Special Functions and Their Applications. By N. N. Lebedev, translated by R. A. Silverman. Pp. xii, 308. 96s. 1965. (Prentice-Hall)

Given a spherical variety X for a group G over a non-archimedean local field k, the Plancherel decomposition for L^2(X) should be related to "distinguished" Arthur parameters into a dual group closely related to that defined by Gaitsgory and Nadler. Motivated by this, we develop, under some assumptions on the spherical variety, a Plancherel formula for L^2(X) up to discrete (modulo center) spectra of its "boundary degenerations", certain G-varieties with more symmetries which model X at infinity. Along the way, we discuss the asymptotic theory of subrepresentations of C^{infty}(X), and establish conjectures of Ichino-Ikeda and Lapid-Mao. We finally discuss global analogues of our local conjectures, concerning the period integrals of automorphic forms over spherical subgroups.

Periods and harmonic analysis on spherical varieties

One of the most remarkable results of this century in mathematics has been the classification — completed in 1980 — of all the finite simple groups. This took over 20 years and occupies almost 5000 pages in the literature, and it is conceivable that there are some errors there, so the details of classification are not really available to us, but the main results can be summarized. There are 17 families of simple groups, the alternating groups and 16 families of “Lie type”. These in turn are broken up into subfamilies in several different ways. There are, first, the historical breakdowns, 6 families of “classical groups”, the projective special linear groups over finite fields, the orthogonal groups, (three types), unitary groups, and the symplectic groups. Then there are the 5 families of groups of “exceptional types”, the groups G 2(q), F 4(q), E 6 (q), E 7 (q), and E 8 (q), the q denoting the order of the finite field over which they are defined. Finally, there are five special families, the groups of “twisted type”, the Suzuki groups Sz[22n+1] the Ree groups 2 G 2[32n+1] and 2 F 2[22n+1], the twisted exceptional groups 2 E 6[22n+1], and the groups 3 D 4(q), the “triality twisted D 4 (q) ’s”. Of course, the most natural classification of these groups is via Lie theory and Dynkin diagrams. A complete discussion can be found in R.W. Carter’s book [Carter].

Finite Groups of Lie Type

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Periods of automorphic forms

An old question of Linnik asks about the equidistribution of integral points on a large sphere. This question proved to be very rich: it is intimately linked to modular forms, to subconvex estimates for L-functions, and to dynamics of torus actions on homogeneous spaces. Indeed, Linnik gave a partial answer using ergodic methods, and his question was completely answered by Duke using harmonic analysis and modular forms. We survey the context of these ideas and their developments over the last decades. Mathematics Subject Classification (2000).11F66; Secondary: 11F67, 11M41

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Equidistribution, L-functions and Ergodic Theory: on some problems of Y.V Linnik

A conjecture on Whittaker-Fourier coefficients of cusp forms

We prove a generalization to the totally real field case of the Waldspurger’s formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger’s formula as a combination of two ingredients – an equality between global distributions, and a dichotomy result for theta correspondence. As applications we generalize the Kohnen–Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindelof hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting.

Central Value Of Automorphic L-Functions

We prove a generalization to the totally real field case of the Waldspurger's formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger's formula in terms of equality between global distributions. As applications we generalize the Kohnen-Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindelof hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting.

Central value of automorphic $L-$functions

We prove certain identities between Bessel functions attached to irreducible unitary representations of PGL2(R) and Bessel functions attached to irreducible unitary representations of the double cover of SL2(R). These identities give a correspondence between such representations which turns out to be the Waldspurger correspondence. In the process we prove several regularity theorems for Bessel distributions which appear in the relative trace formula. In the heart of the proof lies a classical result of Weber and Hardy on a Fourier transform of classical Bessel functions. This paper constitutes the local (real) spectral theory of the relative trace formula for the Waldspurger correspondence for which the global part was developed by Jacquet. CONTENTS

Bessel identities in the waldspurger correspondence over the real numbers

We prove an explicit identity relating the central values of the twist L-functions of weight 0
Maass forms to the Fourier coefficients of weight ±1
2 Maass forms. We also give a bound on
Fourier coefficients of weight ±1
2 Maass forms.

Zhengyu Mao

Papers

A conjecture on Whittaker-Fourier coefficients of cusp forms

Central Value Of Automorphic L-Functions

Central value of automorphic $L-$functions

Bessel identities in the waldspurger correspondence over the real numbers

A generalized Kohnen–Zagier formula for Maass forms