Orthogonal subsets of root systems and the orbit method

Abstract

Abstract. Let k be the algebraic closure of a finite field, G a Chevalley group over k, U the maximal unipotent subgroup of G. To each orthogonal subset D of the root system of G and each set ξ of |D| nonzero scalars in k one can assign the coadjoint orbit of U . It is proved that the dimension of such an orbit does not depend on ξ. An upper bound for this dimension is also given in terms of the Weyl group. §0. Introduction 0.1. In studying irreducible complex representations of finite unipotent groups, the main tool is the orbit method. It was created by Kirillov for nilpotent Lie groups over R, see [12] and [13], and then adapted by Kazhdan [11] to the case of finite groups (see also [14] and the paper [3], where the theory of -adic sheaves for unipotent groups was explained). Here we consider the groups U (q) and U , the maximal unipotent subgroups of Chevalley groups over a finite field F q and its algebraic closure, respectively.The orbit method establishes a bijection between the set of equivalence classes of irreducible representations of U (q) and the set of orbits of the coadjoint representation of U (q). Note that many questions about representations can be interpreted in terms of orbits, and that the problem of giving a complete description of orbits remains unsolved and seems to be very difficult. On the other hand, much information about some special types of orbits, representations, and characters is known.In particular, a description is known for the regular orbits (i.e., those of maximal dimension) of the group UT n of all unipotent triangular matrices of size n × n; see [13]. The subregular orbits (i.e., orbits of second maximal dimension) and the corresponding characters 1 were described in [7] and [8]. As a generalization, Panov considered orbits of the group UT n associated with involutions in the symmetric group. In [16], he obtained a formula for the dimension of such an orbit.It is well known that the group UT n corresponds to the root system of type A n−1 . In order to generalize the results of Panov, we introduced the concept of orbits associated with orthogonal subsets of root systems. In the paper [10], we studied these orbits for the case of classical root systems. (For some special orthogonal subsets of the root systems of types B n and D n , we also obtained a formula involving the corresponding irreducible characters; see [9, Theorem 3.8].)Our main goal in this paper is to extend the results of [10] to the general case of an arbitrary root system, not only the classical one. The paper is organized as follows. In the remaining part of this section, we give the necessary definitions and formulate 2010 Mathematics Subject Classification. Primary 17B22. Key words and phrases. Orthogonal subsets of root systems, coadjoint orbits. 1 The description of the irreducible character corresponding to a given orbit is itself a nontrivial problem; see, e.g., the papers of André and Neto [1] and [2] for the description of the so-called supercharacters. In this paper, we concentrate on orbits, not...

Rook placements in $G_2$ and $F_4$ and associated coadjoint orbits

“…It turns out that almost all coadjoint orbit studied to the moment are associated with certain D and ξ, see, e.g., [An95], [AN06], [Ko12], [Ko13], [Pa08], [IPa09], [Ig09], [Ig11], [Ig12]. On the other hand, C.A.M.…”

Gradedness of the set of rook placements in A n−1

“…Panov in [18] and by me in [6]. One can define analogues of such orbits for other root systems, see [7], [8], [9] for the case of I(n). For arbitrary rook placements in R(n), such orbits were considered in [10]; see also [1], [2], where C. Andre and A. Neto used rook placements to construct so-called supercharacter theory for the group U .…”

On Involutions in the Weyl Group and B-Orbit Closures in the Orthogonal Case

“…(Obviously, f D = f D,ξ 1 , where ξ 1 (α) = 1 for all α ∈ D.) Given an orthogonal subset D ⊆ Φ + , we say that the orbits Ω D = Ω f D and Θ D,ξ = Θ f D,ξ are associated with D. Note that U -orbits associated with orthogonal subsets and their generalizations were studied, in particular, in [Pa], [Ig1], [Ig2], [IV]. Now, let W be the Weyl group of Φ.…”