Let G be a reductive p-adic group. Let Φ be an invariant distribution on G lying in the Bernstein... more Let G be a reductive p-adic group. Let Φ be an invariant distribution on G lying in the Bernstein center Z(G). We prove that Φ is supported on compact elements in G if and only if it defines a constant function on every component of the set Irr(G); in particular, we show that the space of all elements of Z(G) supported on compact elements is a subalgebra of Z(G). Our proof is a slight modification of the argument from Section 2 of , where our result is proven in one direction.
We present a conjecture (and a proof for G = SL(2)) generalizing a result of J. Arthur which expr... more We present a conjecture (and a proof for G = SL(2)) generalizing a result of J. Arthur which expresses a character value of a cuspidal representation of a p-adic group as a weighted orbital integral of its matrix coefficient. It also generalizes a conjecture by the second author proved by Schneider-Stuhler and (independently) the first author. The latter statement expresses an elliptic character value as an orbital integral of a pseudo-matrix coefficient defined via the Chern character map taking value in zeroth Hochschild homology of the Hecke algebra. The present conjecture generalizes the construction of pseudomatrix coefficient using compactly supported Hochschild homology, as well as a modification of the category of smooth representations, the so called compactified category of smooth G-modules. This newly defined "compactified pseudo-matrix coefficient" lies in a certain space K on which the weighted orbital integral is a conjugation invariant linear functional, our conjecture states that evaluating a weighted orbital integral on the compactified pseudo-matrix coefficient one recovers the corresponding character value of the representation. We also discuss the properties of the averaging map from K to the space of invariant distributions, partly building on works of Waldspurger and Beuzart-Plessis.
This paper is a continuation of in which the first two authors have introduced the spherical Heck... more This paper is a continuation of in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori-Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik's double affine Hecke algebra. Furthermore, using these results, we give an explicit description of the affine Satake isomorphism, generalizing Macdonald's formula for the spherical function in the finite-dimensional case. The results of this paper have been previously announced in .
This paper is the written version of D. Kazhdan's plenary talk at ICM 2022. It is dedicated to an... more This paper is the written version of D. Kazhdan's plenary talk at ICM 2022. It is dedicated to an exposition of recent results and (mostly) conjectures attempting to construct an analog of the theory of automorphic functions on moduli spaces of bundles on curves over local fields (both archimedian and nonarchimedian). The talk is based on joint works of D. Kazhdan with A.Braverman, P.Etingof, E.Frenkel and A.Polishchuk.
Let G be a reductive group and X a smooth projective curve. We prove that, for G classical and σ ... more Let G be a reductive group and X a smooth projective curve. We prove that, for G classical and σ an arbitrary G-local system on X, the space Op gen G,σ of generic extended oper structures on σ is homologically contractible. This contractibility result is crucial for the proof of the geometric Langlands conjecture. gen G,σ .
Let F be a local non-archimedian field and G be the group of F -points of a split connected reduc... more Let F be a local non-archimedian field and G be the group of F -points of a split connected reductive group over F . In we define an algebra J (G) of functions on G which contains the Hecke algebra H(G) and is contained in the Harish-Chandra Schwartz algebra C(G). We consider J (G) as an algebraic analog the algebra C(G). Given a parabolic subgroup P of G with a Levi subgroup M and the unipotent radical U P we write X P := G/U P . Let S c (X P ) be the space of locally constant functions on X P with compact support and S cusp,c (X P ) ⊂ S c (X P ) be subspace of functions whose right shifts span a cuspidal representation of M . In this paper we study the space S(X P ) := J (S c (X P )) and show that (a) For two parabolics P, Q with the same Levi subrgoups the normalized intertwining operator I Q,P defines and isomorphism of (J (G), H(M ))-bimodules S(X P ) cusp and S(X Q ) cusp . (b) When P is a Borel subgroup the space S(X P ) is equal to the Schwartz space studied in .
We prove a version of the Gindikin-Karpelevich formula for untwisted affine Kac-Moody groups over... more 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 are valid in positive characteristic; we believe that generalizing to the case of positive characteristic should be essentially straightforward but we have not checked the details). 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 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 .
Let G be a reductive p-adic group. Let Φ be an invariant distribution on G lying in the Bernstein... more Let G be a reductive p-adic group. Let Φ be an invariant distribution on G lying in the Bernstein center Z(G). We prove that Φ is supported on compact elements in G if and only if it defines a constant function on every component of the set Irr(G); in particular, we show that the space of all elements of Z(G) supported on compact elements is a subalgebra of Z(G). Our proof is a slight modification of the argument from Section 2 of , where our result is proven in one direction.
Let G be a reductive algebraic group over a local field K or a global field F . It is well known ... more Let G be a reductive algebraic group over a local field K or a global field F . It is well known 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 adelic group G(AF ). The purpose of this talk 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 G aff (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, Iwahori-Hecke algebra of G aff , affine Eisenstein series and Tamagawa measure.
We define the spherical Hecke algebra for an (untwisted) affine Kac-Moody group over a local non-... more 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 this algebra, 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 .
We define the spherical Hecke algebra for an (untwisted) affine Kac-Moody group over a local non-... more 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 . 1 This group-scheme is non-reduced if G is not semi-simple
We use geometry of the wonderful compactification to obtain a new proof of the relation between D... more We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne-Lusztig (or Alvis-Curtis) duality for p-adic groups and homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for G = GL(n) and by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct, we describe the Serre functor for representations of a p-adic group. To Sasha Beilinson, with admiration and best wishes for his birthday.
The geometric Langlands program is an exciting direction of research, although a lot of progress ... more The geometric Langlands program is an exciting direction of research, although a lot of progress has been made there are still open questions and gaps. One of the necessary steps for this program is the proof of the existence and the cohomological triviality (more precisely, the O- contractibility) of the space of rational opers. In this paper, we prove the homotopical contractibility of the space of rational opers for classical groups.
This paper is a continuation of the series [EK1-5]. We show that the image of a Kac-Moody Lie bia... more This paper is a continuation of the series [EK1-5]. We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in [EK1,EK2] 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 in [KL]. In particular, we answer positively question 8.1 from [Dr1]: we show that 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). This answers question 8.3 in [Dr1], of Drinfeld and Gelfand. We also prove the Drinfeld-Kohno theorem for the algebra g(A) (it was previously proved by Varchenko [V] using integral formulas for solutions of the KZ equations). Acnowledgments. The authors were supported by the NSF grant DMS-9700477. 2. Generalized Kac-Moody algebras Throughout the paper, k denotes an algebraically closed field of characteristic zero. All vector spaces in this paper will be over k. 2.1. We recall definitions from [K]. Let A = (a ij) be an n-by-n matrix with entries in k, and (h, Π, Π ∨) be a realization of A. This means that h is a vector space of dimension 2n − rank(A), Π = {α 1 , ..., α n } ⊂ h * , Π ∨ = {h 1 , ..., h n } ⊂ h are linearly independent, and α i (h j) = a ji. Definition. The Lie algebrag(A) is generated by h, e 1 , ..., e n , f 1 , ..., f n with defining relations [h, h ′ ] = 0, h, h ′ ∈ h; [h, e i ] = α i (h)e i ; [h, f i ] = −α i (h)f i ; [e i , f j ] = δ ij h i .
Let k be a perfect field and let X ⊂ P N be a hypersurface of degree d defined over k and contain... more Let k be a perfect field and let X ⊂ P N be a hypersurface of degree d defined over k and containing a linear subspace L defined over k with codim P N L = r. We show that X contains a linear subspace L 0 defined over k with codim P N L ≤ dr. We conjecture that the intersection of all linear subspaces (over k) of minimal codimension r contained in X, has codimension bounded above only in terms of r and d. We prove this when either d ≤ 3 or r ≤ 2.
For a split reductive group L G over a global field, we determine the spectrum of the spherical H... more For a split reductive group L G over a global field, we determine the spectrum of the spherical Hecke algebra coming from the unramified Eisenstein series for the minimal parabolic L B. This is done using a certain decomposition of the Springer stack T˚pBzG{Bq for the Langlands dual group G in the additive group of cobordisms of cohomologically proper derived quotient stacks. Contents
Consider a complex affine variety $\tilde V$ and a real analytic Zariski-dense submanifold V of $... more Consider a complex affine variety $\tilde V$ and a real analytic Zariski-dense submanifold V of $\tilde V$. We compare modules over the ring $O (\tilde V)$ of regular functions on $\tilde V$ with modules over the ring $C^\infty (V)$ of smooth complex valued functions on V. Under a mild condition on the tangent spaces, we prove that $C^\infty (V)$ is flat as a module over $O (\tilde V)$. From this we deduce a comparison theorem for the Hochschild homology of finite type algebras over $O (\tilde V)$ and the Hochschild homology of similar algebras over $C^\infty (V)$. We also establish versions of these results for functions on $\tilde V$ (resp. V) that are invariant under the action of a finite group G. When V is compact, we check that $C^\infty (V)$ has finite rank as module over $C^\infty (V)^G$.
Lie Groups, Geometry, and Representation Theory, 2018
Let G be a split reductive p-adic group, I ⊂ G be an Iwahori subgroup, H(G) be the Hecke algebra ... more Let G be a split reductive p-adic group, I ⊂ G be an Iwahori subgroup, H(G) be the Hecke algebra and C(G) ⊃ H(G) be the Harish-Chandra Schwartz algebra. The purpose of this note is to define (in spectral terms) a subalgebra J (G) of C(G), containing H(G), which we consider as an algebraic version of C(G). We show that the subalgebra J (G) I×I ⊂ J (G) is isomorphic to the Lusztig's asymptotic Hecke algebra J and explain a relation between the algebra J (G) and the Schwartz space of the basic affine space studied in [2].
Let k be a perfect field and let X⊂ℙ^N be a hypersurface of degree d defined over k and containin... more Let k be a perfect field and let X⊂ℙ^N be a hypersurface of degree d defined over k and containing a linear subspace L defined over an algebraic closure k with codim_ℙ^NL=r. We show that X contains a linear subspace L_0 defined over k with codim_ℙ^NL≤ dr. Furthermore, we propose an explicit algorithm for finding L_0 (with a worse estimate on codimension) from L and its Galois conjugates. We also prove a similar result for a certain version of the Schmidt rank for quartic polynomials.
This paper is a continuation of "Quantization of Lie bialgebras I-IV". The goal of this... more 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.
Let G be a reductive p-adic group. Let Φ be an invariant distribution on G lying in the Bernstein... more Let G be a reductive p-adic group. Let Φ be an invariant distribution on G lying in the Bernstein center Z(G). We prove that Φ is supported on compact elements in G if and only if it defines a constant function on every component of the set Irr(G); in particular, we show that the space of all elements of Z(G) supported on compact elements is a subalgebra of Z(G). Our proof is a slight modification of the argument from Section 2 of , where our result is proven in one direction.
We present a conjecture (and a proof for G = SL(2)) generalizing a result of J. Arthur which expr... more We present a conjecture (and a proof for G = SL(2)) generalizing a result of J. Arthur which expresses a character value of a cuspidal representation of a p-adic group as a weighted orbital integral of its matrix coefficient. It also generalizes a conjecture by the second author proved by Schneider-Stuhler and (independently) the first author. The latter statement expresses an elliptic character value as an orbital integral of a pseudo-matrix coefficient defined via the Chern character map taking value in zeroth Hochschild homology of the Hecke algebra. The present conjecture generalizes the construction of pseudomatrix coefficient using compactly supported Hochschild homology, as well as a modification of the category of smooth representations, the so called compactified category of smooth G-modules. This newly defined "compactified pseudo-matrix coefficient" lies in a certain space K on which the weighted orbital integral is a conjugation invariant linear functional, our conjecture states that evaluating a weighted orbital integral on the compactified pseudo-matrix coefficient one recovers the corresponding character value of the representation. We also discuss the properties of the averaging map from K to the space of invariant distributions, partly building on works of Waldspurger and Beuzart-Plessis.
This paper is a continuation of in which the first two authors have introduced the spherical Heck... more This paper is a continuation of in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori-Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik's double affine Hecke algebra. Furthermore, using these results, we give an explicit description of the affine Satake isomorphism, generalizing Macdonald's formula for the spherical function in the finite-dimensional case. The results of this paper have been previously announced in .
This paper is the written version of D. Kazhdan's plenary talk at ICM 2022. It is dedicated to an... more This paper is the written version of D. Kazhdan's plenary talk at ICM 2022. It is dedicated to an exposition of recent results and (mostly) conjectures attempting to construct an analog of the theory of automorphic functions on moduli spaces of bundles on curves over local fields (both archimedian and nonarchimedian). The talk is based on joint works of D. Kazhdan with A.Braverman, P.Etingof, E.Frenkel and A.Polishchuk.
Let G be a reductive group and X a smooth projective curve. We prove that, for G classical and σ ... more Let G be a reductive group and X a smooth projective curve. We prove that, for G classical and σ an arbitrary G-local system on X, the space Op gen G,σ of generic extended oper structures on σ is homologically contractible. This contractibility result is crucial for the proof of the geometric Langlands conjecture. gen G,σ .
Let F be a local non-archimedian field and G be the group of F -points of a split connected reduc... more Let F be a local non-archimedian field and G be the group of F -points of a split connected reductive group over F . In we define an algebra J (G) of functions on G which contains the Hecke algebra H(G) and is contained in the Harish-Chandra Schwartz algebra C(G). We consider J (G) as an algebraic analog the algebra C(G). Given a parabolic subgroup P of G with a Levi subgroup M and the unipotent radical U P we write X P := G/U P . Let S c (X P ) be the space of locally constant functions on X P with compact support and S cusp,c (X P ) ⊂ S c (X P ) be subspace of functions whose right shifts span a cuspidal representation of M . In this paper we study the space S(X P ) := J (S c (X P )) and show that (a) For two parabolics P, Q with the same Levi subrgoups the normalized intertwining operator I Q,P defines and isomorphism of (J (G), H(M ))-bimodules S(X P ) cusp and S(X Q ) cusp . (b) When P is a Borel subgroup the space S(X P ) is equal to the Schwartz space studied in .
We prove a version of the Gindikin-Karpelevich formula for untwisted affine Kac-Moody groups over... more 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 are valid in positive characteristic; we believe that generalizing to the case of positive characteristic should be essentially straightforward but we have not checked the details). 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 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 .
Let G be a reductive p-adic group. Let Φ be an invariant distribution on G lying in the Bernstein... more Let G be a reductive p-adic group. Let Φ be an invariant distribution on G lying in the Bernstein center Z(G). We prove that Φ is supported on compact elements in G if and only if it defines a constant function on every component of the set Irr(G); in particular, we show that the space of all elements of Z(G) supported on compact elements is a subalgebra of Z(G). Our proof is a slight modification of the argument from Section 2 of , where our result is proven in one direction.
Let G be a reductive algebraic group over a local field K or a global field F . It is well known ... more Let G be a reductive algebraic group over a local field K or a global field F . It is well known 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 adelic group G(AF ). The purpose of this talk 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 G aff (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, Iwahori-Hecke algebra of G aff , affine Eisenstein series and Tamagawa measure.
We define the spherical Hecke algebra for an (untwisted) affine Kac-Moody group over a local non-... more 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 this algebra, 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 .
We define the spherical Hecke algebra for an (untwisted) affine Kac-Moody group over a local non-... more 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 . 1 This group-scheme is non-reduced if G is not semi-simple
We use geometry of the wonderful compactification to obtain a new proof of the relation between D... more We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne-Lusztig (or Alvis-Curtis) duality for p-adic groups and homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for G = GL(n) and by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct, we describe the Serre functor for representations of a p-adic group. To Sasha Beilinson, with admiration and best wishes for his birthday.
The geometric Langlands program is an exciting direction of research, although a lot of progress ... more The geometric Langlands program is an exciting direction of research, although a lot of progress has been made there are still open questions and gaps. One of the necessary steps for this program is the proof of the existence and the cohomological triviality (more precisely, the O- contractibility) of the space of rational opers. In this paper, we prove the homotopical contractibility of the space of rational opers for classical groups.
This paper is a continuation of the series [EK1-5]. We show that the image of a Kac-Moody Lie bia... more This paper is a continuation of the series [EK1-5]. We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in [EK1,EK2] 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 in [KL]. In particular, we answer positively question 8.1 from [Dr1]: we show that 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). This answers question 8.3 in [Dr1], of Drinfeld and Gelfand. We also prove the Drinfeld-Kohno theorem for the algebra g(A) (it was previously proved by Varchenko [V] using integral formulas for solutions of the KZ equations). Acnowledgments. The authors were supported by the NSF grant DMS-9700477. 2. Generalized Kac-Moody algebras Throughout the paper, k denotes an algebraically closed field of characteristic zero. All vector spaces in this paper will be over k. 2.1. We recall definitions from [K]. Let A = (a ij) be an n-by-n matrix with entries in k, and (h, Π, Π ∨) be a realization of A. This means that h is a vector space of dimension 2n − rank(A), Π = {α 1 , ..., α n } ⊂ h * , Π ∨ = {h 1 , ..., h n } ⊂ h are linearly independent, and α i (h j) = a ji. Definition. The Lie algebrag(A) is generated by h, e 1 , ..., e n , f 1 , ..., f n with defining relations [h, h ′ ] = 0, h, h ′ ∈ h; [h, e i ] = α i (h)e i ; [h, f i ] = −α i (h)f i ; [e i , f j ] = δ ij h i .
Let k be a perfect field and let X ⊂ P N be a hypersurface of degree d defined over k and contain... more Let k be a perfect field and let X ⊂ P N be a hypersurface of degree d defined over k and containing a linear subspace L defined over k with codim P N L = r. We show that X contains a linear subspace L 0 defined over k with codim P N L ≤ dr. We conjecture that the intersection of all linear subspaces (over k) of minimal codimension r contained in X, has codimension bounded above only in terms of r and d. We prove this when either d ≤ 3 or r ≤ 2.
For a split reductive group L G over a global field, we determine the spectrum of the spherical H... more For a split reductive group L G over a global field, we determine the spectrum of the spherical Hecke algebra coming from the unramified Eisenstein series for the minimal parabolic L B. This is done using a certain decomposition of the Springer stack T˚pBzG{Bq for the Langlands dual group G in the additive group of cobordisms of cohomologically proper derived quotient stacks. Contents
Consider a complex affine variety $\tilde V$ and a real analytic Zariski-dense submanifold V of $... more Consider a complex affine variety $\tilde V$ and a real analytic Zariski-dense submanifold V of $\tilde V$. We compare modules over the ring $O (\tilde V)$ of regular functions on $\tilde V$ with modules over the ring $C^\infty (V)$ of smooth complex valued functions on V. Under a mild condition on the tangent spaces, we prove that $C^\infty (V)$ is flat as a module over $O (\tilde V)$. From this we deduce a comparison theorem for the Hochschild homology of finite type algebras over $O (\tilde V)$ and the Hochschild homology of similar algebras over $C^\infty (V)$. We also establish versions of these results for functions on $\tilde V$ (resp. V) that are invariant under the action of a finite group G. When V is compact, we check that $C^\infty (V)$ has finite rank as module over $C^\infty (V)^G$.
Lie Groups, Geometry, and Representation Theory, 2018
Let G be a split reductive p-adic group, I ⊂ G be an Iwahori subgroup, H(G) be the Hecke algebra ... more Let G be a split reductive p-adic group, I ⊂ G be an Iwahori subgroup, H(G) be the Hecke algebra and C(G) ⊃ H(G) be the Harish-Chandra Schwartz algebra. The purpose of this note is to define (in spectral terms) a subalgebra J (G) of C(G), containing H(G), which we consider as an algebraic version of C(G). We show that the subalgebra J (G) I×I ⊂ J (G) is isomorphic to the Lusztig's asymptotic Hecke algebra J and explain a relation between the algebra J (G) and the Schwartz space of the basic affine space studied in [2].
Let k be a perfect field and let X⊂ℙ^N be a hypersurface of degree d defined over k and containin... more Let k be a perfect field and let X⊂ℙ^N be a hypersurface of degree d defined over k and containing a linear subspace L defined over an algebraic closure k with codim_ℙ^NL=r. We show that X contains a linear subspace L_0 defined over k with codim_ℙ^NL≤ dr. Furthermore, we propose an explicit algorithm for finding L_0 (with a worse estimate on codimension) from L and its Galois conjugates. We also prove a similar result for a certain version of the Schmidt rank for quartic polynomials.
This paper is a continuation of "Quantization of Lie bialgebras I-IV". The goal of this... more 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.
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