My early work on symmetric spaces concerned some algebraic results about their structure. This included a classification of the real symmetric spaces and their fine structure (see [3]). While I was finishing my thesis I realized that the bulk of the work on the real symmetric spaces and their representations had been accomplished. I decided to commence a study of the representations associated to symmetric spaces over other base fields. Following Harish Chandras example in the groups case, a study of p-adic symmetric spaces and their representations seemed to me the most natural next case. However at that time almost nothing was known about these p-adic symmetric spaces and their structure. This made it especially difficult to study their representations, since most problems in representation theory ultimately rest on questions about the structure and geometry of the underlying spaces. It seemed to me that understanding the structure and geometry of symmetric varieties over non algebraically closed fields would be a necessary first step to studying the representations associated with these p-adic symmetric spaces.

Historically, advances in representation theory were only obtained after enough was known about the structure and geometry of the underlying spaces. A good example of this is the study of representations related to real symmetric spaces. The structure of the Riemannian symmetric spaces (a subset of the semisimple symmetric spaces) is relatively simple and a Plancherel formula for these symmetric spaces was given by Harish Chandra in the mid 50's. The structure of the general semisimple symmetric spaces is a lot more complicated and consequently no results about their representations were obtained until the late 70's, early 80's, when a number of results about the structure and geometry of these symmetric spaces were published. Soon after that many results about the representations associated to these semisimple symmetric spaces followed.

With this in mind I started working on the structure of symmetric spaces over non algebraically closed base fields with the aim of eventually studying the representations of p-adic symmetric spaces. This study of the structure has led me to a number of algebraic and combinatorial results, including a generalization of the fundamental work of Borel and Tits on ``reductive groups'' (see [1]) to symmetric spaces (see [7]). Most of these results actually hold for arbitrary base fields. I also gave a partial classification of the p-adic symmetric spaces (see [19]) and proved a number of properties about the related maximal k-split tori (see [4,15,18]). A number of other results are given in [3,5,6,7,8,14,16,17,23,24]. A discussion of my current work on the structure of p-adic symmetric spaces is in the symmetric spaces section.

About two and a half years ago I finally moved on to the representation theory. This has led so far to a few encouraging results about the multiplicities in the Plancherel decomposition of the Hilbert space L²(G_k /H_k) of square integrable functions on the p-adic symmetric space G_k /H_k. (see [21,23]). Besides myself there is a growing number of mathematicians working on the representation theory of these symmetric spaces over local fields. They include Jacquet, Rallis, Lai and several others. I anticipate that this area will prove a fertile ground for many interesting representation theoretical problems. See the representation theory page for more detail.

I have always been very interested in questions about invariant theory and geometry of symmetric spaces. In recent years I have studied some problems in these areas as well (see [20,22,24,28,29]) and they now form an active part of my current research. See the invariant theory page for more detail. Another aspect of these symmetric spaces which interests me concerns computational questions related to these symmetric varieties. As with Lie groups, a lot of the structure of the symmetric spaces can be described combinatorially and the corresponding computations could potentionally be done by a computer. To build such a symbolic manipulation package one needs first an algorithm which describes this structure and which can be implemented on a computer. In [11] I gave the first algorithms for computations related to symmetric varieties. In [13] I extended this algorithm to include real symmetric spaces as well. Recently, jointly with my students and coworkers, we gave several additional algorithms (see [31,33,34]). I I hope to implement these algorithms and develop such a computer algebra package over the next few years.

Besides the study of symmetric k-varieties and their applications, I have been active in a few other areas as well.

• Hilbert Flag Varieties: Jointly with G. F. Helminck I studied a geometric realization of infinite dimensional analogues of the finite dimensional representations of the general linear group. This required a detailed analysis of the structure of the flag varieties involved and the line bundles over them (see[9,10]). In general the action of the restricted linear group can not be lifted to the line bundles and thus leads to central extensions of this group. These representations are of importance in quantum field theory and in the framework of integrable systems. We also discussed a number of applications of these flag varieties to integrable systems and mathematical physics (see[12]). Recent results include a Kahler structure on these Hilbert flag varieties (see[25]).
• Hamiltonian cycles in Cayley graphs: Jointly with Carla Savage (NCSU, computer science department) I studied Hamiltonian cycles in Cayley graphs (see[26]). We have shown that if one takes an arbitrary basis of reflections for the Coxeter group, then in the corresponding Cayley graph one can always find a Hamiltonian path. This required a detailed study of the related Coxeter groups. We intend to study other related problems.
• Alternating forms: Jointly with Arjeh Cohen I studied Trilinear alternating forms on a vector space of dimension 7 (see [27]).

References:

1. A. Borel and J. Tits, Groupes reductifs, Inst. Hautes Etudes Sci. Publ. Math. 27 (1965), 55-152.
2. F. Bruhat and J. Tits, Groupes reductifs sur un corps local, Inst. Hautes Etudes Sci. Publ. Math. 41 (1972), 5-252.
3. A. G. Helminck, Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces, Adv. in Math. 71 (1988), 21-91.
4. A. G. Helminck, Tori Invariant under an Involutorial Automorphism I, Adv. in Math. 85 (1991), 1-38.
5. A. G. Helminck, On the orbits of affine symmetric spaces under the action of a parabolic subgroup, Contemporary Math. 88 (1989), 435-447.
6. A. G. Helminck, On groups with a Cartan involution, Proceedings of the Hyderabad conference on Algebraic Groups (Hyderabad, India), National Board for Higher Mathematics, 1992, pp. 151-192.
7. A. G. Helminck and S. P. Wang, On rationality properties of involutions of reductive groups, Adv. in Math. 99 (1993), 26-96.
8. A. G. Helminck, Symmetric k-varieties, Algebraic Groups and Their Generalizations: Classical Methods (Providence, RI), vol. 56, Proc. Sympos. Pure Math. no. Part 1, Amer. Math. Soc, 1994, pp. 233-279.
9. A. G. Helminck and G.F. Helminck, Holomorphic line bundles on Hilbert flag varieties, Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods (Providence, RI), vol. 56, Proc. Sympos. Pure Math., no. Part 2, Amer. Math. Soc, 1994, pp. 349-375.
10. A. G. Helminck and G.F. Helminck, The structure of Hilbert Flag Varieties, Publ. Res. Inst. Math. Sci., Kyoto Univ. 30 (1994), 401-442.
11. A. G. Helminck, Computing B-orbits on G/H, J. Symb. Comp. Vol. 21, (1996) 169-209.
12. A. G. Helminck and G.F. Helminck, Infinite dimensional flag manifolds in integrable systems, Acta Appl. Math. 41 (1995), 99-121.
13. A. G. Helminck, Computing orbits of minimal parabolic k-subgroups acting on symmetric k-varieties. J. Symb. Comp. To appear.
14. A. G. Helminck and G.F. Helminck, A class of parabolic k-subgroups associated with symmetric k-varieties. Trans. Amer. Math. Soc. 350, (1998), 4669-4691.
15. A. G. Helminck, Tori Invariant under an Involutorial Automorphism II, Adv. in Math. 131, no. 1, (1997), 1-92.
16. A. G. Helminck, J. Hilgert, A. Neumann and G. Olafsson, A Conjugacy Theorem for Symmetric Spaces, Mathematische Annalen, 313, (1999), 785-791.
17. A. G. Helminck, On the conjugacy of Cartan subspaces. To appear.
18. A. G. Helminck, Tori Invariant under an Involutorial Automorphism III, Real groups. (61 pages), under revision.
19. A. G. Helminck, On the classification of k-involutions, Adv. in Math. 153, no. 1, (2000), 1-117.
20. A. G. Helminck and G. Schwarz, Orbits and invariants associated with commuting involutions. Duke Math. Journal, Vol. 106 (2001), No.2, 237-280.
21. A. G. Helminck and G.F. Helminck, Multiplicities for representations related to p-adic symmetric varieties, To appear.
22. A. G. Helminck and Michel Brion, On orbit closures of symmetric subgroups in flag varieties. Can. Journal Math., Vol 52 (2), (2000), pp. 265-292.
23. A. G. Helminck and G.F. Helminck, H_k-fixed distributionvectors for representations related to p-adic symmetric varieties, In preparation.
24. A. G. Helminck, Combinatorics related to orbit closures of symmetric subgroups in flag varieties. Invariant theory in all characteristics, CRM Proc. Lecture Notes, 35, Amer. Math. Soc., (2004), 71-90.
25. A. G. Helminck and G.F. Helminck, Hilbert Flag Varieties and their Kahler structure, J. Physics A., 35 (2002), no. 40, 8531-8550.
26. A. G. Helminck and G.F. Helminck, Spherical distribution vectors, Acta Appl. Math., 73 (2002), no. 1-2, 39-57.
27. A. G. Helminck and G.F. Helminck, Multiplicity one for representations corresponding to spherical distribution vectors of class r, (31 pages), Acta Appl. Math., To Appear.
28. A. G. Helminck and G. Schwarz, Orbits and invariants associated with a pair of spherical varieties, Acta Appl. Math., 73 (2002), no. 1-2, 103-113.
29. A. G. Helminck and G. Schwarz, Smoothness of quotients associated with a pair of commuting involutions, Can. Journal Math., (17 pages), To Appear.
30. A. G. Helminck and L. Wu, Classification of involutions of SL(2, k), Comm. in Algebra, 30 (2002), no. 1, 193--203.
31. A. G. Helminck and J. Fowler, Algorithms for computations in local symmetric spaces, (23 pages), To Appear.
32. A. G. Helminck, C. Dometrius and L. Wu, Classification of involutions of SL(n, k), (33 pages), To Appear.
33. A. G. Helminck and R. Haas, Computing admissible sequences for twisted involutions in Weyl groups, (10 pages) To Appear.
34. A. G. Helminck and R. Haas, Twisted involutions and other computations for Weyl groups: algorithms and data structures, (14 pages), To Appear.
35. A. G. Helminck and Carla Savage, Hamiltonian cycles in Cayley graphs. In preparation.
36. A. G. Helminck and A.M. Cohen, Trilinear alternating forms on a vector space of dimension 7. Comm. in Algebra, 16 (1) (1988), 1-25.