168 research outputs found
On Shalika germs
Let be a reductive group over a local field satisfying the
assumptions of \cite{Deb1}, the subset of regular elements.
Let be a maximal torus. We write . Let be Haar measures on and . They define an invariant measure
on . Let be the space of complex valued locally constant
functions on with compact support. For any
we define . Let be the set of
conjugacy classes of unipotent elements in . For any we fix
an invariant measure on . As well known \cite {R} for any
the integral is
absolutely convergent. Shalika \cite{Sh} has shown that there exist functions
on such that for
any {\it near} to where the notion of {\it near}
depends on . For any positive real number one defines an open
-invariant subset of and a subspace as in
\cite{Deb1}. In this paper I show that for any the
equality is true for all .Comment: 3 page
A construction of projective bases for irreducible representations of multiplicative groups of division algebras over local fields
Let be a local non-archimedian field of positive characteristic, be a
skew-field with center and be the multiplicative group of
. The goal of this paper is to provide a canonical decomposition of any
complex irreducible representation of 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 be a vector space over a field . We show the
existence of a function such that for any field
, a finite-dimensional -vector space and a polynomial
of degree such that for all
. Our proof of this theorem is based on the application of results on
Gowers norms for finite fields . We don't know a direct proof in the case
when
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 over
a p-adic field, is again stable. We generalize this theorem to representations
whose generic stabilizer subgroup is connected and reductive (assuming that
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 -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 matrices
over a p-adic field (for odd ).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
Quantization of Poisson algebraic groups and Poisson homogeneous spaces
This paper consists of two parts. In the first part we show that any Poisson
algebraic group over a field of characteristic zero and any Poisson Lie group
admits a local quantization. This answers positively a question of Drinfeld. In
the second part we apply our techniques of quantization to obtain some
nontrivial examples of quantization of Poisson homogeneous spaces.Comment: 9 pages, amstex. The revised version contains a new referenc
Representations of affine Kac-Moody groups over local and global fields: a survey of some recent results
Let G be a reductive algebraic group over a local field K or a global field
F. It is well know that there exists a non-trivial and interesting
representation theory of the group G(K) as well as the theory of automorphic
forms on the corresponding adelic group. The purpose of this paper is to give a
survey of some recent constructions and results, which show that there should
exist an analog of the above theories in the case when G is replaced by the
corresponding affine Kac-Moody group (which is essentially built from the
formal loop group G((t)) of G). Specifically we discuss the following topics :
affine (classical and geometric) Satake isomorphism, affine Iwahori-Hecke
algebra, affine Eisenstein series and Tamagawa measure.Comment: To appear in the Proceedings of 6th European Congress of
Mathematician
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}
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