Let $k$ be a field, let $H \subset G$ be (possibly disconnected) reductive groups over $k$, and let $\Gamma $ be a finitely generated group. Vinberg and Martin have shown that the induced morphism $\underline{\operatorname{Hom}}_{k\textrm{gp}}(\Gamma , H)//H \to \underline{\operatorname{Hom}}_{k\textrm{gp}}(\Gamma , G)//G$ is finite. In this note, we generalize this result (with a significantly different proof) by replacing $k$ with an arbitrary locally Noetherian scheme, answering a question of Dat. Along the way, we use Bruhat–Tits theory to establish a few apparently new results about integral models of reductive groups over discrete valuation rings.
Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher.
Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?
Some links on this page may take you to nonfederal websites. Their policies may differ from this site.

Abstract Free, publiclyaccessible full text available June 13, 2025 
Abstract We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies
of the algebra of symmetric functions embedded in the elliptic Hall algebra ℰ of Burban and Schiffmann.As a corollary, we deduce an explicit raising operator formula for the ∇ operator applied to any LLT polynomial.In particular, we obtain a formula for$\mathrm{\Lambda}({X}^{m,n})\subset \mathcal{E}$ \Lambda(X^{m{,}n})\subset\mathcal{E} which serves as a starting point for our proof of the Loehr–Warrington conjecture in a companion paper to this one.${\nabla}^{m}{s}_{\lambda}$ \nabla^{m}s_{\lambda} Free, publiclyaccessible full text available April 23, 2025 
Abstract We construct a moduli space $\textsf {LP}_{G}$ of $\operatorname {SL}_{2}$parameters over ${\mathbb {Q}}$, and show that it has good geometric properties (e.g., explicitly parametrized geometric connected components and smoothness). We construct a Jacobson–Morozov morphism$\textsf {JM}\colon \textsf {LP}_{G}\to \textsf {WDP}_{G}$ (where $\textsf {WDP}_{G}$ is the moduli space of Weil–Deligne parameters considered by several other authors). We show that $\textsf {JM}$ is an isomorphism over a dense open of $\textsf {WDP}_{G}$, that it induces an isomorphism between the discrete loci $\textsf {LP}^{\textrm {disc}}_{G}\to \textsf {WDP}_{G}^{\textrm {disc}}$, and that for any ${\mathbb {Q}}$algebra $A$ it induces a bijection between Frobenius semisimple equivalence classes in $\textsf {LP}_{G}(A)$ and Frobenius semisimple equivalence classes in $\textsf {WDP}_{G}(A)$ with constant (up to conjugacy) monodromy operator.

Abstract We show that an orthogonal root number of a tempered
L parameter decomposes as the product of two other numbers: the orthogonal root number of the principal parameter and the value on a central involution of Langlands’s central character for$$\varphi $$ $\phi $ . The formula resolves a conjecture of Gross and Reeder and computes root numbers of Weil–Deligne representations arising in a conjectural description of the Plancherel measure.$$\varphi $$ $\phi $ 
We study the geometry of smooth projective surfaces defined by Frobenius forms, a class of homogenous polynomials in prime characteristic recently shown to have minimal possible Fpure threshold among forms of the same degree. We call these surfaces extremal surfaces, and show that their geometry is reminiscent of the geometry of smooth cubic surfaces, especially nonFrobenius split cubic surfaces. For instance, extremal surfaces have many lines but no triangles, hence many “star points” analogous to Eckardt points on a cubic surface. We generalize the classical notion of a double six for cubic surfaces to a double 2d on an extremal surface of degree d. We show that, asymptotically in d, smooth extremal surfaces have at least (1/16)d^{14} double 2d's. A key element of the proofs is the large automorphism group of an extremal surface, which we show to act transitively on many associated sets, such as the set of triples of skew lines on the extremal surface.more » « lessFree, publiclyaccessible full text available May 15, 2025

Free, publiclyaccessible full text available May 7, 2025

Free, publiclyaccessible full text available May 1, 2025

Free, publiclyaccessible full text available March 21, 2025

The goal of this paper is extend Kottwitz’s theory of B(G) for global fields. In particular, we show how to extend the definition of “B(G) with adelic coefficients” from tori to all connected reductive groups. As an application, we give an explicit construction of certain transfer factors for nonregular semisimple elements of nonquasisplit groups. This generalizes some results of Kaletha and Taibi. These formulas are used in the stabilization of the cohomology of Shimura and Igusa varieties.more » « lessFree, publiclyaccessible full text available March 15, 2025

Supercuspidal representations are usually infinitedimensional, so the size of such a representation cannot be measured by its dimension; the formal degree is a better alternative. Hiraga, Ichino, and Ikeda conjectured a formula for the formal degree of a supercuspidal in terms of its Lparameter only. Our first main result is to compute the formal degrees of the supercuspidal representations constructed by Yu. Our second result, using the first, is to verify that Kaletha’s regular supercuspidal Lpackets satisfy the conjecture.more » « lessFree, publiclyaccessible full text available January 22, 2025