Existence and Weyl's law for spherical cusp forms

Let Phi be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Phi is a Dirichlet series in r complex variables s_1,...,s_r, initially converging for Re(s_i) sufficiently large, which has meromorphic continuation to C^r and satisfies functional equations under the transformations of C^r corresponding to the Weyl group of Phi. Two constructions of such series are available, one based on summing products of n-th order Gauss sums, the second based on aver...

Let $G$ be the group of $\mathbb R$--points of a semisimple algebraic group $\mathcal G$ defined over $\mathbb Q$. Assume that $G$ is connected and noncompact. We study Fourier coefficients of Poincar\' e series attached to matrix coefficients of integrable discrete series. We use these results to construct explicit automorphic cuspidal realizations, which have appropriate Fourier coefficients $\neq 0$, of integrable discrete series in families of congruence subgroups. In the...

The description of irreducible representations of a group G can be seen as a question in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G x G by left and right multiplication. For a split p-adic reductive group G over a local non-archimedean field, unramified irreducible smooth representations are in bijection with semisimple conjugacy classes in the ``Langlands dual'' group. We generalize this description to ...

This work contains a proof of a non-trivial explicit quantitative bound in the eigenvalue aspect for the sup-norm of a SL(3,Z) Hecke-Maass cusp form restricted to a compact set.

This paper is concerned with a compatible family of 4-dimensional \ell-adic representations \rho_{\ell} of G_\Q:=\Gal(\bar \Q/\Q) attached to the space of weight 3 cuspforms S_3 (\Gamma) on a noncongruence subgroup \Gamma \subset \SL. For this representation we prove that: 1.)It is automorphic: the L-function L(s, \rho_{\ell}^{\vee}) agrees with the L-function for an automorphic form for \text{GL}_4(\mathbb A_{\Q}), where \rho_{\ell}^{\vee} is the dual of \rho_{\ell}. 2.) For...

We give new proofs of two basic results in number theory: The law of quadratic reciprocity and the sign of the Gauss sum. We show that these results are encoded in the relation between the discrete Fourier transform and the action of the Weyl element in the Weil representation modulo p,q and pq.

In this work we shall generalize the Selberg trace formula to a non-unitary finite-dimensional complex representation $\chi:\Gamma\rightarrow\operatorname{GL}(V)$ of a uniform lattice $\Gamma$ of a real Lie group $G$.

We introduce a new trace formula of Kuznetsov type involving the central standard L-values and the Whittaker periods of cuspidal automorphic representations of PGL_n(Q) which are spherical at the archimedean place. As an application, we show a simultaneous non-vanishing of standard L-values at n-1 points on the critical strip for infinitely many Hecke-Maass cuspidal newforms.

Let $G$ be a reductive $p$-adic group. We give a short proof of the fact that $G$ always admits supercuspidal complex representations. This result has already been established by A. Kret using the Deligne-Lusztig theory of representations of finite groups of Lie type. Our argument is of a different nature and is self-contained. It is based on the Harish-Chandra theory of cusp forms and it ultimately relies on the existence of elliptic maximal tori in $G$.

The main result of this article proves the nonvanishing of cuspidal cohomology for $GL(n)$ over a number field which is Galois over its maximal totally real subfield. The proof uses the internal structure of a strongly-pure weight that can possibly support cuspidal cohomology and the foundational work of Borel, Labesse, and Schwermer.