Let F be a finite extension of Qp and G be GL(2,F). When V is the tensor product of three admissible, irreducible, finite dimensional representations of G, the space of G-invariant linear forms has dimension at most one. When a non zero linear form exists, one wants to find an element of V which is not in its kernel: this is a test vector. Gross and Prasad found explicit test vectors when the three representations are unramified principal series, and when they are all unramified twists of the Steinberg representation. In this paper we decribe explicit test vectors when two of the representations are principal series.
Let F be a finite extension of Qp and G be GL(2,F). When V is the tensor product of three infinite dimensional, irreducible, admissible representations of G, the space of G-invariant linear forms has dimension 0 or 1. When a non-zero linear form exists, one wants to find an element of V which is not in its kernel : this is a test vector. Gross and Prasad found explicit test vectors when the three representations are unramified principal series, and when the three representations are unramified twists of the Steinberg representation. In this paper, we find an explicit test vector when two of th…
S. Orlik and M. Strauch have studied locally analytic principal series representation for general $p$-adic reductive groups generalizing an earlier work of P. Schneider for $GL(2)$ and related the condition of irreducibility of such locally analytic representation with that of a suitable Verma module. In this article, we take the case of $GL(n)$ and study the globally analytic principal series representation under the action of the pro-$p$ Iwahori subgroup of $GL(n,\mathbb{Z}_p)$, following the notion of globally analytic representations introduced by M. Emerton. Furthermore, we relate the con…
For a given Dirichlet character $χ(n) = e^{i θ_n}$, we prove central limit theorems for the series $\sum_{p'} \cos θ_{p'}$ for non-principal characters, and $\sum_{p' } \cos (t \log p')$ for principal characters, where $p'$ are integers based on a variant of Cramér's random model for the primes. For non-principal characters, we use these results to show that the Generalized Riemann Hypothesis for the associated $L$-function is true with probability equal to one. For principal characters we propose how to extend these arguments to $\Re (s) = t \to \infty$.
We describe explicitly the whole structures of the $(g,K)$-modules of principal series representations of $Sp(3,R)$. We apply this result to determine the holonomic system characterizing those Whittaker functions.
Let $G_n$ denote either the group $SO(2n+1, F)$, $Sp(2n, F)$, or $GSpin(2n+1, F)$ over a non-archimedean local field of characteristic different than two. We determine all composition factors of degenerate principal series of $G_n$, using methods based on the Aubert involution and known results on irreducible subquotiens of the generalized principal series of particular type.
This paper initiates the study by analytic methods of the generalized principal series Maass forms on $GL(3)$. These forms occur as an infinite sequence of one-parameter families in the two-parameter spectrum of $GL(3)$ Maass forms, analogous to the relationship between the holomorphic modular forms and the spherical Maass cusp forms on $GL(2)$. We develop a Kuznetsov trace formula attached to these forms at each weight and use it to prove an arithmetically-weighted Weyl law, demonstrating the existence of forms which are not self-dual. Previously, the only such forms that were known to exist …
The Kuznetsov and Petersson trace formulae for $GL(2)$ forms may collectively be derived from Poincaré series in the space of Maass forms with weight. Having already developed the spherical spectral Kuznetsov formula for $GL(3)$, the goal of this series of papers is to derive the spectral Kuznetsov formulae for non-spherical Maass forms and use them to produce the corresponding Weyl laws; this appears to be the first proof of the existence of such forms not coming from the symmetric-square construction. Aside from general interest in new types of automorphic forms, this is a necessary step in …
We study the Schneider-Vigneras functor attaching a module over the Iwasawa algebra $Λ(N_0)$ to a $B$-representation for irreducible modulo $π$ principal series of the group $\mathrm{GL}_n(F)$ for any finite field extension $F|\mathbb{Q}_p$.
The p-adic local Langlands correspondence for GL2(Qp) attaches to any 2-dimensional irreducible p-adic representation V of the absolute Galois groups of Qp an admissible unitary representation Pi(V) of GL2(Qp). The unitary principal series of GL2(Qp) are those Pi(V) corresponding to trianguline representations. In this article, for p>2, using the machinery of Colmez, we determine the space of locally analytic vectors for all non-exceptional unitary principal series of GL2(Qp) by proving a conjecture of Emerton.
We provide an irreducibility test and factoring algorithm (with some qualifications) for formal power series in the unique factorization domain $R[[X]]$, where $R$ is any principal ideal domain. We also classify all integral domains arising as quotient rings of $R[[X]]$. Our main tool is a generalization of the $p$-adic Weierstrass preparation theorem to the context of complete filtered commutative rings.
The work of Bernstein-Zelevinsky and Zelevinsky gives a good understanding of irreducible subquotients of a reducible principal series representation of $GL_n(F)$, $F$ a $p$-adic field (without specifying their multiplicities which is done by a Kazhdan-Lusztig type conjecture). In this paper we make a proposal of a similar kind for principal series representations of $GL_n({\mathbb R})$. Our investigation on principal series representations naturally led us to consider the Steinberg representation for real groups, which has curiuosly not been paid much attention to in the subject (unlike the $…
Let $G$ be an unramified group over a $p$-adic field. This article introduces a base change homomorphism for Bernstein centers of depth-zero principal series blocks for $G$ and proves the corresponding base change fundamental lemma. This result is used in the approach to Shimura varieties with $Γ_1(p)$-level structure initiated by M. Rapoport and the author in [HR2].
In earlier work, the first named author generalized the construction of Darmon-style $\mathcal{L}$-invariants to cuspidal automorphic representations of semisimple groups of higher rank, which are cohomological with respect to the trivial coefficient system and Steinberg at a fixed prime. In this paper, assuming that the Archimedean component of the group has discrete series we show that these automorphic $\mathcal{L}$-invariants can be computed in terms of derivatives of Hecke-eigenvalues in $p$-adic families. Our proof is novel even in the case of modular forms, which was established by Bert…
We prove a conjecture of Udo Riese about the minimal ring of definition for principal series Weil characters of $\text{SL}_2(\mathbb{F}_p)$, for $p$ an odd prime. More precisely, we show that the $(p+1)/2$-dimensional Weil characters can be realized over the ring of integers of $\mathbb{Q}(\sqrt{\varepsilon p})$, where $\varepsilon =(-1)^{(p-1)/2}$, and we provide explicit integral models over these quadratic rings. We do so by studying the Galois action on the integral models of Weil characters recently discovered by Yilong Wang.
In this paper we view pro-$p$ Iwahori subgroups $I$ as rigid analytic groups $\Bbb{I}$ for large enough $p$. This is done by endowing $I$ with a natural $p$-valuation, and thereby generalizing results of Lazard for $\text{GL}_n$. We work with a general connected reductive split group over some $p$-adic field (with simply connected derived group) and study the $\Bbb{I}$-analytic vectors in principal series representations. Our main result is an irreducibility criterion which generalizes results of Clozel and Ray in the $\text{GL}_n$-case.
Let G be a connected reductive quasisplit algebraic group over a field L which is a finite extension of the p-adic numbers. We construct an exact sequence modelled on (the dual of) the BGG resolution involving locally analytic principal series representations for G(L). This leads to an exact sequence involving spaces of overconvergent p-adic automorphic forms for certain groups compact modulo centre at infinity.
Let F be a nonarchimedean locally compact field with residue characteristic p and G(F) the group of F-rational points of a connected reductive group. Following Schneider and Stuhler, one can realize, in a functorial way, any smooth complex finitely generated representation of G(F) as the 0-homology of a certain coefficient system on the semi-simple building of G(F). It is known that this method does not apply in general for smooth mod p representations of G(F), even when G= GL(2). However, we prove that a principal series representation of GL(n,F) over a field with arbitrary characteristic can…
For a local field $F$ we consider tamely ramified principal series representations $V$ of $G={\rm GL}_{d+1}(F)$ with coefficients in a finite extension $K$ of ${\mathbb Q}_p$. Let $I_0$ be a pro-$p$-Iwahori subgroup in $G$, let ${\mathcal H}(G,I_0)$ denote the corresponding pro-$p$-Iwahori Hecke algebra. If $V$ is locally unitary, i.e. if the ${\mathcal H}(G,I_0)$-module $V^{I_0}$ admits an integral structure, then such an integral structure can be chosen in a particularly well organized manner, in particular its modular reduction can be made completely explicit.
By the unfolding method, Rankin-Selberg L-functions for ${\rm GL}(n)\times{\rm GL}(m)$ can be expressed in terms of period integrals. These period integrals actually define invariant forms on tensor products of the relevant automorphic representations. By the multiplicity-one theorems due to Sun-Zhu and Chen-Sun such invariant forms are unique up to scalar multiples and can therefore be related to invariant forms on equivalent principal series representations. We construct meromorphic families of such invariant forms for spherical principal series representations of ${\rm GL}(n,\mathbb{R})$ an…
We consider an $n$-fold Brylinski-Deligne cover of a reductive group over a $p$-adic field. Since the space of Whittaker functionals of an irreducible genuine representation of such a cover is not one-dimensional, one can consider a local coefficients matrix arising from an intertwining operator, which is the natural analogue of the local coefficients in the linear case. In this paper, we concentrate on genuine principal series and establish some fundamental properties of such a local coefficients matrix, including the investigation of its arithmetic invariants. As a consequence, we prove a fo…
Let $p\geq 3$ be a prime number and $F$ a $p$-adic field. Let $I_1$ denote the pro-$p$-Iwahori subgroup of $\textrm{GL}_n(F)$, and $\mathcal{H}$ the pro-$p$-Iwahori--Hecke algebra of $\textrm{GL}_n(F)$ with respect to $I_1$ (over a coefficient field of characteristic $p$). We compute the structure of $\textrm{H}^1(I_1,π)$ as an $\mathcal{H}$-module, where $π$ is a mod-$p$ principal series representation of $\textrm{GL}_n(F)$. We also give some partial results about the structure of $\textrm{H}^1(I_1,π)$ for a general split reductive group with irreducible root system.
We provide an observation relating several known and conjectured $q$-series identities to the theory of principal subspaces of basic modules for twisted affine Lie algebras. We also state and prove two new families of $q$-series identities. The first family provides quadruple sum representations for Nandi's identities, including a manifestly positive representation for the first identity. The second is a family of new mod 10 identities connected with principal characters of level 4 integrable, highest-weight modules of $\mathrm{D}_4^{(3)}$.
We present sum-sides for principal characters of all standard (i.e., integrable and highest-weight) irreducible modules for the affine Lie algebra $A_2^{(2)}$. We use modifications of five known Bailey pairs; three of these are sufficient to obtain all the necessary principal characters. We then use the technique of Bailey lattice appropriately extended to include "out-of-bounds" values of one of the parameters, namely, $i$. We demonstrate how the sum-sides break into six families depending on the level of the modules modulo 6, confirming a conjecture of McLaughlin--Sills.
