Let F be a non-archimedean locally compact field of residual characteristic p, let G=\operatorname {GL}_{r}(F)$G=\operatorname {GL}_{r}(F)$ and let \widetilde {G}$\widetilde {G}$ be an n-fold metaplectic cover of G with \operatorname {gcd}(n,p)=1$\operatorname {gcd}(n,p)=1$. We study the category \operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$\operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$ of complex smooth representations of \widetilde {G}$\widetilde {G}$ having inertial equivalence class \mathfrak {s}=(\widetilde {M},\mathcal {O})$\mathfrak {s}=(\widetilde {M},\mathcal {O})$, which is a block of the category \operatorname {Rep}(\widetilde {G})$\operatorname {Rep}(\widetilde {G})$, following the ‘type theoretical’ strategy of Bushnell-Kutzko.

Precisely, first we construct a ‘maximal simple type’ (\widetilde {J_{M}},\widetilde {\lambda }_{M})$(\widetilde {J_{M}},\widetilde {\lambda }_{M})$ of \widetilde {M}$\widetilde {M}$ as an \mathfrak {s}_{M}$\mathfrak {s}_{M}$-type, where \mathfrak {s}_{M}=(\widetilde {M},\mathcal {O})$\mathfrak {s}_{M}=(\widetilde {M},\mathcal {O})$ is the related cuspidal inertial equivalence class of \widetilde {M}$\widetilde {M}$. Along the way, we prove the folklore conjecture that every cuspidal representation of \widetilde {M}$\widetilde {M}$ could be constructed explicitly by a compact induction. Secondly, we construct ‘simple types’ (\widetilde {J},\widetilde {\lambda })$(\widetilde {J},\widetilde {\lambda })$ of \widetilde {G}$\widetilde {G}$ and prove that each of them is an \mathfrak {s}$\mathfrak {s}$-type of a certain block \operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$\operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$. When \widetilde {G}$\widetilde {G}$ is either a Kazhdan-Patterson cover or Savin’s cover, the corresponding blocks turn out to be those containing discrete series representations of \widetilde {G}$\widetilde {G}$. Finally, for a simple type (\widetilde {J},\widetilde {\lambda })$(\widetilde {J},\widetilde {\lambda })$ of \widetilde {G}$\widetilde {G}$, we describe the related Hecke algebra \mathcal {H}(\widetilde {G},\widetilde {\lambda })$\mathcal {H}(\widetilde {G},\widetilde {\lambda })$, which turns out to be not far from an affine Hecke algebra of type A, and is exactly so if \widetilde {G}$\widetilde {G}$ is one of the two special covers mentioned above.

We leave the construction of a ‘semi-simple type’ related to a general block \operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$\operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$ to a future phase of the work.

Arthur packets have been defined for pure real forms of symplectic and special orthogonal groups following two different approaches. The first approach, due to Arthur, Moeglin, and Renard uses harmonic analysis. The second approach, due to Adams, Barbasch, and Vogan uses microlocal geometry. We prove that the two approaches produce essentially equivalent Arthur packets. This extends previous work of the authors and J. Adams for the quasisplit real forms.

Let { F}/{ F}_0${ F}/{ F}_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic p\neq 2$p\neq 2$ with Galois automorphism \sigma $\sigma$, and let R be an algebraically closed field of characteristic \ell \notin \{0,p\}$\ell \notin \{0,p\}$. We reduce the classification of \operatorname {GL}_n({ F}_0)$\operatorname {GL}_n({ F}_0)$-distinguished cuspidal R-representations of \operatorname {GL}_n({ F})$\operatorname {GL}_n({ F})$ to the level 0$0$ setting. Moreover, under a parity condition, we give necessary conditions for a \sigma $\sigma$-self-dual cuspidal R-representation to be distinguished. Finally, we classify the distinguished cuspidal {\overline {\mathbb {F}}_{\ell }}${\overline {\mathbb {F}}_{\ell }}$-representations of \operatorname {GL}_n({ F})$\operatorname {GL}_n({ F})$ having a distinguished cuspidal lift to {\overline {\mathbb {Q}}_\ell }${\overline {\mathbb {Q}}_\ell }$.

We extend a comparison theorem of Anandavardhanan–Borisagar between the quotient of the induction of a mod p$p$ character by the image of an Iwahori–Hecke operator and compact induction of a weight to the case of the trivial character. This involves studying the corresponding non-commutative Iwahori–Hecke algebra. We use this to give an Iwahori theoretic reformulation of the (semi-simple) mod p$p$ local Langlands correspondence discovered by Breuil and reformulated functorially by Colmez. This version of the correspondence is expected to have applications to computing the mod p$p$ reductions of semi-stable Galois representations.

In this work, we develop an integral representation for the partial L-function of a pair \pi \times \tau $\pi \times \tau$ of genuine irreducible cuspidal automorphic representations, \pi $\pi$ of the m-fold covering of Matsumoto of the symplectic group \operatorname {\mathrm {Sp}}_{2n}$\operatorname {\mathrm {Sp}}_{2n}$ and \tau $\tau$ of a certain covering group of \operatorname {\mathrm {GL}}_k$\operatorname {\mathrm {GL}}_k$, with arbitrary m, n and k. Our construction is based on the recent extension by Cai, Friedberg, Ginzburg and the author, of the classical doubling method of Piatetski-Shapiro and Rallis, from rank-1$1$ twists to arbitrary rank twists. We prove a basic global identity for the integral and compute the local integrals with unramified data. Our global results are subject to certain conjectures, but when k=1$k=1$ they are unconditional for all m. One possible future application of this work will be a Shimura-type lift of representations from covering groups to general linear groups. In a recent work, we used the present results in order to provide an analytic definition of local factors for representations of the m-fold covering of \operatorname {\mathrm {Sp}}_{2n}$\operatorname {\mathrm {Sp}}_{2n}$.

In this paper, we define compact open subgroups of quasi-split even unitary groups for each even non-negative integer and establish the theory of local newforms for irreducible tempered generic representations with a certain condition on the central characters. To do this, we use the local Gan–Gross–Prasad conjecture, the local Rankin–Selberg integrals and the local theta correspondence.

In this article, we generalize results of Clozel and Ray (for SL_2$SL_2$ and SL_n$SL_n$, respectively) to give explicit ring-theoretic presentation in terms of a complete set of generators and relations of the Iwasawa algebra of the pro-p Iwahori subgroup of a simple, simply connected, split group \mathbf {G}$\mathbf {G}$ over {{\mathbb Q}_p}${{\mathbb Q}_p}$.

Without using the p$p$-adic Langlands correspondence, we prove that for many finite-length smooth representations of \mathrm {GL}_2(\mathbf {Q}_p)$\mathrm {GL}_2(\mathbf {Q}_p)$ on p$p$-torsion modules the \mathrm {GL}_2(\mathbf {Q}_p)$\mathrm {GL}_2(\mathbf {Q}_p)$-linear morphisms coincide with the morphisms that are linear for the normalizer of a parahoric subgroup. We identify this subgroup to be the Iwahori subgroup in the supersingular case, and \mathrm {GL}_2(\mathbf {Z}_p)$\mathrm {GL}_2(\mathbf {Z}_p)$ in the principal series case. As an application, we relate the action of parahoric subgroups to the action of the inertia group of \mathrm {Gal}(\overline {\mathbf {Q}}_p/\mathbf {Q}_p)$\mathrm {Gal}(\overline {\mathbf {Q}}_p/\mathbf {Q}_p)$, and we prove that if an irreducible Banach space representation \Pi$\Pi$ of \mathrm {GL}_2(\mathbf {Q}_p)$\mathrm {GL}_2(\mathbf {Q}_p)$ has infinite \mathrm {GL}_2(\mathbf {Z}_p)$\mathrm {GL}_2(\mathbf {Z}_p)$-length, then a twist of \Pi$\Pi$ has locally algebraic vectors. This answers a question of Dospinescu. We make the simplifying assumption that p > 3$p > 3$ and that all our representations are generic.

We define an involution on the elliptic space of tempered unipotent representations of inner twists of a split simple p$p$-adic group G$G$ and investigate its behaviour with respect to restrictions to reductive quotients of maximal compact open subgroups. In particular, we formulate a precise conjecture about the relation with a version of Lusztig's nonabelian Fourier transform on the space of unipotent representations of the (possibly disconnected) reductive quotients of maximal compact subgroups. We give evidence for the conjecture, including proofs for {\mathsf {SL}}_n${\mathsf {SL}}_n$ and {\mathsf {PGL}}_n${\mathsf {PGL}}_n$.

Let G be a split connected reductive group defined over \mathbb {Z}$\mathbb {Z}$. Let F and F'$F'$ be two non-Archimedean m-close local fields, where m is a positive integer. D. Kazhdan gave an isomorphism between the Hecke algebras \mathrm {Kaz}_m^F :\mathcal {H}\big (G(F),K_F\big ) \rightarrow \mathcal {H}\big (G(F'),K_{F'}\big )$\mathrm {Kaz}_m^F :\mathcal {H}\big (G(F),K_F\big ) \rightarrow \mathcal {H}\big (G(F'),K_{F'}\big )$, where K_F$K_F$ and K_{F'}$K_{F'}$ are the mth usual congruence subgroups of G(F)$G(F)$ and G(F')$G(F')$, respectively. On the other hand, if \sigma $\sigma$ is an automorphism of G of prime order l, then we have Brauer homomorphism \mathrm {Br}:\mathcal {H}(G(F),U(F))\rightarrow \mathcal {H}(G^\sigma (F),U^\sigma (F))$\mathrm {Br}:\mathcal {H}(G(F),U(F))\rightarrow \mathcal {H}(G^\sigma (F),U^\sigma (F))$, where U(F)$U(F)$ and U^\sigma (F)$U^\sigma (F)$ are compact open subgroups of G(F)$G(F)$ and G^\sigma (F),$G^\sigma (F),$ respectively. In this article, we study the compatibility between these two maps in the local base change setting. Further, an application of this compatibility is given in the context of linkage – which is the representation theoretic version of Brauer homomorphism.

Let F$F$ be a totally real field in which p$p$ is unramified and let B$B$ be a quaternion algebra over F$F$ which splits at at most one infinite place. Let \overline {r}:\operatorname {{\mathrm {Gal}}}(\overline {F}/F)\rightarrow \mathrm {GL}_2(\overline {\mathbb {F}}_p)$\overline {r}:\operatorname {{\mathrm {Gal}}}(\overline {F}/F)\rightarrow \mathrm {GL}_2(\overline {\mathbb {F}}_p)$ be a modular Galois representation which satisfies the Taylor–Wiles hypotheses. Assume that for some fixed place v|p$v|p$, B$B$ ramifies at v$v$ and F_v$F_v$ is isomorphic to \mathbb {Q}_p$\mathbb {Q}_p$ and \overline {r}$\overline {r}$ is generic at v$v$. We prove that the admissible smooth representations of the quaternion algebra over \mathbb {Q}_p$\mathbb {Q}_p$ coming from mod p$p$ cohomology of Shimura varieties associated to B$B$ have Gelfand–Kirillov dimension 1$1$. As an application we prove that the degree-two Scholze's functor (which is defined by Scholze [On the p$p$-adic cohomology of the Lubin–Tate tower, Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), 811–863]) vanishes on generic supersingular representations of \mathrm {GL}_2(\mathbb {Q}_p)$\mathrm {GL}_2(\mathbb {Q}_p)$. We also prove some finer structure theorems about the image of Scholze's functor in the reducible case.

Let F$F$ be a non-archimedean local field of characteristic different from 2 and residual characteristic p$p$. This paper concerns the \ell$\ell$-modular representations of a connected reductive group G$G$ distinguished by a Galois involution, with \ell$\ell$ an odd prime different from p$p$. We start by proving a general theorem allowing to lift supercuspidal \overline {\mathbf {F}}_{\ell }$\overline {\mathbf {F}}_{\ell }$-representations of \operatorname {GL}_n(F)$\operatorname {GL}_n(F)$ distinguished by an arbitrary closed subgroup H$H$ to a distinguished supercuspidal \overline {\mathbf {Q}}_{\ell }$\overline {\mathbf {Q}}_{\ell }$-representation. Given a quadratic field extension E/F$E/F$ and an irreducible \overline {\mathbf {F}}_{\ell }$\overline {\mathbf {F}}_{\ell }$-representation \pi$\pi$ of \operatorname {GL}_n(E)$\operatorname {GL}_n(E)$, we verify the Jacquet conjecture in the modular setting that if the Langlands parameter \phi _\pi$\phi _\pi$ is irreducible and conjugate-selfdual, then \pi$\pi$ is either \operatorname {GL}_n(F)$\operatorname {GL}_n(F)$-distinguished or (\operatorname {GL}_{n}(F),\omega _{E/F})$(\operatorname {GL}_{n}(F),\omega _{E/F})$-distinguished (where \omega _{E/F}$\omega _{E/F}$ is the quadratic character of F^\times$F^\times$ associated to the quadratic field extension E/F$E/F$ by the local class field theory), but not both, which extends one result of Sécherre to the case p=2$p=2$. We give another application of our lifting theorem for supercuspidal representations distinguished by a unitary involution, extending one result of Zou to p=2$p=2$. After that, we give a complete classification of the \operatorname {GL}_2(F)$\operatorname {GL}_2(F)$-distinguished representations of \operatorname {GL}_2(E)$\operatorname {GL}_2(E)$. Using this classification we discuss a modular version of the Prasad conjecture for \operatorname {PGL}_2$\operatorname {PGL}_2$. We show that the ‘classical’ Prasad conjecture fails in the modular setting. We propose a solution using non-nilpotent Weil–Deligne representations. Finally, we apply the restriction method of Anandavardhanan and Prasad to classify the \operatorname {SL}_2(F)$\operatorname {SL}_2(F)$-distinguished modular representations of \operatorname {SL}_2(E)$\operatorname {SL}_2(E)$.

We study actions of higher rank lattices \Gamma <G$\Gamma <G$ on hyperbolic spaces and we show that all such actions satisfying mild properties come from the rank-one factors of G. In particular, all non-elementary isometric actions on an unbounded hyperbolic space are of this type.

We prove a general formula that relates the parity of the Langlands parameter of a conjugate self-dual discrete series representation of \operatorname { {GL}}_n$\operatorname { {GL}}_n$ to the parity of its Jacquet-Langlands image. It gives a generalization of a partial result by Mieda concerning the case of invariant 1/n$1/n$ and supercuspidal representations. It also gives a variation of the result on the self-dual case by Prasad and Ramakrishnan.

Let p \geq 5$p \geq 5$ be a prime number, and let G = {\mathrm {SL}}_2(\mathbb {Q}_p)$G = {\mathrm {SL}}_2(\mathbb {Q}_p)$. Let \Xi = {\mathrm {Spec}}(Z)$\Xi = {\mathrm {Spec}}(Z)$ denote the spectrum of the centre Z of the pro-p Iwahori–Hecke algebra of G with coefficients in a field k of characteristic p. Let \mathcal {R} \subset \Xi \times \Xi $\mathcal {R} \subset \Xi \times \Xi$ denote the support of the pro-p Iwahori {\mathrm {Ext}}${\mathrm {Ext}}$-algebra of G, viewed as a (Z,Z)$(Z,Z)$-bimodule. We show that the locally ringed space \Xi /\mathcal {R}$\Xi /\mathcal {R}$ is a projective algebraic curve over {\mathrm {Spec}}(k)${\mathrm {Spec}}(k)$ with two connected components and that each connected component is a chain of projective lines. For each Zariski open subset U of \Xi /\mathcal {R}$\Xi /\mathcal {R}$, we construct a stable localising subcategory \mathcal {L}_U$\mathcal {L}_U$ of the category of smooth k-linear representations of G.

For a connected reductive group G over a nonarchimedean local field F of positive characteristic, Genestier-Lafforgue and Fargues-Scholze have attached a semisimple parameter {\mathcal {L}}^{ss}(\pi )${\mathcal {L}}^{ss}(\pi )$ to each irreducible representation \pi $\pi$. Our first result shows that the Genestier-Lafforgue parameter of a tempered \pi $\pi$ can be uniquely refined to a tempered L-parameter {\mathcal {L}}(\pi )${\mathcal {L}}(\pi )$, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of {\mathcal {L}}^{ss}(\pi )${\mathcal {L}}^{ss}(\pi )$ for unramified G and supercuspidal \pi $\pi$ constructed by induction from an open compact (modulo center) subgroup. If {\mathcal {L}}^{ss}(\pi )${\mathcal {L}}^{ss}(\pi )$ is pure in an appropriate sense, we show that {\mathcal {L}}^{ss}(\pi )${\mathcal {L}}^{ss}(\pi )$ is ramified (unless G is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show \mathcal {L}^{ss}(\pi )$\mathcal {L}^{ss}(\pi )$ is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is {\mathbb {P}}^1${\mathbb {P}}^1$ and a simple application of Deligne’s Weil II.

In this paper, we investigate the twisted GGP conjecture for certain tempered representations using the theta correspondence and establish some special cases, namely when the L-parameter of the unitary group is the sum of conjugate-dual characters of the appropriate sign.

We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman–Wallach representation of \operatorname {\mathrm {GL}}_n(F)$\operatorname {\mathrm {GL}}_n(F)$, where F is an archimedean local field, that quantifies the extent to which this representation may be ramified. We also determine a distinguished vector, the newform, occurring with multiplicity one in this representation, with the complexity of this vector measured in a natural way by the conductor exponent. Finally, we show that the newform is a test vector for \operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n$\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n$ and \operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$ Rankin–Selberg integrals when the second representation is unramified. This theory parallels an analogous nonarchimedean theory due to Jacquet, Piatetski-Shapiro, and Shalika; combined, this completes a global theory of newforms for automorphic representations of \operatorname {\mathrm {GL}}_n$\operatorname {\mathrm {GL}}_n$ over number fields. By-products of the proofs include new proofs of Stade’s formulæ and a new resolution of the test vector problem for archimedean Godement–Jacquet zeta integrals.

Let E/F$E/F$ be a quadratic unramified extension of non-archimedean local fields and \mathbb H$\mathbb H$ a simply connected semisimple algebraic group defined and split over F. We establish general results (multiplicities, test vectors) on {\mathbb H} (F)${\mathbb H} (F)$-distinguished Iwahori-spherical representations of {\mathbb H} (E)${\mathbb H} (E)$. For discrete series Iwahori-spherical representations of {\mathbb H} (E)${\mathbb H} (E)$, we prove a numerical criterion of {\mathbb H} (F)${\mathbb H} (F)$-distinction. As an application, we classify the {\mathbb H} (F)${\mathbb H} (F)$-distinguished discrete series representations of {\mathbb H} (E)${\mathbb H} (E)$ corresponding to degree 1$1$ characters of the Iwahori-Hecke algebra.

In this paper we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation \pi$\pi$ we consider a special small \mathrm{GL}_2(\mathbb{Z}_p)$\mathrm{GL}_2(\mathbb{Z}_p)$-type V in \pi$\pi$ and prove global sup-norm bounds for an average over an orthonormal basis of V. We achieve a non-trivial saving when the dimension of V grows.