Exposing relative endoscopy in unitary symmetric spaces

Transfer operators and Hankel transforms between relative trace formulas, II: Rankin–Selberg theory

Transfer operators and Hankel transforms between relative trace formulas, I: Character theory

We study Banach envelopes for commutative symmetric sequence or function spaces, and noncommutative symmetric spaces of measurable operators. We characterize the class (HC) of quasi-normed symmetric sequence or function spaces E for which their Banach envelopes \(\widehat{E}\) are also symmetric spaces. The class of symmetric spaces satisfying (HC) contains but is not limited to order continuous spaces. Let \(\mathcal {M}\) be a non-atomic, semifinite von Neumann algebra with a faithful, normal, \(\sigma \)-finite trace \(\tau \) and E be as symmetric function space on \([0,\tau (\mathbf 1 ))\) or symmetric sequence space. We compute Banach envelope norms on \(E(\mathcal {M},\tau )\) and \(C_E\) for any quasi-normed symmetric space E. Then we show under assumption that \(E\in (HC)\) that the Banach envelope \(\widehat{E(\mathcal {M},\tau )}\) of \(E(\mathcal {M},\tau )\) is equal to \(\widehat{E}\left( \mathcal {M},\tau \right) \) isometrically. We also prove the analogous result for unitary matrix spaces \(C_E\).

We study the relative trace formula of a reductive group over an algebraic number field. Following Langlands we stabilize the geometric side of the relative trace formula contributed by the elliptic regular double cosets.

The arithmetic Riemann–Roch theorem refines both the algebraic geometric and differential geometric counterparts, and it is stated within the formalism of Arakelov geometry. For some simple Shimura varieties and automorphic vector bundles, the cohomological part of the formula can be understood via the theory of automorphic representations. Functoriality principles from this theory may then be applied to derive relations between arithmetic intersection numbers for different Shimura varieties. In this lectures we explain this philosophy in the case of modular curves and compact Shimura curves. This indicates that there is some relationship between the arithmetic Riemann–Roch theorem and trace type formulae.

The exterior square L-function attached to an automorphic cuspidal representation of GL 2n has a pole if and only if a certain period integral does not vanish on the space of the representation.We conjecture, in the "if" direction, a similar result is true for representations of GL 2 ( D), where D is a division algebra.We prove a partial result which provides evidence for the conjecture.The proof is based on a relative trace formula.

Twisted Linear Periods and a New Relative Trace Formula

We elaborate an explicit version of the relative trace formula on ${\rm PGL}(2)$ over a totally real number field for the toral periods of Hilbert cusp forms along the diagonal split torus. As an application, we prove (i) a spectral equidistribution resul

On the relative trace formula

We use a relative trace formula on $\text{GL}(2)$ to compute a sum of twisted modular $L$-functions anywhere in the critical strip, weighted by a Fourier coefficient and a Hecke eigenvalue. When the weight $k$ or level $N$ is sufficiently large, the sum is nonzero. Specializing to the central point, we show in some cases that the resulting bound for the average is as good as that predicted by the Lindelöf hypothesis in the $k$ and $N$ aspects.

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.Author: Bismut, Jean-Michel Publisher: Princeton University Press Illustration: N Language: ENG Title: Hypoelliptic Laplacian and Orbital Integrals (AM-177) Pages: 00320 (Encrypted EPUB) / 00320 (Encrypted PDF) On Sale: 2011-08-28 SKU-13/ISBN: 9780691151304 Category: Mathematics : Geometry Analytic Category: Mathematics : Mathematical Analysis Category: Mathematics : Matrices

In the last two decades much of the algebraic/combinatorial structure of Lie groups, Lie algebras, and their representations has been implemented in several excellent computer algebra packages, including LiE, GAP4, Chevie, Magma, and Maple. The structure of reductive symmetric spaces or more generally symmetric k-varieties is very similar to that of the underlying Lie group, with a few additional complications. A computer algebra package enabling one to do computations related to these symmetric spaces would be an important tool for researchers in many areas of mathematics, including representation theory, Harish Chandra modules, singularity theory, differential and algebraic geometry, mathematical physics, character sheaves, Lie theory, etc. In this article we lay the groundwork for computing the fine structure of symmetric spaces over the real numbers and other base fields, give a complete set of algorithms for computing the fine structure of symmetric varieties and use this to compute nice bases for the local symmetric varieties.

This note investigates the multiplicity problem of principal curvatures of equifocal hypersurfaces in simply connected rank 1 symmetric spaces. Using Clifford representation theory, and the author also constructs infinitely many equifocal hypersurfaces in the symmetric spaces.

In this paper we calculate an invariant in W(Z2), the Witt ring of nonsingular, symmetric Z2-inner product spaces, for orientation-preserving involutions on compact, closed, connected 4n-dimensional manifolds M. This invariant with the Atiyah-Singer index theorem uniquely determines the orthogonal representation of Z on H2n(M; Z)/TOR. We also give an example to show that this invariant detects actions that the Atiyah-Singer theorem cannot. In this paper we calculate a torsion invariant for orientation-preserving involutions. This invariant and the Atiyah-Singer-Segal G-signature theorem allow one to compute precisely the element of W(Z; Z2) given by the orthogonal representation of Z2 on H2n(M; Z)/TOR. This cannot be done with the Atiyah-Singer-Segal theorem alone. We will return to this in the last section. In [4] Conner and Raymond define an invariant q(T, M) for orientationpreserving actions of Zp, p a prime, T a generator of Zp, on closed, compact, oriented manifolds M of dimension 4n. Briefly the action of Zp on M gives us an orthogonal representation of Z on H2n(M; Q). If we denote by w(T, M) and sgn(T, M) the rational Witt class and signature, respectively, of the inner product (x, y)p(x u y, [M]) on the subspace of fixed vectors then q(T, M) = w(T, M) sgn(T, M)1. In [1] the problem of expressing q(T, M) in terms of fixed point information was solved for P odd. In this paper we compute q(T, M) for p = 2 and give some applications. First some notation and background. W(R) will always denote the Witt group of nonsingular, symmetric, inner-product spaces over R, R a ring. W(Z2) is isomorphic to Z2 and the isomorphism is given by taking the rank of the inner product space mod 2. From now on we identify W(Z2) and Z2 by this isomorphism. If V is a disjoint union of a finite number of closed Received by the editors June 11, 1974. AMS (MOS) subject classifications (1970). Primary 57D85; Secondary IOC05.

Local Orbital Integrals of a Relative Trace Formula