Eitan Sayag

Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis's theorem

In the first part of the paper we generalize a descent technique due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over an arbitrary local field F of characteristic zero. Our main tool is the Luna Slice Theorem. In the second part of the paper we apply this technique to symmetric pairs. In particular we prove that the pairs (GL(n+k,F), GL(n,F) x GL(k,F)) and (GL(n,E), GL(n,F)) are Gelfand pairs for any local field F and its quadratic extension E. In the non-Archimedean case, the first result was proven earlier by Jacquet and Rallis and the second by Flicker. We also prove that any conjugation invariant distribution on GL(n,F) is invariant with respect to transposition. For non-Archimedean F the latter is a classical theorem of Gelfand and Kazhdan.

(GL n+1( F), GL n( F)) is a Gelfand pair for any local field F

Let F be an arbitrary local field. Consider the standard embedding of GL(n,F) into GL(n+1,F) and the two-sided action of GL(n,F) \times GL(n,F) on GL(n+1,F). In this paper we show that any GL(n,F) \times GL(n,F)-invariant distribution on GL(n+1,F) is invariant with respect to transposition. We show that this implies that the pair (GL(n+1,F),GL(n,F)) is a Gelfand pair. Namely, for any irreducible admissible representation

Finite Word Length Effects on Transmission Rate in Zero Forcing Linear Precoding for Multichannel DSL

(\pi,E)

Vanishing at infinity on homogeneous spaces of reductive type

of (GL(n+1,F),

z-Finite distributions on p-adic groups

dimHom_{GL(n,F)}(E,\cc) \leq 1.

Existence of Klyachko models for GL(n,R) and GL(n,C)

For the proof in the archimedean case we develop several new tools to study invariant distributions on smooth manifolds.

INVARIANT FUNCTIONALS ON SPEH REPRESENTATIONS

Crosstalk interference is the limiting factor in transmission over copper lines. Crosstalk cancellation techniques show great potential for enabling the next leap in DSL transmission rates. An important issue when implementing crosstalk cancelation techniques in hardware is the effect of finite word length on performance. In this paper, we provide an analysis of the performance of linear zero-forcing precoders, used for crosstalk compensation, in the presence of finite word length errors. We quantify analytically the tradeoff between precoder word length and transmission rate degradation. More specifically, we prove a simple formula for the transmission-rate loss as a function of the number of bits used for precoding, the signal-to-noise ratio, and the standard line parameters. We demonstrate, through simulations on real lines, the accuracy of our estimates. Moreover, our results are stable in the presence of channel estimation errors. Lastly, we show how to use these estimates as a design tool for DSL linear crosstalk precoders. For example, we show that for standard VDSL2 precoded systems, 14 bit representation of the precoder entries results in capacity loss below 1% for lines over 300 m.

Representation theory, complex analysis, and integral geometry

By the collective name of {\it lattice counting} we refer to a setup introduced in Duke-Rudnick-Sarnak that aim to establish a relationship between arithmetic and randomness in the context of affine symmetric spaces. In this paper we extend the geometric setup from symmetric to real spherical spaces and continue to develop the approach with harmonic analysis which was initiated in Duke-Rudnick-Sarnak.