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 article, we refine a conjecture of Hiraga, Ichino, and Ikeda which relates the formal degree of a discrete series representation 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 at x=q, the cardinality of the residue field, to the quotient of this local gamma factor by the gamma factor of the Steinberg parameter. 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 article.

https://abel.math.harvard.edu/~gross/preprints/adjointgamma3.pdf

Arithmetic invariants of discrete Langlands parameters

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

/pdf/refined-anisotropic-k-types-and-supercuspidal-30dhf4dywv.pdf

Refined anisotropic k-types and supercuspidal representations

We prove the local Langlands conjecture for $GSp_4(F)$ where $F$ is a non-archimedean local field of characteristic zero.

The Local Langlands Conjecture for GSp(4)

For every genuine irreducible admissible smooth representation $\pi$ of the metaplectic group
$\widetilde{{\rm Sp}}(2n)$ over a $p$-adic field, and every smooth oscillator representation $\omega_\psi$
of $\widetilde{{\rm Sp}}(2n)$, we prove that the tensor product $\pi\otimes\omega_\psi$ is multiplicity free
as a smooth representation of the symplectic group ${\rm Sp}(2n)$. Similar results are proved for general
linear groups and unitary groups.

pdf/multiplicity-one-theorems-for-fourier-jacobi-models-30bpob6c0x.pdf

Multiplicity one theorems for Fourier-Jacobi models

This paper studies the behavior of Jiu-Kang Yu's tame supercuspidal representations relative to involutions of reductive p-adic groups. Symmetric space methods are used to illuminate various aspects of Yu's construction. Necessary conditions for a tame supercuspidal representation of G to be distinguished by (the fixed points of) an involution of G are expressed in terms of properties of the G-orbit of the associated G-datum. When these conditions are satisfied, the question of whether a tame supercuspidal representation is distinguished reduces to the question of whether certain cuspidal representations of finite groups of Lie type are distinguished relative to particular quadratic characters. As an application of the main results, we obtain necessary and sufficient conditions for equivalence of two of Yu's supercuspidal representations associated to distinct G-data.

Distinguished tame supercuspidal representations

https://www.degruyter.com/view/j/crll.2004.2004.issue-575/crll.2004.080/crll.2004.080.pdf

Murnaghan-Kirillov theory for supercuspidal representations of tame general linear groups

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, Fourier transforms of orbital integrals, and certain signs and cardinalities that are described explicitly in terms of the datum associated to the representation and of the element at which the character is evaluated.

Supercuspidal characters of reductive p-adic groups

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 describe an explicit region on which the local character expansion is valid. We assume neither that the group is connected, nor that the underlying field has characteristic zero

https://www.math.toronto.edu/jkorman/Articles/Loc-char-exp/loc-char-exp.pdf

The local character expansion near a tame, semisimple element

/pdf/good-product-expansions-for-tame-elements-of-p-adic-groups-1k0ykp88gu.pdf

Good product expansions for tame elements of p-adic groups

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