Root data with group actions, (with Joshua Lansky). Submitted for publication. E-print available at arXiv:1707.01935.

- Abstract: Suppose k is a field, G is a connected reductive algebraic k-group, T is a maximal k-torus in G, and Γ is a finite group that acts on (G,T). From the above, one obtains a root datum Ψ on which Gal(k) × Γ acts. Provided that Γ preserves a positive system in Ψ, not necessarily invariant under Gal(k), we construct an inverse to this process. That is, given a root datum on which Gal(k)×Γ acts appropriately, we show how to construct a pair (G,T), on which Γ acts as above.
Although the pair (G,T) and the action of Γ are canonical only up to an equivalence relation, we construct a particular pair for which G is k-quasisplit and Γ fixes a Gal(k)-stable pinning of G. Using these choices, we can define a notion of taking “Γ-fixed points” at the level of equivalence classes, and this process is compatible with a general “restriction” process for root data with Γ-action.

On Kostant sections and topological nilpotence, (with Jessica Fintzen and Sandeep Varma). Submitted for publication. E-print available at arXiv:1611.08566.

- Abstract: Let G denote a connected, quasi-split reductive group over a field F that is complete with respect to a discrete valuation and that has a perfect residue field. Under mild hypotheses, we produce a subset of the Lie algebra g(F) that picks out a G(F)-conjugacy class in every stable, regular, topologically nilpotent conjugacy class in g(F). This generalizes an earlier result obtained by DeBacker and one of the authors under stronger hypotheses. We then show that if F is p-adic, then the characteristic function of this set behaves well with respect to endoscopic transfer.

Character relations for a lifting of representations of finite reductive groups, (with Michael Cassel, Joshua Lansky, Emma Morgan, and Yifei Zhao), Involve 9 (2016), no. 5, pp. 805–812. E-print available at arXiv:1205.6448.

- Abstract: Given a connected reductive group over a finite field k, and a semisimple k-automorphism ε of of finite order, let G denote the connected part of the group of ε-fixed points. Then there exists a lifting from packets of representations of G(k) to packets for (k). In the case of Deligne-Lusztig representations, we show that this lifting satisfies a character relation analogous to that of Shintani.

Liftings of representations of finite reductive groups II: Explicit conorm functions, (with Joshua Lansky). Under revision. E-print available at arXiv:1109.0794.

- Abstract: Let k be a field. Suppose that is a connected reductive k-quasisplit group, Γ is a group of k-automorphisms of satisfying a quasi-semisimplicity condition, and G is the connected part of the group of fixed points, also assumed k-quasisplit. In an earlier work, the authors constructed a canonical map from the set of stable semisimple conjugacy classes in the dual G
^{*}(k) to the set of such classes in^{*}(k). We describe several situations where this map can be refined to an explicit function on points, or where it factors through such a function.

Liftings of representations of finite reductive groups I: Semisimple conjugacy classes, (with Joshua Lansky), Canad. J. Math.. 66 (2014), no. 6, pp. 1201–1224. DOI: http://dx.doi.org/10.4153/CJM-2014-013-6. E-print available at arXiv:1106.0786.

- Abstract: Suppose that is a connected reductive group defined over a field k, Γ is a group of k-automorphisms of satisfying a quasi-semisimplicity condition, and G is the connected part of the group of fixed points. Then G is reductive. If both and G are k-quasisplit, then we can consider their duals
^{*}and G^{*}. We show the existence and give an explicit formula for a natural map from stable conjugacy classes in G^{*}(k) to those in^{*}(k). If k is finite, then our groups are automatically quasisplit, and our result specializes to give a map from semisimple conjugacy classes in G^{*}(k) to those in^{*}(k). Since such classes parametrize packets of irreducible representations of G(k) and (k), one obtains a mapping of such packets.

Extensions of representations of p-adic groups, (with Dipendra Prasad), Nagoya Math. J. 208 (2012), pp. 171–199. E-print available at arXiv:1108.3668.

- Abstract: We compute extensions between certain irreducible representations of p-adic groups.

Supercuspidal
characters
of
of
SL_{2}
over
a
p-adic
field,
(with
Stephen
DeBacker,
P.
J.
Sally,
Jr.,
and
Loren
Spice),
in
Harmonic
analysis
on
reductive,
p-adic
groups,
Robert
S.
Doran,
Paul
J.
Sally,
Jr.,
and
Loren
Spice,
eds.,
Contemporary
Mathematics,
vol. 543,
pp. 19–69.
American
Mathematical
Society,
Providence,
RI,
2011.
E-print
available
at
arXiv:1012.5548.

- Abstract: We compute explicit formulas for all supercuspidal characters of SL
_{2}over a p-adic field of odd residual characteristic.

Depth-zero base change for ramified U(2, 1), (with Joshua Lansky), Trans. Amer. Math. Soc., 362 (2010), 5569–5599. E-print available at arXiv:0807.1528.

- Abstract: We give an explicit description of L-packets and quadratic base change for depth-zero representations of ramified unitary groups in two and three variables. We show that this base change lifting is compatible with a certain lifting of families of representations of finite groups. We conjecture that such a compatibility is valid in much greater generality.

Supercuspidal characters of reductive p-adic groups (with Loren Spice), Amer. J. Math. 131 (2009), no. 4, 1137–1210. E-print available at arXiv:0707.3313.

- Abstract: We compute the characters of many supercuspidal representations of reductive p-adic groups. Specifically, we deal with representations that arise via Yu’s construction from data satisfying a certain compactness condition. Each character is expressed in terms of a depth-zero character of a smaller group, the (linear) characters appearing in Yu’s construction, and Fourier transforms of orbital integrals, in addition to certain explicitly computed signs and cardinalities.

Good product expansions for tame elements of p-adic groups (with Loren Spice), Int. Math. Res. Pap. vol. 2008, 95 pages. E-print available at arXiv:math.RT/0611554.

- Abstract: We show that, under fairly general conditions, many elements of a p-adic group can be well approximated by a product whose factors have properties that are helpful in performing explicit character computations.

The local character expansion near a tame, semisimple element (with Jonathan Korman), Amer. J. Math., 129 (2007), no. 2, 381–403.

- Abstract: Consider the character of an irreducible admissible representation of a p-adic reductive group. The Harish-Chandra-Howe local expansion expresses this character near a semisimple element as a linear combination of Fourier transforms of nilpotent orbital integrals. Under mild hypotheses (we assume neither that the group is connected, nor that the underlying field has characteristic zero), we describe an explicit region on which the local character expansion is valid.

On certain multiplicity one theorems (with Dipendra Prasad), Israel J. Math, 153 (2006), 221–245.

- Abstract: We prove several multiplicity one theorems. For k a local field not of characteristic two, and V a symplectic space over k, any irreducible admissible representation of the symplectic similitude group GSp(V ) decomposes with multiplicity one when restricted to the symplectic group Sp(V ). We prove the analogous result for GO(V ) and O(V ), where V is an orthogonal space over k. When k is non-archimedean, we prove the uniqueness of Fourier-Jacobi models for representations of GSp(4), and the existence of such models for supercuspidal representations of GSp(4).

Depth-zero base change for unramified U(2, 1), (with Joshua Lansky), J. Number Theory 114 (2005), no. 2, pp. 324–360. Printer’s error corrected in vol. 121 (2006), no. 1, 186.

- Abstract: We give an explicit description of L-packets and quadratic base change for depth-zero representations of unramified unitary groups in two and three variables. We show that this base change is compatible with unrefined minimal K-types.

Discrete series representations of unipotent p-adic groups, (with Alan Roche), J. Lie Theory 15 (2005), 261–267.

- Abstract: For a certain class of locally profinite groups, we show that an irreducible smooth discrete series representation is necessarily supercuspidal and, more strongly, can be obtained by induction from a linear character of a suitable open and compact modulo center subgroup. If F is a non-Archimedean local field, then our class of groups includes the groups of F-points of unipotent algebraic groups defined over F. We therefore recover earlier results of van Dijk and Corwin.

Injectivity, projectivity, and supercuspidal representations, (with Alan Roche), J. London Math. Soc. (2) 70 (2004), no. 2, 356–368.

- Abstract: Let G be a reductive p-adic group. Consider the category of smooth (complex) representations of G in which a (fixed) closed cocompact subgroup of the centre acts by a (fixed) character. It is well known that the supercuspidal representations in this category are both injective and projective. We show that conversely an admissible injective or projective object is necessarily supercuspidal.

Murnaghan-Kirillov theory for supercuspidal representations of tame general linear groups, (with Stephen DeBacker), J. Reine Angew. Math. 575 (2004), 1–35.

- Abstract: This paper exploits the formalism of Moy and Prasad to sharpen and extend a result of Murnaghan. Let F be a nonarchimedean local field of residual characteristic p > n. We show that the character of a supercuspidal representation of GL
_{n}(F) can be expressed on a large set as its formal degree times the Fourier transform of an elliptic orbital integral. We prove a similar result for tame, very supercuspidal representations of more general reductive groups. Finally, we examine a consequence for local character expansions.

Discrete
series
characters
of
division
algebras
and
GL_{n}
over
a
p-adic
field
(with
L.
Corwin
and
P.
J.
Sally,
Jr.),
in
Contributions
to
Automorphic
Forms,
Geometry,
and
Number
Theory,
pp. 57–64.
Edited
by
H.
Hida,
D.
Ramakrishnan,
and
F.
Shahidi.
Johns
Hopkins
University
Press,
2004.

- Abstract: Let D be a central division algebra over a nonarchimedean local field F. Assume that the degree of D is prime to the residual characteristic of F. Then we present explicit formulas for all irreducible characters of the multiplicative group of D. Proofs will appear elsewhere.

A generalization of a result of Kazhdan and Lusztig, (with Stephen DeBacker), Proc. Amer. Math. Soc., 132 (2004), no. 6, 1861–1868.

- Abstract: Kazhdan and Lusztig show that every topologically nilpotent, regular semisimple orbit in the Lie algebra of a simple, split group over the field ℂ((t)) is, in some sense, close to a regular nilpotent orbit. We generalize this result to a setting that includes most quasisplit p-adic groups.

Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive p-adic group (with Stephen DeBacker), Mich. Math. J. 50 (2002), No. 2, 263–286. (An early version of this work was distributed under the title “Moy-Prasad filtrations and harmonic analysis”.) MR:2003g:22016.

- Abstract: Let F denote a complete nonarchimedean local field with perfect residue field. Let G be a connected reductive group defined over F. This paper exploits the formalism of Moy and Prasad to sharpen and extend familiar harmonic analysis results for the Lie algebra of G. We show that the G-orbits of the Moy-Prasad filtration lattices are asymptotic to the set of nilpotent elements. In the Lie algebra, we define G-domains in terms of the filtration lattices and explore their properties. We then show that the domain where the local expansion for G-invariant distributions is valid behaves well with respect to parabolic induction.

A construction of types, Analyse harmonique sur le groupe Sp_{4}, (CIRM,
Luminy, June, 1998), Paul Sally, ed. University of Chicago Lecture Notes in
Representation Theory, 1999.

An intertwining result for p-adic groups, (with Alan Roche), Canad. J. Math., 52 (2000), no. 3, 449–467.

- Abstract: For a reductive p-adic group G, we compute the supports of the Hecke algebras for the K-types for G lying in a certain frequently-occurring class. When G is classical, we compute the intertwining between any two such K-types.

Refined anisotropic K-types and supercuspidal representations, Pacific J. Math., 185 (1998), no. 1, 1–32. MR:2000f:22019. Zbl 924.22015.

- Abstract: Let F be a nonarchimedean local field, and G a connected reductive group defined over F. We classify the representations of G(F) that contain any anisotropic unrefined minimal K-type satisfying a certain tameness condition. We show that these representations are induced from compact (mod center) subgroups, and we construct corresponding refined minimal K-types.

Self-contragredient
supercuspidal
representations
of
GL_{n},
Proc. Amer. Math. Soc.,
125
(1997),
No.
8,
2471–2479.
MR:97j:22038.
Zbl
886.22011.

- Abstract: Let F be a non-archimedean local field of residual characteristic p. Then GL
_{n}(F) has tamely ramified self-contragredient supercuspidal representations if and only if n or p is even. When such representations exist, they do so in abundance.

The Poster Session: A Tool for Education, Assessment, and Recruitment (with Ethel R. Wheland, Timothy W. O’Neil, and Kathy J. Liszka), Mathematics and Computer Education, 43 (Spring, 2009), no. 2, 141–150.

- Abstract: We describe the benefits and mechanics of running a student poster session.

Reading encrypted diplomatic correspondence: An undergraduate research project, (with Ryan Fuoss, Michael Levin, and Amanda Youell), Cryptologia, 32 (2008), Issue 1, pp. 1–12.

- Abstract: We describe the cryptanalysis of a collection of sixteenth-century Spanish diplomatic correspondence, performed by undergraduates who do not know Spanish.

Undergraduate research in mathematics at the University of Akron, Proceeding of the Conference on Promoting Undergraduate Research in Mathematics (Chicago, 2006), Joseph A. Gallian, ed., American Mathematical Society, pp. 145–148.

- Abstract: The University of Akron has been active in undergraduate research in mathematics for several years. We describe the history of this activity, particularly during the last two summers.

Groups
of
order
p^{4}
made
less
difficult
(with
Michael
Garlow
and
Ethel
R.
Wheland),
preprint.
E-print
available
at
arXiv:1611.00461.

- Abstract: Using only elementary methods, we classify the groups of order p
^{4}, for p an odd prime.

The Neighborhood Covering Heuristic (NCH) Approach for the General Mixed Integer Programming Problem, (with A. A. Sterns, Douglas Kline, and Scott Collins (who has since become mononymous)), Final Report completed for the Navy Personnel Research, Science, and Technology Division, Contract N00014-03-M-0254, Office of Naval Research, 2004.

- Abstract: We present a new approach to the mixed integer programming problem. We compare this approach to the traditional branch and bound method on thousands of randomly generated problems.