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David Kazhdan

Publications and source records attributed to David Kazhdan.

At least 109 records · Page 6Linked to original sources
arXiv2012

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.

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arXiv2011

Affine Gindikin-Karpelevich formula via Uhlenbeck spaces

We prove a version of the Gindikin-Karpelevich formula for untwisted affine Kac-Moody groups over a local field of positive characteristic. The proof is geometric and it is based on the results of [1] about intersection cohomology of certain Uhlenbeck-type moduli spaces (in fact, our proof is conditioned upon the assumption that the results of [1] are valid in positive characteristic). In particular, we give a geometric explanation of certain combinatorial differences between finite-dimensional and affine case (observed earlier by Macdonald and Cherednik), which here manifest themselves by the fact that the affine Gindikin-Karpelevich formula has an additional term compared to the finite-dimensional case. Very roughy speaking, that additional term is related to the fact that the loop group of an affine Kac-Moody group (which roughly speaking should be thought of as some kind of "double loop group") does not behave well from algebro-geometric point of view; however it has a better behaved version which has something to do with algebraic surfaces. A uniform (i.e. valid for all local fields) and unconditional (but not geometric) proof of the affine Gindikin-Karpelevich formula is going to appear in [2].

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arXiv2010

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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arXiv2010

The Tannakian Formalism and the Langlands Conjectures

Let H be a connected reductive group over an algebraically closed field of characteristic zero, and let G be an abstract group. In this note we show that every homomorphism from the Grothendieck semiring of H to that of G which maps irreducible representations to irreducibles, comes from a group homomorphism from G to H. We also connect this result with the Langlands conjectures.

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arXiv2009

On endoscopic transfer of Deligne-Lusztig functions

In this paper we prove the fundamental lemma for Deligne-Lusztig functions. Namely, for every Deligne-Lusztig function $ϕ$ on a $p$-adic group $G$ we write down an explicit linear combination $ϕ^H$ of Deligne-Lusztig functions on an endoscopic group $H$ such that $ϕ$ and $ϕ^H$ have ``matching orbital integrals''. In particular, we prove a conjecture of Kottwitz. More precisely, we do it under some mild restriction on $p$.

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arXiv2009

Motivic Poisson summation

We develop a "motivic integration" version of the Poisson summation formula for function fields, with values in the Grothendieck ring of definable exponential sums. We also study division algebras over the function field, and obtain relations among the motivic Fourier transforms of a test function at different completions. We use these to prove, in a special case, a motivic version of a theorem of Deligne-Kazhdan-Vigneras.

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arXiv2008

Quantization of Lie bialgebras, part VI: quantization of generalized Kac-Moody algebras

This paper is a continuation of the series of papers "Quantization of Lie bialgebras (QLB) I-V". We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in QLB-I,II is isomorphic to the Drinfeld-Jimbo quantization of this Lie bialgebra, with the standard quasitriangular structure. This implies that when the quantization parameter is formal, then the category O for the quantized Kac-Moody algebra is equivalent, as a braided tensor category, to the category O over the corresponding classical Kac-Moody algebra, with the tensor category structure defined by a Drinfeld associator. This equivalence is a generalization of the functor constructed previously by G.Lusztig and the second author. In particular, we answer positively a question of Drinfeld whether the characters of irreducible highest weight modules for quantized Kac-Moody algebras are the same as in the classical case. Moreover, our results are valid for the Lie algebra g(A) corresponding to any symmetrizable matrix A (not necessarily with integer entries), which answers another question of Drinfeld. We also prove the Drinfeld-Kohno theorem for the algebra g(A) (it was previously proved by Varchenko using integral formulas for solutions of the KZ equations).

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arXiv2007

Geometric and unipotent crystals II: From unipotent bicrystals to crystal bases

For each reductive algebraic group G we introduce and study unipotent bicrystals which serve as a regular version of birational geometric and unipotent crystals introduced earlier by the authors. The framework of unipotent bicrystals allows, on the one hand, to study systematically such varieties as Bruhat cells in G and their convolution products and, on the other hand, to give a new construction of many normal Kashiwara crystals including those for G^\vee-modules, where G^\vee is the Langlands dual groups. In fact, our analogues of crystal bases (which we refer to as crystals associated to G^\vee-modules) are associated to G^\vee-modules directly, i.e., without quantum deformations. One of the main results of the present paper is an explicit construction of the crystal B_0 for the coordinate ring of the (Langlands dual) flag variety based on the positive unipotent bicrystal on the open Bruhat cell. Our general tropicalization procedure assigns to each strongly positive unipotent bicrystal a normal Kashiwara crystal B equipped with the multiplicity erasing homomorphism B--> B_0 and the combinatorial central charge B--> Z which is invariant under all crystal operators.

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arXiv2006

Integration in valued fields

We develop a theory of integration over valued fields of residue characteristic zero. In particular we obtain new and base-field independent foundations for integration over local fields of large residue characteristic, extending results of Denef,Loeser, Cluckers. The method depends on an analysis of definable sets up to definable bijections. We obtain a precise description of the Grothendieck semigroup of such sets in terms of related groups over the residue field and value group. This yields new invariants of all definable bijections, as well as invariants of measure preserving bijections.

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arXiv2006

Geometric and unipotent crystals

In this paper we introduce geometric crystals and unipotent crystals which are algebro-geometric analogues of Kashiwara's crystal bases. Given a reductive group G, let I be the set of vertices of the Dynkin diagram of G and T be the maximal torus of G. The structure of a geometric G-crystal on an algebraic variety X consists of a rational morphism γ:X-->T and a compatible family e_i:G_m\times X-->X, i\in I of rational actions of the multiplicative group G_m satisfying certain braid-like relations. Such a structure induces a rational action of W on X. Quite surprisingly, many interesting rational actions of the group W come from geometric crystals. Also all the known examples of the action of W which appear in the construction of gamma-functions for the representations of ^LG in the recent work by A. Braverman and D. Kazhdan come from geometric crystals. There are many examples of positive geometric crystals on (G_m)^l, i.e., those geometric crystals for which the actions e_i and the morphism γare given by positive rational expressions. To each positive geometric crystal X we associate a Kashiwara's crystal corresponding to the Langlands dual group ^LG. An emergence of ^LG in the "crystal world" was observed earlier by G. Lusztig. Another application of geometric crystals is a construction of trivialization which is an W-equivariant isomorhism X-->γ^{-1}(e) \times T for any geometric SL_n-crystal. Unipotent crystals are geometric analogues of normal Kashiwara crystals. They form a strict monoidal category. To any unipotent crystal built on a variety X we associate a certain gometric crystal.

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arXiv2005

Some examples of Hecke algebras over 2-dimensional local fields

Let K be a local non-archimedian field, F=K((t)) and let G be a split semi-simple group. The purpose of this paper is to study certain analogs of spherical (and Iwahori) Hecke algebras for representations of the group G(F) and its central extension by means of K*. For instance our spherical Hecke algebra corresponds to the subgroup G(A) where A is the subring $\calO_K((t))$ where $\calO_K\subset K$ is the ring of integers. It turns out that for generic level the spherical Hecke algebra is trivial; however, on the critical level it is quite large. On the other hand we expect that the size of the corresponding Iwahori-Hecke algebra does not depend on a choice of a level (details will be considered in another publication).

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arXiv2005

Algebraic groups over a 2-dimensional local field: some further constructions

In math.RT/0302174 we developed a framework to study representations of groups of the form $G((t))$, where $G$ is an algebraic group over a local field $K$. The main feature of this theory is that natural representations of groups of this kind are not on vector spaces, but rather on pro-vector spaces. In this paper we present some further constructions related to this theory. The main results include: 1) General theorems insuring representability of covariant functors, 2) Study of the functor of semi-invariants, which is an analog of the functor of semi-infinite cohomology for infinite-dimensional Lie algebras, 3) Construction of representations from the moduli space of $G$-bundles on algebraic curve over $K$.

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arXiv2005

Algebraic groups over a 2-dimensional local field: irreducibility of certain induced representations

Let $G$ be a split reductive group over a local field $\bK$, and let $G((t))$ be the corresponding loop group. In \cite{GK} we have introduced the notion of a representation of (the group of $\bK$-points) of $G((t))$ on a pro-vector space. In addition, we have defined an induction procedure, which produced $G((t))$-representations from usual smooth representations of $G$. We have conjectured that the induction of a cuspidal irreducible representation of $G$ is irreducible. In this paper we prove this conjecture for $G=SL_2$.

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arXiv2002

Minimal representations: spherical vectors and automorphic functionals

In the first part of this paper we study minimal representations of simply connected simple split groups of type $D_k$ or $E_k$ over local non-archimedian fields. Our main result is an explicit formula for the spherical vectors in these representations. In the case of real and complex groups such a formula was obtained recently in hep-th/0107222. We also use our techniques to study the structure of the space of smooth vectors in the minimal representation recovering some results of Magaard and Savin. In the second part we consider groups as above defined over a global field. In this situation we describe the form of the automorphic functional on the minimal representation of the corresponding adelic group.

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arXiv2002

Normalized intertwining operators and nilpotent elements in the Langlands dual group

Let $F$ be a local non-archimedian field and let $G$ be a group of points of a split reductive group over $F$. For a parabolic subgroup $P$ of $G$ we set $X_P=G/[P,P]$. For any two parabolics $P$ and $Q$ with the same Levi component $M$ we construct an explicit unitary isomorphism $L^2(X_P)\to L^2(X_Q)$ (which depends on a choice of an additive character of $F$). The formula for the above isomorphism involves the action of the principal nilpotent element in the Langlands dual group of $M$ on the unipotent radicals of the corresponding dual parabolics. We use the above isomorphisms to define a new space $\calS(G,M)$ of functions on $X_P$ (which depends only on $P$ and not on $M$). We explain how this space may be applied in order to reformulate in a slightly more elegant way the construction of $L$-functions associated with the standard representation of a classical group due to Gelbart, Piatetski-Shapiro and Rallis.

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