude by showing how wave matrices of the infinite Toda chain hierarchy can be constructed starting from an infinite dimensional symmetric space.}, journal={JOURNAL OF GEOMETRY AND PHYSICS}, author={Helminck, G. F. and Helminck, A. G.}, year={2014}, month={Nov}, pages={60–74} }
 @article{helminck_helminck_panasenko_2013, title={Integrable deformations in the algebra of pseudodifferential operators from a Lie algebraic perspective}, volume={174}, ISSN={["0040-5779"]}, DOI={10.1007/s11232-013-0011-7}, number={1}, journal={THEORETICAL AND MATHEMATICAL PHYSICS}, author={Helminck, G. F. and Helminck, A. G. and Panasenko, E. A.}, year={2013}, month={Jan}, pages={134–153} }
 @article{haas_helminck_2012, title={Algorithms for Twisted Involutions in Weyl Groups}, volume={19}, ISSN={["1005-3867"]}, DOI={10.1142/s100538671200017x}, abstractNote={ Let (W, Σ) be a finite Coxeter system, and θ an involution such that θ (Δ) = Δ, where Δ is a basis for the root system Φ associated with W, and [Formula: see text] the set of θ-twisted involutions in W. The elements of [Formula: see text] can be characterized by sequences in Σ which induce an ordering called the Richardson-Spinger Bruhat poset. The main algorithm of this paper computes this poset. Algorithms for finding conjugacy classes, the closure of an element and special cases are also given. A basic analysis of the complexity of the main algorithm and its variations is discussed, as well experience with implementation. }, number={2}, journal={ALGEBRA COLLOQUIUM}, author={Haas, R. and Helminck, A. G.}, year={2012}, month={Jun}, pages={263–282} }
 @article{haas_helminck_2011, title={Admissible Sequences for Twisted Involutions in Weyl Groups}, volume={54}, ISSN={["0008-4395"]}, DOI={10.4153/cmb-2011-075-1}, abstractNote={Abstract}, number={4}, journal={CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES}, author={Haas, Ruth and Helminck, Aloysius G.}, year={2011}, month={Dec}, pages={663–675} }
 @article{helminck_helminck_opimakh_2011, title={Equivalent forms of multi component Toda hierarchies}, volume={61}, ISSN={["1879-1662"]}, DOI={10.1016/j.geomphys.2010.11.013}, abstractNote={In this paper we consider various sets of commuting directions in the Z×Z-matrices. For each k≥1, we decompose the Z×Z-matrices in k×k-blocks. The set of basic commuting directions splits then roughly speaking half in a set of directions that are upper triangular w.r.t. this decomposition and half in a collection of directions that possess a lower triangular form. Next we consider deformations of each set in respectively the upper k×k-block triangular Z×Z-matrices and the strictly lower k×k-block triangular Z×Z-matrices that preserve the commutativity of the generators of each subset and for which the evolution w.r.t. the parameters of the opposite set is compatible. It gives rise to an integrable hierarchy consisting of a set of evolution equations for the perturbations of the basic directions. They amount to a tower of differential and difference equations for the coefficients of these perturbed matrices. The equations of the hierarchy are conveniently formulated in so-called Lax equations for these perturbations. They possess a minimal realization for which it is shown that the relevant evolutions of the perturbation commute. These Lax equations are shown in a purely algebraic way to be equivalent to zero curvature equations for a collection of finite band matrices. As the name zero curvature equations suggests there is a Cauchy problem related to these equations. Therefore a description of the relevant infinite Cauchy problems is given together with a discussion of its solvability and uniqueness. There exists still another form of the nonlinear equations of the hierarchy: the bilinear form. It requires the notion of wave matrices and a description of the related linearizations and then we can show how this bilinear form is equivalent with the Lax form. We conclude with the construction of solutions of the hierarchy.}, number={4}, journal={JOURNAL OF GEOMETRY AND PHYSICS}, author={Helminck, G. F. and Helminck, A. G. and Opimakh, A. V.}, year={2011}, month={Apr}, pages={847–873} }
 @article{helminck_helminck_opimakh_2011, title={Equivalent forms of multi component Toda hierarchies (Reprinted from JOURNAL OF GEOMETRY AND PHYSICS, vol 61, pg 847, 2011)}, volume={61}, ISSN={["1879-1662"]}, DOI={10.1016/j.geomphys.2011.06.012}, abstractNote={In this paper we consider various sets of commuting directions in the Z×Z-matrices. For each k≥1, we decompose the Z×Z-matrices in k×k-blocks. The set of basic commuting directions splits then roughly speaking half in a set of directions that are upper triangular w.r.t. this decomposition and half in a collection of directions that possess a lower triangular form. Next we consider deformations of each set in respectively the upper k×k-block triangular Z×Z-matrices and the strictly lower k×k-block triangular Z×Z-matrices that preserve the commutativity of the generators of each subset and for which the evolution w.r.t. the parameters of the opposite set is compatible. It gives rise to an integrable hierarchy consisting of a set of evolution equations for the perturbations of the basic directions. They amount to a tower of differential and difference equations for the coefficients of these perturbed matrices. The equations of the hierarchy are conveniently formulated in so-called Lax equations for these perturbations. They possess a minimal realization for which it is shown that the relevant evolutions of the perturbation commute. These Lax equations are shown in a purely algebraic way to be equivalent to zero curvature equations for a collection of finite band matrices. As the name zero curvature equations suggests there is a Cauchy problem related to these equations. Therefore a description of the relevant infinite Cauchy problems is given together with a discussion of its solvability and uniqueness. There exists still another form of the nonlinear equations of the hierarchy: the bilinear form. It requires the notion of wave matrices and a description of the related linearizations and then we can show how this bilinear form is equivalent with the Lax form. We conclude with the construction of solutions of the hierarchy.}, number={9}, journal={JOURNAL OF GEOMETRY AND PHYSICS}, author={Helminck, G. F. and Helminck, A. G. and Opimakh, A. V.}, year={2011}, month={Sep}, pages={1755–1781} }
 @article{helminck_schwarz_2011, title={On generalized Cartan subspaces}, volume={16}, ISSN={["1083-4362"]}, DOI={10.1007/s00031-011-9151-8}, number={3}, journal={TRANSFORMATION GROUPS}, author={Helminck, Aloysius G. and Schwarz, Gerald W.}, year={2011}, month={Sep}, pages={783–805} }
 @article{helminck_2010, title={On Orbit Decompositions for Symmetric k-Varieties}, volume={278}, ISBN={["978-0-8176-4874-9"]}, ISSN={["0743-1643"]}, DOI={10.1007/978-0-8176-4875-6_6}, abstractNote={Orbit decompositions play a fundamental role in the study of symmetric k-varieties and their applications to representation theory and many other areas of mathematics, such as geometry, the study of automorphic forms and character sheaves. Symmetric k-varieties generalize symmetric varieties and are defined as the homogeneous spaces G k /H k , where G is a connected reductive algebraic group defined over a field k of characteristic not 2, H the fixed point group of an involution σ and G k (resp., H k ) the set of k-rational points of G (resp., H). In this contribution we give a survey of results on the various orbit decompositions which are of importance in the study of these symmetric k-varieties and their applications with an emphasis on orbits of parabolic k-subgroups acting on symmetric k-varieties. We will also discuss a number of open problems.}, journal={SYMMETRY AND SPACES}, author={Helminck, A. G.}, year={2010}, pages={83–127} }
 @article{helminck_helminck_opimakh_2010, title={THE RELATIVE FRAME BUNDLE OF AN INFINITE-DIMENSIONAL FLAG VARIETY AND SOLUTIONS OF INTEGRABLE HIERARCHIES}, volume={165}, ISSN={["0040-5779"]}, DOI={10.1007/s11232-010-0133-0}, number={3}, journal={THEORETICAL AND MATHEMATICAL PHYSICS}, author={Helminck, G. F. and Helminck, A. G. and Opimakh, A. V.}, year={2010}, month={Dec}, pages={1610–1636} }
 @article{beun_helminck_2009, title={On the Classification of Orbits of Symmetric Subgroups Acting on Flag Varieties of SL(2, k)}, volume={37}, ISSN={["1532-4125"]}, DOI={10.1080/00927870802466983}, abstractNote={Symmetric k-varieties are a generalization of symmetric spaces to general fields. Orbits of a minimal parabolic k-subgroup acting on a symmetric k-variety are essential in the study of symmetric k-varieties and their representations. In this article, we present the classification of these orbits for the group SL(2,k) for a number of base fields k, including finite fields and the 𝔭-adic numbers. We use the characterization in Helminck and Wang (1993), which requires one to first classify the orbits of the θ-stable maximal k-split tori under the action of the k-points of the fixed point group.}, number={4}, journal={COMMUNICATIONS IN ALGEBRA}, author={Beun, Stacy L. and Helminck, Aloysius G.}, year={2009}, pages={1334–1352} }
 @article{helminck_schwarz_2009, title={Real double coset spaces and their invariants}, volume={322}, ISSN={["1090-266X"]}, DOI={10.1016/j.jalgebra.2009.01.028}, abstractNote={Let G be a real form of a complex reductive group. Suppose that we are given involutions σ and θ of G. Let H=Gσ denote the fixed group of σ and let K=Gθ denote the fixed group of θ. We are interested in calculating the double coset space H\G/K. We use moment map and invariant theoretic techniques to calculate the double cosets, especially the ones that are closed. One salient point of our results is a stratification of a quotient of a compact torus over which the closed double cosets fiber as a collection of trivial bundles.}, number={1}, journal={JOURNAL OF ALGEBRA}, author={Helminck, Aloysius G. and Schwarz, Gerald W.}, year={2009}, month={Jul}, pages={219–236} }
 @article{daniel_helminck_2008, title={Algorithms for computations in local symmetric spaces}, volume={36}, ISSN={["1532-4125"]}, DOI={10.1080/00927870801940434}, abstractNote={In the last two decades much of the algebraic/combinatorial structure of Lie groups, Lie algebras, and their representations has been implemented in several excellent computer algebra packages, including LiE, GAP4, Chevie, Magma, and Maple. The structure of reductive symmetric spaces or more generally symmetric k-varieties is very similar to that of the underlying Lie group, with a few additional complications. A computer algebra package enabling one to do computations related to these symmetric spaces would be an important tool for researchers in many areas of mathematics, including representation theory, Harish Chandra modules, singularity theory, differential and algebraic geometry, mathematical physics, character sheaves, Lie theory, etc. In this article we lay the groundwork for computing the fine structure of symmetric spaces over the real numbers and other base fields, give a complete set of algorithms for computing the fine structure of symmetric varieties and use this to compute nice bases for the local symmetric varieties.}, number={5}, journal={COMMUNICATIONS IN ALGEBRA}, author={Daniel, Jennifer R. and Helminck, Aloysius G.}, year={2008}, month={May}, pages={1758–1788} }
 @article{gagliardi_helminck_2007, title={Algorithms for computing characters for symmetric spaces}, volume={99}, ISSN={["1572-9036"]}, DOI={10.1007/s10440-007-9171-5}, number={3}, journal={ACTA APPLICANDAE MATHEMATICAE}, author={Gagliardi, Daniel and Helminck, Aloysius G.}, year={2007}, month={Dec}, pages={339–365} }
 @article{daniel_helminck_2007, title={Computing the fine structure of real reductive symmetric spaces}, volume={42}, ISSN={["0747-7171"]}, DOI={10.1016/j.jsc.2006.08.003}, abstractNote={Much of the structure of Lie groups has been implemented in several computer algebra packages, including LiE , GAP4, Chevie, Magma and Maple. The structure of reductive symmetric spaces is very similar to that of the underlying Lie group and a computer algebra package for computations related to symmetric spaces would be an important tool for researchers in many areas of mathematics. Until recently only very few algorithms existed for computations in symmetric spaces due to the fact that their structure is much more complicated than that of the underlying group. In recent work, Daniel and Helminck [Daniel, J.R., Helminck, A.G., 2004. Algorithms for computations in local symmetric spaces. Comm. Algebra (in press)] gave a complete set of algorithms for computing the fine structure of Riemannian symmetric spaces. In this paper we make the first step in extending these results to general real reductive symmetric spaces and give a number of algorithms for computing some of their fine structure. This case is a lot more complicated since it involves the intricate relations of five root systems and their Weyl groups instead of just two as in the Riemannian case. We show first that this fine structure can be obtained from the setting of a complex reductive Lie group with a pair of commuting involutions. Then we proceed to give a number of algorithms for computing the fine structure of the latter.}, number={5}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, author={Daniel, Jennifer R. and Helminck, Aloysius G.}, year={2007}, month={May}, pages={497–510} }
 @article{helminck_wu_dometrius_2006, title={Involutions of SL(n,k), (n < 2)}, volume={90}, ISSN={["1572-9036"]}, DOI={10.1007/s10440-006-9032-7}, number={1-2}, journal={ACTA APPLICANDAE MATHEMATICAE}, author={Helminck, Aloysius G. and Wu, Ling and Dometrius, Christopher E.}, year={2006}, month={Jan}, pages={91–119} }
 @article{helminck_helminck_2005, title={Multiplicity one for representations corresponding to spherical distribution vectors of class rho}, volume={86}, ISSN={["1572-9036"]}, DOI={10.1007/s10440-005-0461-5}, abstractNote={In this paper one considers a unimodular second countable locally compact group G and the homogeneous space X:=H\G, where H is a closed unimodular subgroup of G. Over X complex vector bundles are considered such that H acts on the fibers by a unitary representation ρ with closed image. The natural action of G on the space of square integrable sections is unitary and possesses an integral decomposition in so-called spherical distributions of class ρ. The uniqueness of this decomposition can be characterized by a number of equivalent properties. Uniqueness is shown to hold for a class of semidirect products. In the case that H is compact, the multiplicity free decomposition is shown to be equivalent with the commutativity of a suitable convolution algebra. As an example, one takes for X a symmetric k-variety $\mathcal{H}_{k}\backslash \mathcal{G}_{k}$ , with k a locally compact field of characteristic not equal to two, and for ρ a character of ℋk, whose square is trivial. Here $\mathcal{G}$ is a reductive algebraic group defined over k and ℋ is the fixed point group of an involution σ of $\mathcal{G}$ defined over k. It is shown then that the natural representation ℒ of G k on the Hilbert space $L^{2}(\mathcal{H}_{k}\backslash \mathcal{G}_{k})$ is multiplicity free if ℋ is anisotropic. Next a criterion is derived that leads to multiplicity one also in the noncompact situation. Finally, in the non-Archimedean case, a general procedure is given that might lead to showing that a pair $(\mathcal{G}_{k},\mathcal{H}_{k})$ is a generalized Gelfand pair. Here $\mathcal{G}$ and ℋ are suitable algebraic groups defined over k.}, number={1-2}, journal={ACTA APPLICANDAE MATHEMATICAE}, author={Helminck, AG and Helminck, GF}, year={2005}, month={Mar}, pages={21–48} }
 @article{helminck_schwarz_2004, title={Smoothness of quotients associated with a pair of commuting involutions}, volume={56}, ISSN={["1496-4279"]}, DOI={10.4153/CJM-2004-043-7}, abstractNote={Abstract}, number={5}, journal={CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES}, author={Helminck, AG and Schwarz, GW}, year={2004}, month={Oct}, pages={945–962} }
 @article{helminck_wu_2002, title={Classification of involutions of SL(2, k)}, volume={30}, ISSN={["0092-7872"]}, DOI={10.1081/AGB-120006486}, abstractNote={ABSTRACT In this paper we give a simple characterization of the isomorphy classes of involutions of with k any field of characteristic not 2. We also classify the isomorphy classes of involutions for k algebraically closed, the real numbers, the -adic numbers and finite fields. We determine in which cases the corresponding fixed point group H is k -anisotropic. In those cases the corresponding symmetric k -variety consists of semisimple elements.}, number={1}, journal={COMMUNICATIONS IN ALGEBRA}, author={Helminck, AG and Wu, L}, year={2002}, pages={193–203} }
 @article{helminck_helminck_2002, title={Hilbert flag varieties and their Kahler structure}, volume={35}, ISSN={["0305-4470"]}, DOI={10.1088/0305-4470/35/40/312}, abstractNote={In this paper, we introduce the infinite-dimensional flag varieties associated with integrable systems of the KdV- and Toda-type and discuss the structure of these manifolds. As an example we treat the Fubini–Study metric on the projective space associated with a separable complex Hilbert space and conclude by showing that all flag varieties introduced before possess a Kahler structure.}, number={40}, journal={JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL}, author={Helminck, GF and Helminck, AG}, year={2002}, month={Oct}, pages={8531–8550} }
 @article{helminck_schwarz_2002, title={Orbits and invariants associated with a pair of spherical varieties: Some examples}, volume={73}, ISSN={["0167-8019"]}, DOI={10.1023/A:1019726804264}, number={1-2}, journal={ACTA APPLICANDAE MATHEMATICAE}, author={Helminck, AG and Schwarz, GW}, year={2002}, month={Aug}, pages={103–113} }
 @article{helminck_helminck_2002, title={Spherical distribution vectors}, volume={73}, ISSN={["1572-9036"]}, DOI={10.1023/A:1019762302447}, abstractNote={In this paper we consider a locally compact second countable unimodular group G and a closed unimodular subgroup H. Let ρ be a finite-dimensional unitary representation of H with closed image. For the unitary representation of G obtained by inducing ρ from H to G a decomposition in Hilbert subspaces of a certain space of distributions is given. It is shown that the representations relevant for this decomposition are determined by so-called (ρ,H) spherical distributions, which leads to a description of the decomposition on the level of these distributions.}, number={1-2}, journal={ACTA APPLICANDAE MATHEMATICAE}, author={Helminck, AG and Helminck, GF}, year={2002}, month={Aug}, pages={39–57} }
 @article{helminck_schwarz_2001, title={Orbits and invariants associated with a pair of commuting involutions}, volume={106}, ISSN={["0012-7094"]}, DOI={10.1215/s0012-7094-01-10622-4}, abstractNote={Let σ, θ be commuting involutions of the connected reductive algebraic group G where σ, θ and G are defined over a (usually algebraically closed) field k, char k = 2. We have fixed point groups H := G and K := G and an action (H × K ) × G → G, where ((h, k), g) → hgk−1, h ∈ H, k ∈ K, g ∈ G. Let G//(H × K ) denote Spec O(G)H×K (the categorical quotient). Let A be maximal among subtori S of G such that θ(s) = σ(s) = s−1 for all s ∈ S. There is the associated Weyl group W := WH×K (A). We show: • The inclusion A → G induces an isomorphism A/W ∼ → G//(H × K ). In particular, the closed (H × K )-orbits are precisely those which intersect A. • The fibers of G → G//(H × K ) are the same as those occurring in certain associated symmetric varieties. In particular, the fibers consist of finitely many orbits. We investigate: • The structure of W and its relation to other naturally occurring Weyl groups and to the action of σθ on the A-weight spaces of . • The relation of the orbit type stratifications of A/W and G//(H × K ). Along the way we simplify some of Richardson’s proofs for the symmetric case σ = θ, and at the end we quickly recover results of Berger, Flensted-Jensen, Hoogenboom and Matsuki [Ber57, FJ78, Hoo84, Mat97] for the case k = .}, number={2}, journal={DUKE MATHEMATICAL JOURNAL}, author={Helminck, AG and Schwarz, GW}, year={2001}, month={Feb}, pages={237–279} }
 @article{helminck_2000, title={Computing orbits of minimal parabolic k-subgroups acting on symmetric k-varieties}, volume={30}, ISSN={["0747-7171"]}, DOI={10.1006/jsco.2000.0395}, abstractNote={In this paper we present an algorithm to compute the orbits of a minimal parabolic k -subgroup acting on a symmetric k -variety and most of the combinatorial structure of the orbit decomposition. This algorithm can be implemented in LiE, GAP4, Magma, Maple or in a separate program. These orbits are essential in the study of symmetric k -varieties and their representations. In a similar way to the special case of a Borel subgroup acting on the symmetric variety, (see A. G. Helminck. Computing B -orbits on G/H. J. Symb. Comput.,21 , 169?209, 1996.) one can use the associated twisted involutions in the restricted Weyl group to describe these orbits (see A. G. Helminck and S. P. Wang. On rationality properties of involutions of reductive groups. Adv. Math., 99, 26?96, 1993). However, the orbit structure in this case is much more complicated than the special case of orbits of a Borel subgroup. We will first modify the characterization of the orbits of minimal parabolic k -subgroups acting on the symmetric k -varieties given in Helminck and Wang (1993), to illuminate the similarity to the one for orbits of a Borel subgroup acting on a symmetric variety in Helminck (1996). Using this characterization we show how the algorithm in Helminck (1996) can be adjusted and extended to compute these twisted involutions as well.}, number={5}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, author={Helminck, AG}, year={2000}, month={Nov}, pages={521–553} }
 @article{brion_helminck_2000, title={On orbit closures of symmetric subgroups in flag varieties}, volume={52}, ISSN={["1496-4279"]}, DOI={10.4153/CJM-2000-012-9}, abstractNote={Abstract}, number={2}, journal={CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES}, author={Brion, M and Helminck, AG}, year={2000}, month={Apr}, pages={265–292} }
 @article{helminck_2000, title={On the classification of k-involutions}, volume={153}, ISSN={["0001-8708"]}, DOI={10.1006/aima.1998.1884}, abstractNote={Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, θ an involution of G defined over k, H a k-open subgroup of the fixed point group of θ, and Gk (resp. Hk) the set of k-rational points of G (resp. H). The variety Gk/Hk is called a symmetric k-variety. These varieties occur in many problems in representation theory, geometry, and singularity theory. Over the last few decades the representation theory of these varieties has been extensively studied for k=R and C. As most of the work in these two cases was completed, the study of the representation theory over other fields, like local fields and finite fields, began. The representations of a homogeneous space usually depend heavily on the fine structure of the homogeneous space, like the restricted root systems with Weyl groups, etc. Thus it is essential to study first this structure and the related geometry. In this paper we give a characterization of the isomorphy classes of these symmetric k-varieties together with their fine structure of restricted root systems and also a classification of this fine structure for the real numbers, p-adic numbers, finite fields and number fields.}, number={1}, journal={ADVANCES IN MATHEMATICS}, author={Helminck, AG}, year={2000}, month={Jul}, pages={1–117} }
 @article{helminck_hilgert_neumam_olafsson_1999, title={A conjugacy theorem for symmetric spaces}, volume={313}, ISSN={["0025-5831"]}, DOI={10.1007/s002080050282}, number={4}, journal={MATHEMATISCHE ANNALEN}, author={Helminck, AG and Hilgert, J and Neumam, A and Olafsson, G}, year={1999}, month={Apr}, pages={785–791} }
 @article{helminck_helminck_1998, title={A class of parabolic k-subgroups associated with symmetric k-varieties}, volume={350}, ISSN={["1088-6850"]}, DOI={10.1090/S0002-9947-98-02029-7}, abstractNote={Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, σ an involution of G defined over k, H a k-open subgroup of the fixed point group of σ, Gk (resp. Hk) the set of k-rational points of G (resp. H) and Gk/Hk the corresponding symmetric k-variety. A representation induced from a parabolic k-subgroup of G generically contributes to the Plancherel decomposition of L2(Gk/Hk) if and only if the parabolic k-subgroup is σ-split. So for a study of these induced representations a detailed description of the Hk-conjucagy classes of these σ-split parabolic k-subgroups is needed. 
In this paper we give a description of these conjugacy classes for general symmetric kvarieties. This description can be refined to give a more detailed description in a number of cases. These results are of importance for studying representations for real and p�-adic symmetric k-varieties.}, number={11}, journal={TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Helminck, AG and Helminck, GF}, year={1998}, month={Nov}, pages={4669–4691} }
 @article{helminck_1997, title={Tori invariant under an involutorial automorphism .2.}, volume={131}, ISSN={["0001-8708"]}, DOI={10.1006/aima.1997.1633}, abstractNote={The geometry of the orbits of a minimal parabolick-subgroup acting on a symmetrick-variety is essential in several areas, but its main importance is in the study of the representations associated with these symmetrick-varieties (see for example [5, 6, 20, and 31]). Up to an action of the restricted Weyl group ofG, these orbits can be characterized by theHk-conjugacy classes of maximalk-split tori, which are stable underk-involutionθassociated with the symmetrick-variety. HereHis a openk-subgroup of the fixed point group ofθ. This is the second in a series of papers in which we characterize and classify theHk-conjugacy classes of maximalk-split tori. The first paper in this series dealt with the case of algebraically closed fields. In this paper we lay the foundation for a characterization and classification for the case of nonalgebraically closed fields. This includes a partial classification in the cases, where the base field is the real numbers, p-adic numbers, finite fields, and number fields.}, number={1}, journal={ADVANCES IN MATHEMATICS}, author={Helminck, AG}, year={1997}, month={Oct}, pages={1–92} }