arXiv subjects

David Kazhdan

Publications and source records attributed to David Kazhdan.

120 records · Page 7Linked to original sources

arXiv2002

Gamma sheaves on reductive groups

The purpose of this paper is to introduce and study certain irreducible perverse l-adic sheaves on a reductive group G over a finite field (we call them gamma-sheaves). One can construct such a sheaf starting with (almost) every finite-dimensional representation of the Langlands dual group. We present conjecture connecting the above sheaves with generalized gamma-functions introduced in our previous paper. We also conjecture that the convolution functor with the above sheaves enjoys certain nice properties (in particular, we compute the convolution of a gamma-sheaf with a character sheaf). We prove the above conjectures for G of semi-simple rank 0 or 1 and (partially) for G=GL(n).

math.AG↗

arXiv2001

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 $n$).

math.AG↗

arXiv2000

Gamma-functions of representations and lifting

Let G be the group of points of a quasi-split reductive algebraic group over a local field F. It follows from the local Langlands conjectures that to every non-trivial additive character of F and every representation of the Langlands dual group of F one can associate certain meromorphic function on the set of isomorphism classed of irreducible representations of G (which we call gamma-functions). In this paper we describe a general framework for an explicit construction of these gamma-functions. We make this idea precise in certain special cases. As a byproduct we give explicit conjectural formulas for the representation $l_E(θ)$ of the group GL(n,F) associated to a character $θ$ of the multiplicative group $E^*$ of a separable extension E of F of degree n. We also describe a conjectural analogue of the Poisson summation formula associated to a representation of the Langlands dual group, which implies the existence of the meromorphic continuation and functional equation for the corresponding automorphic L-functions.

math.AG↗

arXiv2000

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.

math.AG↗

arXiv1999

Generalized character sums associated to regular prehomogeneous vector spaces

In this paper we consider the analogue of the Sato's functional equation for the prehomogeneous vector spaces over finite fields. The corresponding character sums depend on a relative invariant on such a space and an irreducible representation of the group of components of the stabilizer of a generic point. The proof is based on the Picard-Lefschetz formula in $l$-adique cohomology.

math.AG↗

arXiv1998

Versal deformations of formal arcs

Let X be a complex algebraic variety, and L(X) be the scheme of formal arcs in X. Let f be an arc whose image is not contained in the singularities of X. We show that the formal neighborhood of f in L(X) admits a decomposition into a product of an infinite-dimensional smooth piece, and a piece isomorphic to the formal neighborhood of a closed point of a scheme of finite type.

math.AG↗

arXiv1998

On the Schwartz space of the basic affine space

Let G be the group of points of a split reductive algebraic group over a local field k and let X=G/U where U is a maximal unipotent subgroup of G. In this paper we construct certain canonical G-invariant space S(X) (called the Schwartz space of X) of functions on X, which is an extension of the space of smooth compactly supported functions on X. We show that the space of all elements of S(X) invariant under the Iwahori subgroup of G coincides with space generated by the elements of the so called periodic Lusztig's basis, introduced recently by G.Lusztig. We also give an interpretation of this space in terms of certain equivariant K-group (this was also done by G.Lusztig). Finally we present a global analogue of S(X) which allows us to give a somewhat untraditional treatment of the theory of principal Eisenstein series.

math.AG↗

arXiv1998

Quantization of Lie bialgebras, IV

This paper is a continuation of "Quantization of Lie bialgebras, III" (q-alg/9610030, revised version). In QLB-III, we introduced the Hopf algebra F(R)_\z associated to a quantum R-matrix R(z) with a spectral parameter, and a set of points \z=(z_1,...,z_n). This algebra is generated by entries of a matrix power series T_i(u), i=1,...,n,subject to Faddeev-Reshetikhin-Takhtajan type commutation relations, and is a quantization of the group GL_N[[t]]. In this paper we consider the quotient F_0(R)_\z of F(R)_\z by the relations \qdet_R(T_i)=1, where \qdet_R is the quantum determinant associated to R (for rational, trigonometric, or elliptic R-matrices). This is also a Hopf algebra, which is a quantization of the group SL_N[[t]]. This paper was inspired by the pioneering paper of I.Frenkel and Reshetikhin. The main goal of this paper is to study the representation theory of the algebra F_0(R)_\z and of its quantum double, and show how the consideration of coinvariants of this double (quantum conformal blocks) naturally leads to the quantum Knizhnik-Zamolodchikov equations of Frenkel and Reshetikhin. Our construction for the rational R-matrix is a quantum analogue of the standard derivation of the Knizhnik-Zamolodchikov equations in the Wess-Zumino-Witten model of conformal field theory, and for the elliptic R-matrix is a quantum analogue of the construction of Kuroki and Takebe. Our result is a generalization of the construction of Enriques and Felder, which appeared while this paper was in preparation. Enriques and Felder gave a derivation of the quantum KZ equations from coinvariants in the case of the rational R-matrix and N=2.

math.QA↗

arXiv1998

Quantization of Lie bialgebras, V

This paper is a continuation of "Quantization of Lie bialgebras I-IV". The goal of this paper is to define and study the notion of a quantum vertex operator algebra in the setting of the formal deformation theory and give interesting examples of such algebras. In particular, we construct a quantum vertex operator algebra from a rational, trigonometric, or elliptic R-matrix, which is a quantum deformation of the affine vertex operator algebra. The simplest vertex operator in this algebra is the quantum current of Reshetikhin and Semenov-Tian-Shansky.

math.QA↗

arXiv1998

Quantization of Lie bialgebras, III

In this paper we construct explicitly the quantization of Lie bialgebras of $\g$-valued functions on a punctured rational or elliptic curve, where $\g$ is a finite dimensional simple Lie algebra. by reducing the problem of quantization of the algebra of $\g$-valued functions on a curve with many punctures to the case of one puncture.

q-alg↗

arXiv1998

Quantization of Lie bialgebras, II

This paper is a continuation of "Quantization of Lie bialgebras, I" (q-alg/9606005). We show that the quantization procedure defined in "Quantization of Lie bialgebras, I" is given by universal acyclic formulas and defines a functor from the category of Lie bialgebras to the category of quantized universal enveloping algebras. We also show that this functor defines an equivalence between the category of Lie bialgebras over k[[h]] and the category quantized universal enveloping (QUE) algebras.

q-alg↗

arXiv1995

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.

q-alg↗