Let $F$ be a non-archimedean local field of residual characteristic $p \neq 2$. Let $G$ be a (connected) reductive group over $F$ that splits over a tamely ramified field extension of $F$. We revisit Yu's construction of smooth complex representations of $G(F)$ from a slightly different perspective and provide a proof that the resulting representations are supercuspidal. We also provide a counterexample to Proposition 14.1 and Theorem 14.2 in Yu [Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14 (2001), 579–622], whose proofs relied on a typo in a reference.

The main aim of this article is to show that normalised standard intertwining operator between induced representations of p-adic groups, at a very specific point of evaluation, has an arithmetic origin. This result has applications to Eisenstein cohomology and the special values of automorphic L-functions.

Let G be a reductive p-adic group which splits over an unramified extension of the ground field. Hiraga, Ichino and Ikeda [24] conjectured that the formal degree of a square-integrable G-representation $\pi $ can be expressed in terms of the adjoint $\gamma $ -factor of the enhanced L-parameter of $\pi $ . A similar conjecture was posed for the Plancherel densities of tempered irreducible G-representations.

We prove these conjectures for unipotent G-representations. We also derive explicit formulas for the involved adjoint $\gamma $ -factors.

Let $G$ be a $p$ -adic group that splits over an unramified extension. We decompose $\text{Rep}_{\unicode[STIX]{x1D6EC}}^{0}(G)$ , the abelian category of smooth level $0$ representations of $G$ with coefficients in $\unicode[STIX]{x1D6EC}=\overline{\mathbb{Q}}_{\ell }$ or $\overline{\mathbb{Z}}_{\ell }$ , into a product of subcategories indexed by inertial Langlands parameters. We construct these categories via systems of idempotents on the Bruhat–Tits building and Deligne–Lusztig theory. Then, we prove compatibilities with parabolic induction and restriction functors and the local Langlands correspondence.

In a recent paper, Gross and Reeder study the arithmetic properties of discrete Langlands parameters for semi-simple -adic groups, and they conjecture that a special class of these – the simple wild parameters – should correspond to -packets consisting of simple supercuspidal representations. We provide a construction of this correspondence, and show that the simple wild -packets satisfy many expected properties. In particular, they admit a description in terms of the Langlands dual group, and contain a unique generic element for a fixed Whittaker datum. Moreover, we prove their stability on an open subset of the regular semi-simple elements, and show that they satisfy a natural compatibility with respect to unramified base-change.

When $F$ is a $p$-adic field, and $G\,=\,\mathbb{G}\left( F \right)$ is the group of $F$-rational points of a connected algebraic $F$-group, the complex vector space $\mathcal{H}\left( G \right)$ of compactly supported locally constant distributions on $G$ has a natural convolution product that makes it into a $\mathbb{C}$-algebra (without an identity) called the Hecke algebra. The Hecke algebra is a partial analogue for $p$-adic groups of the enveloping algebra of a Lie group. However, $\mathcal{H}\left( G \right)$ has drawbacks such as the lack of an identity element, and the process $G\,\mapsto \,\mathcal{H}\left( G \right)$ is not a functor. Bernstein introduced an enlargement ${{\mathcal{H}}^{\hat{\ }}}\left( G \right)$ of $\mathcal{H}\left( G \right)$. The algebra ${{\mathcal{H}}^{\hat{\ }}}\left( G \right)$ consists of the distributions that are left essentially compact. We show that the process $G\,\mapsto \,{{\mathcal{H}}^{\hat{\ }}}\left( G \right)$ is a functor. If $\tau \,:\,G\,\to \,H$ is a morphism of $p$-adic groups, let $F\left( \tau \right):\,{{\mathcal{H}}^{\hat{\ }}}\left( G \right)\,\to \,{{\mathcal{H}}^{\hat{\ }}}\left( H \right)$ be the morphism of $\mathbb{C}$-algebras. We identify the kernel of $F\left( \tau \right)$ in terms of $\text{Ker}\left( \tau \right)$. In the setting of $p$-adic Lie algebras, with $\mathfrak{g}$ a reductive Lie algebra, $\mathfrak{m}$ a Levi, and $\tau \,:\,\mathfrak{g}\,\to \,\mathfrak{m}$ the natural projection, we show that $F\left( \tau \right)$ maps $G$-invariant distributions on $\mathcal{G}$ to ${{N}_{G}}\left( \mathfrak{m} \right)$-invariant distributions on $\mathfrak{m}$. Finally, we exhibit a natural family of $G$-invariant essentially compact distributions on $\mathfrak{g}$ associated with a $G$-invariant non-degenerate symmetric bilinear form on $\mathfrak{g}$ and in the case of $SL\left( 2 \right)$ show how certain members of the family can be moved to the group.

We study basic properties of the category of smooth representations of a p-adic group G with coefficients in any commutative ring R in which p is invertible. Our main purpose is to prove that Hecke algebras are Noetherian whenever R is; this question arose naturally with Bernstein's fundamental work for R = ℂ, in which case he proved this Noetherian property. In a first step, we prove that Noetherianity would follow from a generalization of the so-called second adjointness property between parabolic functors, also due to Bernstein for complex representations. Then, to attack this second adjointness, we introduce and study ‘parahoric functors’ between representations of groups of integral points of smooth integral models of G and of their ‘Levi’ subgroups. Applying our general study to Bruhat-Tits parahoric models, we get second adjointness for minimal parabolic groups. For non-minimal parabolic subgroups, we have to restrict to classical and linear groups, and use smooth models associated with Bushnell-Kutzko and Stevens semi-simple characters. The same strategy should apply to ‘tame’ groups, using Yu's smooth models and generic characters.

For the special orthogonal group G = SO(2n + 1) over a p-adic field, we consider a discrete series representation of a standard Levi subgroup of G. We prove that the Arthur R-group and the classical R-group of $\pi$ are isomorphic. If $\pi$ is generic, we consider the Aubert involution $\hat{\pi}$. Under the assumption that $\hat{\pi}$ is unitary, we prove that the Arthur R-group of $\hat{\pi}$ is isomorphic to the R-group of $\hat{\pi}$ defined by Ban (Ann. Sci. École Norm. Sup. 35 (2002), 673–693; J. Algebra 271 (2004), 749–767). This is done by establishing the connection between the A-parameters of $\pi$ and $\hat{\pi}$. We prove that the A-parameter of $\hat{\pi}$ is obtained from the A-parameter of $\pi$ by interchanging the two $\textit{SL}(2,\mathbb{C})$ components.

The goal of this paper is to carry out some explicit calculations of the actions of Hecke operators on spaces of algebraic modular forms on certain simple groups. In order to do this, we give the coset decomposition for the supports of these operators. We present the results of our calculations along with interpretations concerning the lifting of forms. The data we have obtained is of interest both from the point of view of number theory and of representation theory. For example, our data, together with a conjecture of Gross, predicts the existence of a Galois extension of Q with Galois group G2(F5) which is ramified only at the prime 5. We also provide evidence of the existence of the symmetric cube lifting from PGL2 to PGSp4.

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

In the previous part of this paper, we constructed a large family of Hecke algebras on some classical groups G defined over p-adic fields in order to understand their admissible representations. Each Hecke algebra is associated to a pair (JΣ, ρΣ) of an open compact subgroup JΣ and its irreducible representation ρΣ which is constructed from given data Σ = (Γ, P′0, ϱ). Here, Γ is a semisimple element in the Lie algebra of G, P′0 is a parahoric subgroup in the centralizer of Γ in G, and ϱ is a cuspidal representation on the finite reductive quotient of P′0. In this paper, we explicitly describe those Hecke algebras when P′0 is a minimal parahoric subgroup, ϱ is trivial and ρΣ is a character.

This article studies unipotent orbital integrals on symplectic and orthogonal groups from the point of view of endoscopy. It begins by partitioning stable unipotent classes into packets and goes on to propose a transfer of these packets. It then discusses (in rough form) the associated transfer factors. Some supporting calculations in split odd orthogonal groups are given.