Mark Reeder’s research while affiliated with Boston College and other places

Let G be a compact Lie group with Weyl group W. We give a formula for the character of W on the zero weight space of any finite-dimensional representation of G. The formula involves weighted partition functions, generalizing Kostant’s partition function. On the elliptic set of W, the partition functions are trivial. On the elliptic regular set, the character formula is a monomial product of certain coroots, up to a constant equal to 0 or ± 1. This generalizes Kostant’s formula for the trace of a Coxeter element on a zero weight space. If the long element w0 = − 1, our formula gives a method for determining all representations of G for which the zero weight space is irreducible.

Let $\mathfrak g$ be a simple complex Lie algebra of finite dimension. This paper gives an inequality relating the order of an automorphism of $\mathfrak g$ to the dimension of its fixed-point subalgebra, and characterizes those automorphisms of $\mathfrak g$ for which equality occurs. This is amounts to an inequality/equality for Thomae's function on the group of automorphisms of $\mathfrak g$. The result has applications to characters of zero weight spaces, graded Lie algebras, and inequalities for adjoint Swan conductors.

Let G be a simple complex Lie group with Weyl group W. We give a formula for the character of W on the zero weight space of any finite dimensional representation of G. The formula involves partition functions, generalizing Kostant's partition function. On the elliptic set of W the partition functions are trivial. On the elliptic regular set, the character formula is a monomial product of certain co-roots, up to a constant equal to 0 or $\pm 1$. For a Coxeter element we recover Kostant's formula for this trace. If the long element $w_0=-1$, our formula leads to a method for determining all representations of G for which the zero weight space is irreducible.

Let W be the Weil group of a p-adic field and let g be a simple complex Lie algebra. We prove lower bounds for Swan conductors of representations of W by automorphisms of g and give necessary and sufficient conditions for equality. We also relate these bounds to the Local Langlands Correspondence and to the representation theory of p-adic groups. © The Author 2017. Published by Oxford University Press. All rights reserved.

This is an expository account of recent new constructions of supercuspidal representations of reductive p-adic groups, and their role in the local Langlands correspondence.

Let G be a reductive p-adic group. We give a new construction of small-depth "epipelagic" supercuspidal representations of G(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 repre-sentations, 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 Vin-berg 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).

We complete the classification of positive rank gradings on Lie algebras of simple algebraic groups over an algebraically closed field k whose characteristic is zero or not too small, and we determine the little Weyl groups in each case. We also classify the stable gradings and prove Popov's conjecture on the existence of a Kostant section.

Let G be a reductive p-adic group. Given a compact-mod-center maximal torus SG and sufficiently regular character χ of S, one can define, following Adler, Yu and others, a supercuspidal representation π(S,χ) of G. For S unramified, we determine when π(S,χ) is generic, and which generic characters it contains.

The centralizer C(w) of an elliptic element w in a Weyl group has a natural symplectic representa-tion on the group of w-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 L-packets -as well as to elucidating the structure of the centralizer C(w) itself. We give the structure of each elliptic centralizer in W (E 8) 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 (H 4) ⊂ W (E 8).

The local Langlands correspondence can be used as a tool for making verifiable predictions about irreducible complex representations of p-adic groups and their Langlands parameters, which are homomorphisms from the local Weil-Deligne group to the L-group. In this paper we refine a conjecture of Hiraga-Ichino-Ikeda, which relates the formal degree of a discrete series representa-tion to the value of the local gamma factor of its parameter. We attach a rational function in x with rational coefficients to each discrete parameter, which specializes to this local gamma value when x = q, the cardinality of the residue field. The order of this rational function at x = 0 is also an important invariant of the parameter -it leads to a conjectural inequality for the Swan conductor of a discrete parameter acting on the adjoint representation of the L-group. We verify this conjecture in many cases. When we impose equality, we obtain a prediction for the existence of simple wild parameters and simple supercuspidal representations, both of which are found and described in this paper.