Mark Reeder

Arithmetic invariants of discrete Langlands parameters

Let G be a reductive algebraic group over the local field k. The local Langlands conjecture predicts that the irreducible complex representations π of the locally compact group G(k) can be parametrized by objects of an arithmetic nature: homomorphisms φ from the Weil-Deligne group of k to the complex L-group of G, together with an irreducible representation ρ of the component group of the centralizer of φ. In light of this conjecture which has been established for algebraic tori, as well as for some nonabelian groups like GLn(k) [21],[23], and SLn(k) [25] it is reasonable to predict how representation theoretic invariants of π = π(φ, ρ) relate to the arithmetic invariants of its parameters (φ, ρ). An early example of this was the paper [17], which predicts branching laws for the restriction of irreducible representations of the group SOn(k) to the subgroup SOn−1(k), using the e-factor of a distinguished symplectic representation of the L-group of SOn × SOn−1. These conjectures have now been verified in several cases; see [19] and [20].

Epipelagic representations and invariant theory

LetG be a reductive p-adic group. We give a new construction of small-depth “epipelagic” supercuspidal representations ofG(k), using stable orbits in Geometric Invariant Theory (GIT). In contrast to previously known methods, this construction works without any restrictions on the residue characteristic p. We construct appropriate Langlands parameters for these representations, with some restrictions on p. The GIT arises from graded Moy-Prasad filtrations, which we show are isomorphic to the graded Lie algebras whose GIT was analyzed by Vinberg and Levy. This leads to a classification of stable orbits in Moy-Prasad filtrations, as well as epipelagic supercuspidal representations, in terms of Z-regular elliptic automorphisms σ of the absolute root system of G. Transferred to the L-group of G, σ generates the image of tame inertia under the corresponding (wild) Langlands parameter. For unramified groups and sufficiently large p we also classify the Moy-Prasad filtrations which have semi-stable orbits, which solves the long-outstanding problem of classifying non-degenerate K-types for G(k).

Euler–Poincaré Pairings and Elliptic Representations of Weyl Groups and p-Adic Groups

The space of elliptic virtual representations of a p-adic group is endowed with a natural inner product EP( , ), defined analytically by Kazhdan and homologically by Schneider–Stuhler. Arthur has computed EP in terms of analytic R-groups. For Iwahori spherical representations, we show that EP can also be expressed in terms of a corresponding inner product on space of elliptic virtual representations of Weyl groups. This leads to an explicit description of both elliptic representation theories, in terms of the Kazhdan–Lusztig and Springer correspondences

Isogenies of Hecke algebras and a Langlands correspondence for ramified principal series representations

This paper gives a Langlands classification of constituents of ramified principal series representations for split p-adic groups with connected center.

Torsion automorphisms of simple Lie algebras

An automorphism σ of a simple finite dimensional complex Lie algebra g is called torsion, if σ has finite order in the group Aut(g) of all automorphisms of g. The torsion automorphisms of g were classified by Victor Kac in [11], as an application of his results on infinite dimensional Lie algebras. Those torsion automorphisms contained in the identity component G = Aut(g)◦ are called inner; they were classified in 1927 by Elie Cartan [5] who used (and perhaps introduced) the affine Weyl group and the geometry of alcoves for this purpose. This paper extends Cartan’s method to cover all torsion automorphisms of g, thereby recovering Kac’s classification directly from the geometry of the affine Weyl group, without the use of infinite dimensional Lie algebras. The desire for such a treatment arose from the work of Benedict Gross and myself on adjoint gamma factors of discrete Langlands parameters, which involves the characteristic polynomials of torsion automorphisms. Jean-Pierre Serre pointed us to Cartan’s paper, which led to the reformulation of Kac’s classification presented here. Gross’ insights, examples, predictions and requests have helped form this paper, through many discussions and his careful reading of an earlier version. In particular, Gross suggested that the inner case be treated in detail, before studying general torsion automorphisms. Throughout, I make frequent use Kostant’s theory of the principal PGL2, along with conjugacy results of Segal and Steinberg. I include some facts about centers and component groups of centralizers that may not have appeared in the literature, and the last section gives a twisted analogue (Prop. 4.1) of a result of Kostant on principal elements. These complements are used in [8].

From Laplace to Langlands via representations of orthogonal groups

In the late 1960s, Robert Langlands proposed a new and far-reaching connection between the representation theory of Lie groups over real and p-adic fields, and the structure of the Galois groups of these fields [24]. Even though this local Langlands correspondence remains largely conjectural, the relation that it predicts between representation theory and number theory has profoundly changed our views of both fields. Moreover, we now know enough about the correspondence to address, and sometimes solve, traditional problems in representation theory that were previously inaccessible. Roughly speaking, the local Langlands correspondence predicts that complex irreducible representations of a reductive group G over a local field k should be

Supercuspidal L-packets of positive depth and twisted Coxeter elements

The local Langlands correspondence is a conjectural connection between representations of groups G(k) for connected reductive groups G over a padic field k and certain homomorphisms (Langlands parameters) from the Galois (or Weil-Deligne group) of k into a complex Lie group G which is dual, in a certain sense, to G and which encodes the splitting structure of G over k. More introductory remarks on the local Langlands correspondence can be found in [21]. WhenG = GL1 this correspondence should reduce to local abelian class field theory. For G = GLn, the Langlands correspondence is uniquely determined by local factors [24] and was shown to exist in [23] and [25]. So far this correspondence is not completely explicit, but much progress has been made in this direction; see [9], [10], for example. For groups other than GLn or PGLn, the theory is much less advanced; new phenomena appear, arising on the arithmetic side from the difference between conjugacy and stable conjugacy and on the dual side from nontrivial monodromy of Langlands parameters. This means that a single Langlands parameter φ should determine not just one, but a finite set of representations Π(φ); these are the “L-packets” of the title. However, since local factors have not been defined in general, there is no precise characterization of an L-packet for general groups. One can, at present, only hope to define finite sets of representations Π(φ) attached to Langlands parameters φ, and show that they have properties expected (or perhaps unexpected) of L-packets. (See [14, chap. 3] for some of these properties.) One is thereby proposing a definition of local factors for the representations in the sets Π(φ) (cf. [4, chap.3]). This paper is a sequel to [14]. The aim of both papers is to verify, in an explicit and natural way, the local Langlands correspondence for the simplest kinds of non-abelian extensions of k, and the simplest kinds of

EXTERIOR POWERS OF THE ADJOINT REPRESENTATION

Exterior powers of the adjoint representation of a complex s emisim- ple Lie algebra are decomposed into irreducible representa tions, to varying degrees of satisfaction.

Elliptic centralizers in Weyl groups and their coinvariant representations

The centralizer C(w) of an elliptic element w in a Weyl group has a natural symplectic representation on the group ofw-coinvariants in the root lattice. We give the basic properties of this representation, along with applications to p-adic groups classifying maximal tori and computing inducing data in Lpackets as well as to elucidating the structure of the centralizer C(w) itself. We give the structure of each elliptic centralizer in W (E8) in terms of its coinvariant representation, and we refine Springer’s theory for elliptic regular elements to give explicit complex reflections generating C(w). The case where w has order three is examined in detail, with connections to mathematics of the 19th century. A variation of the methods recovers the subgroup W (H4) ⊂ W (E8).

Zero weight spaces and the Springer correspondence

Abstract It is known that an irreducible representation of SL n ( C ) whose highest weight is a partition of n affords in its zero weight space the irreducible representation of S n corresponding to the dual partition. We generalize this to groups of type D and E .