Related papers: Special Values of L-functions for Orthogonal Group…

Assuming GRH and the Ramanujan-Petersson conjecture we prove explicit bounds for $L(1,f)$ for a large class of $L$-functions $L(s,f)$, which includes $L$-functions attached to automorphic cuspidal forms on $GL(n)$. The proof generalizes…

We introduce a family of L-series specialising to both L-series associated to certain Dirichlet characters over F_q[T] and to integral values of Carlitz-Goss zeta function associated to F_q[T]. We prove, with the use of the theory of…

We compute the cohomology with trivial coefficients of Lie algebras $\mathfrak{m}_0$ and $\mathfrak{m}_2$ of maximal class over the field $\mathbb{Z}_2$. In the infinite-dimensional case, we show that the cohomology rings…

The formal degree conjecture relates the formal degree of an irreducible square-integrable representation of a reductive group over a local field to the special value of the adjoint $\gamma$-factor of its $L$-parameter. In this paper, we…

In this article, we will prove that the formal degree conjecture is compatible with the Deligne-Kazhdan correspondence for quasi-split groups, assuming that the local Langlands correspondence is compatible with the Deligne-Kazhdan…

The cohomology groups of Lie superalgebras and, more generally, of color Lie algebras, are introduced and investigated. The main emphasis is on the case where the module of coefficients is non-trivial. Two general propositions are proved,…

A complete classification is established for continuous and SL(n) covariant matrix-valued valuations on Lp(Rn,|x|2dx). The assumption of matrix symmetry is eliminated. For n>2, such valuation is uniquely characterized by the moment matrix…

This is a companion article to my papers on Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebras gl(m|n) (much revised!) and q(n). The goal is to develop the general theory of tilting modules for Lie superalgebras,…

Let $\Pi$ be a cohomological cuspidal automorphic representation of ${\rm GL}_{2n}(\mathbb A)$ over a totally real number field $F$. Suppose that $\Pi$ has a Shalika model. We define a rational structure on the Shalika model of $\Pi_f$.…

In this paper, we relate $L(1,\pi,\mathrm{Ad}^\circ)$ to the congruence ideals for cohomological cuspidal automorphic representations $\pi$ of $\mathrm{GL}_n$ over any number field. We then use this result to deduce relationships between…

We consider arbitrary algebraic families of lower order deformations of nondegenerate toric exponential sums over a finite field. We construct a relative polytope with the aid of which we define a ring of coefficients consisting of p-adic…

A theory of Clebsch-Gordan coefficients for $SL(2, C)$ is given using only rational numbers. Features include orthogonality relations, recurrence relations, and Regge's symmetry group. Results follow from elementary representation theory…

Let G be the split special orthogonal group of degree 2n+1 over a field F of char F \ne 2. Then we describe G-orbits on the triple flag varieties G/P\times G/P\times G/P and G/P\times G/P\times G/B with respect to the diagonal action of G…

In this paper we will study certain models of irreducible admissible representations of the split special orthogonal group $SO(2n+1)$ over a nonarchimedean local field. If $n=1$, these models were considered by Waldspurger. If $n=2$, they…

We give a uniform interpretation of the classical continuous Chebyshev's and Hahn's orthogonal polynomials of discrete variable in terms of Feigin's Lie algebra gl(N), where N is any complex number. One can similarly interpret Chebyshev's…

We show that the additive Borcherds lift of vector-valued non-holomorphic Eisenstein series are orthogonal non-holomorphic Eisenstein series for $O(2, l)$. Using this we give another proof that they have a meromorphic continuation,…

We pursue the study of $\mathrm L^1$ full groups of graphings and of the closures of their derived groups, which we call derived $\mathrm L^1$ full groups. Our main result shows that aperiodic probability measure-preserving actions of…

In this paper we prove rationality results of critical values for $L$-functions attached to representations in the residual spectrum of $\mathrm{GL}_4(\mathbb{A})$. We use the Jacquet-Langlands correspondence to describe their partial…

The Green functions were first introduced by Green to compute the character table of GLn(q) in 1955. They were later generalized by Deligne and Lusztig for an arbitrary finite group of Lie type G(q) using l-adic cohomological methods…

We classify all connected subgroups of SO(2,n) that act irreducibly on $\R^{2,n}$. Apart from $SO_0(2,n)$ itself these are $U(1,n/2)$, $SU(1,n/2)$, if $n$ even, $S^1\cdot SO(1,n/2)$ if $n$ even and $n\ge 2$, and $SO_0(1,2)$ for $n=3$. Our…