On Shalika germs

Let $G$ be a reductive group over a local field $F$ satisfying the assumptions of \cite{Deb1}, $G_{reg}\subset G$ the subset of regular elements. Let $T\subset G$ be a maximal torus. We write $T_{reg}=T\cap G_{reg}$. Let $dg ,dt$ be Haar measures on $G$ and $T$. They define an invariant measure $dg/dt$ on $G/T$. Let $\mathcal{H}$ be the space of complex valued locally constant functions on $G$ with compact support. For any $f\in \mathcal{H} ,t\in T_{reg}$ we define $I_t(f)=\int_{G/T}f(\bar gt\bar g^{-1})dg/dt$. Let $P$ be the set of conjugacy classes of unipotent elements in $G$. For any $\Omega \in P$ we fix an invariant measure $\omega$ on $\Omega$. As well known \cite {R} for any $f\in \mathcal{H}$ the integral $$I_\Omega (f)=\int_\Omega f\omega$$ is absolutely convergent. Shalika \cite{Sh} has shown that there exist functions $\tilde{j}_\Omega (t),\Omega \in P$ on $T\cap G_{reg}$ such that $$I_t(f) = \sum_{\Omega \in P} \tilde{j}_\Omega(t) I_\Omega(f)\qquad\qquad (\star)$$ for any $f\in \mathcal{H} ,t\in T$ {\it near} to $e$ where the notion of {\it near} depends on $f$. For any positive real number $r$ one defines an open $Ad$-invariant subset $G_r$ of $G$ and a subspace $\mathcal{H}_r$ as in \cite{Deb1}. In this paper I show that for any $f\in \mathcal{H}_r$ the equality $(\star)$ is true for all $t\in T_{reg}\cap G_r$.Comment: 3 page

A construction of projective bases for irreducible representations of multiplicative groups of division algebras over local fields

Let $F$ be a local non-archimedian field of positive characteristic, $D$ be a skew-field with center $F$ and $G=D^{\star}$ be the multiplicative group of $D$. The goal of this paper is to provide a canonical decomposition of any complex irreducible representation $V$ of $G$ in a direct sum of one-dimensional subspaces

Fourier transform over finite field and identities between Gauss sums

This is a sequel to math.AG/0003009. Here we study identities for the Fourier transform of "elementary functions" over finite field containing "exponents" of monomial rational functions. It turns out that these identities are governed by monomial identities between Gauss sums. We show that similar to the case of complex numbers such identities correspond to linear relations between certain divisors on the space of multiplicative characters.Comment: 29 pages, AMSLate

On ranks of polynomials

Let $V$ be a vector space over a field $k, P:V\to k, d\geq 3$. We show the existence of a function $C(r,d)$ such that $rank (P)\leq C(r,d)$ for any field $k,char (k)>d$, a finite-dimensional $k$-vector space $V$ and a polynomial $P:V\to k$ of degree $d$ such that $rank(\partial P/\partial t)\leq r$ for all $t\in V-0$. Our proof of this theorem is based on the application of results on Gowers norms for finite fields $k$. We don't know a direct proof in the case when $k=\mathbb C$

Generalization of a theorem of Waldspurger to nice representations

A theorem of Waldspurger states that the Fourier transform of a stable distribution on the Lie algebra of a simply-connected semisimple group $G$ over a p-adic field, is again stable. We generalize this theorem to representations whose generic stabilizer subgroup is connected and reductive (assuming that $G$ is simple). In this more general situation the Fourier transform of a stable distribution is stable up to a sign that we describe explicitly. The proof is based on the $p$-adic stationary phase principle and on the global techniques introduced by Kottwitz for stabilization of the trace formula. As an application of our main theorem, we find the explicit diagonalization of the gamma-matrix for the prehomogeneous space of symmetric $n\times n$ matrices over a p-adic field (for odd $n$).Comment: 38 pages, AMSLate

Yoneda lemma for complete Segal spaces

In this note we formulate and give a self-contained proof of the Yoneda lemma for infinity categories in the language of complete Segal spaces.Comment: revised version, comments are welcom

Geometric approach to parabolic induction

In this note we construct a "restriction" map from the cocenter of a reductive group G over a local non-archimedean field F to the cocenter of a Levi subgroup. We show that the dual map corresponds to parabolic induction and deduce that parabolic induction preserves stability. We also give a new (purely geometric) proof that the character of normalized parabolic induction does not depend on a parabolic subgroup. In the appendix, we use a similar argument to extend a theorem of Lusztig-Spaltenstein on induced unipotent classes to all infinite fields.Comment: 29 pages, a grant acknowledgement is change

Endoscopic decomposition of characters of certain cuspidal representations

We construct an endoscopic decomposition for local L-packets associated to irreducible cuspidal Deligne-Lusztig representations. Moreover, the obtained decomposition is compatible with inner twistings.Comment: 12 pages, seriously revised versio

The spherical Hecke algebra for affine Kac-Moody groups I

We define the spherical Hecke algebra for an (untwisted) affine Kac-Moody group over a local non-archimedian field. We prove a generalization of the Satake isomorphism for these algebras, relating it to integrable representations of the Langlands dual affine Kac-Moody group. In the next publication we shall use these results to define and study the notion of Hecke eigenfunction for the group $G_{\aff}

Polynomial functions as splines

Let $V$ be a vector space over a finite field $k$. We give a condition on a subset $A \subset V$ that allows for a local criterion for checking when a function $f:A \to k$ is a restriction of a polynomial function of degree $<m$ on $V$. In particular, we show that high rank hypersurfaces of $V$ of degree $\ge m$ satisfy this condition. In addition we show that the criterion is robust (namely locally testable in the theoretical computer science jargon).Comment: Added a theorem on subspace splinin