Showing papers on "Reductive group published in 2018"
TL;DR: In this article, the Mirkovic-Vilonen conjecture was shown to hold in full generality for good characteristic on a connected reductive group over an algebraically closed field of good characteristic, satisfying some mild conditions.
Abstract: Let G be a connected reductive group over an algebraically closed field F of good characteristic, satisfying some mild conditions. In this paper we relate tilting objects in the heart of Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of G, and Iwahori-constructible F-parity sheaves on the affine Grassmannian of the Langlands dual group. As applications we deduce in particular the missing piece for the proof of the Mirkovic-Vilonen conjecture in full generality (i.e. for good characteristic), a modular version of an equivalence of categories due to Arkhipov-Bezrukavnikov-Ginzburg, and an extension of this equivalence.
39 citations
01 Jun 2018
TL;DR: In this paper, the authors introduce and survey a Betti form of the geometric Langlands conjecture, parallel to the de Rham form developed by Beilinson-Drinfeld and Arinkin-Gaitsgory, and the Dolbeault form of Donagi-Pantev, inspired by the work of Kapustin-Witten in supersymmetric gauge theory.
Abstract: We introduce and survey a Betti form of the geometric Langlands conjecture, parallel to the de Rham form developed by Beilinson-Drinfeld and Arinkin-Gaitsgory, and the Dolbeault form of Donagi-Pantev, and inspired by the work of Kapustin-Witten in supersymmetric gauge theory. The conjecture proposes an automorphic category associated to a compact Riemann surface X and complex reductive group G is equivalent to a spectral category associated to the underlying topological surface S and Langlands dual group G^. The automorphic category consists of suitable C-sheaves on the moduli stack Bun_G(X) of G-bundles on X, while the spectral category consists of suitable O-modules on the character stack Loc_G^(S) of G^-local systems on S. The conjecture is compatible with and constrained by the natural symmetries of both sides coming from modifications of bundles and local systems. On the one hand, cuspidal Hecke eigensheaves in the de Rham and Betti sense are expected to coincide, so that one can view the Betti conjecture as offering a different "integration measure" on the same fundamental objects. On the other hand, the Betti spectral categories are more explicit than their de Rham counterparts and one might hope the conjecture is less challenging. The Betti program also enjoys symmetries coming from topological field theory: it is expected to extend to an equivalence of four-dimensional topological field theories, and in particular, the conjecture for closed surfaces is expected to reduce to the case of the thrice-punctured sphere. Finally, we also present ramified, quantum and integral variants of the conjecture, and highlight connections to other topics, including representation theory of real reductive groups and quantum groups.
35 citations
TL;DR: In this article, a connected reductive group defined over a non-archimedean local field F of characteristic p>0 is globalized using Poincare series and the notion of distinction with respect to a unipotent subgroup (indeed more general subgroups) is preserved.
Abstract: Let H be a connected reductive group defined over a non-archimedean local field F of characteristic p>0. Using Poincare series, we globalize supercuspidal representations of H(F) in such a way that we have control over ramification at all other places, and such that the notion of distinction with respect to a unipotent subgroup (indeed more general subgroups) is preserved. In combination with the work of Vincent Lafforgue on the global Langlands correspondence, we present some applications, such as the stability of Langlands-Shahidi \gamma-factors and the local Langlands correspondence for classical groups.
31 citations
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TL;DR: This paper showed that every smooth irreducible complex representation of G(k) contains an $\mathfrak{s}$-type of the form constructed by Kim and Yu.
Abstract: Let k be a non-archimedean local field with residual characteristic p. Let G be a connected reductive group over k that splits over a tamely ramified field extension of k. Suppose p does not divide the order of the Weyl group of G. Then we show that every smooth irreducible complex representation of G(k) contains an $\mathfrak{s}$-type of the form constructed by Kim and Yu and that every irreducible supercuspidal representation arises from Yu's construction.
This improves an earlier result of Kim, which held only in characteristic zero and with a very large and ineffective bound on p. By contrast, our bound on p is explicit and tight, and our result holds in positive characteristic as well. Moreover, our approach is more explicit in extracting an input for Yu's construction from a given representation.
25 citations
TL;DR: In this article, the authors study the cohomology of certain Galois gerbes over number fields and give an explicit constriction of the pairing between an adelic L-packet and the corresponding S-group that is the essential ingredient in the description of the discrete automorphic spectrum of a reductive group.
Abstract: We study the cohomology of certain Galois gerbes over number fields. This cohomology provides a bridge between refined local endoscopy, as introduced in Kaletha (Ann Math (2) 184(2):559–632, 2016), and classical global endoscopy. As particular applications, we express the canonical adelic transfer factor that governs the stabilization of the Arthur–Selberg trace formula as a product of normalized local transfer factors, we give an explicit constriction of the pairing between an adelic L-packet and the corresponding S-group (based on the conjectural pairings in the local setting) that is the essential ingredient in the description of the discrete automorphic spectrum of a reductive group, and we give a proof of some expectations of Arthur.
25 citations
TL;DR: In this paper, the Iwahori-Hecke algebraic action on the Gelfand-Graev representation of a p-adic field is described. But the action is restricted to an explicit Hecke algebra module.
Abstract: Let G be a split reductive group over a p-adic field F. Let B be a Borel subgroup and U the maximal unipotent subgroup of B. Let \(\psi \) be a Whittaker character of U. Let I be an Iwahori subgroup of G. We describe the Iwahori–Hecke algebra action on the Gelfand–Graev representation \((\mathrm {ind}_{U}^{G}\psi )^I\) by an explicit Hecke algebra module.
21 citations
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TL;DR: In this article, a local Langlands correspondence for irreducible unipotent representations of a connected reductive group over a non-archimedean local field was established.
Abstract: Let G be a connected reductive group over a non-archimedean local field K, and assume that G splits over an unramified extension of K. We establish a local Langlands correspondence for irreducible unipotent representations of G. It comes as a bijection between the set of such representations and the collection of enhanced L-parameters for G, which are trivial on the inertia subgroup of the Weil group of K. We show that this correspondence has many of the expected properties, for instance with respect to central characters, tempered representations, the discrete series, cuspidality and parabolic induction.
The core of our strategy is the investigation of affine Hecke algebras on both sides of the local Langlands correspondence. When a Bernstein component of G-representations is matched with a Bernstein component of enhanced L-parameters, we prove a comparison theorem for the two associated affine Hecke algebras.
This generalizes work of Lusztig in the case of adjoint K-groups.
19 citations
TL;DR: For an odd prime p, a minimal set of topological generators of the pro-p Iwahori subgroup of a split reductive group G over ℤp was given in this paper.
Abstract: For an odd prime p, we determine a minimal set of topological generators of the pro-p Iwahori subgroup of a split reductive group G over ℤp. In the simple adjoint case and for any sufficiently large regular prime p, we also construct Galois extensions of ℚ with Galois group between the pro-p and the standard Iwahori subgroups of G.
19 citations
TL;DR: In this paper, a bound for the height of these γ which is polynomial with respect to the determinant and denominator is established. But this bound is not applicable to the case of the Zilber-Pink conjecture on unlikely intersections in Shimura varieties.
Abstract: Let G be a reductive group defined over Q and let S be a Siegel set in G ( R ) . The Siegel property tells us that there are only finitely many γ ∈ G ( Q ) of bounded determinant and denominator for which the translate γ. S intersects S . We prove a bound for the height of these γ which is polynomial with respect to the determinant and denominator. The bound generalises a result of Habegger and Pila dealing with the case of GL 2 , and has applications to the Zilber–Pink conjecture on unlikely intersections in Shimura varieties. In addition we prove that if H is a subset of G , then every Siegel set for H is contained in a finite union of G ( Q ) -translates of a Siegel set for G .
17 citations
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TL;DR: In this article, the abelian-nonabelian correspondence for quasimap $I$-functions was proved for affine l.c.i.d. variety with an action by a complex reductive group.
Abstract: We prove the abelian-nonabelian correspondence for quasimap $I$-functions. That is, if $Z$ is an affine l.c.i. variety with an action by a complex reductive group $G$, we prove an explicit formula relating the quasimap $I$-functions of the GIT quotients $Z//_{\theta} G$ and $Z//_{\theta} T$ where $T$ is a maximal torus of $G$. We apply the formula to compute the $J$-functions of some Grassmannian bundles on Grassmannian varieties and Calabi-Yau hypersurfaces in them.
15 citations
TL;DR: In this paper, the uniqueness of supercuspidal support in the modular representation theory of p-adic groups was studied, and it was shown that any counterexample to the same property for a finite reductive group lifts to a countererxample for the corresponding unramified padic group.
TL;DR: In this article, the authors compute the Hodge and de Rham cohomology of the classifying space BG for reductive groups G over many fields, including fields of small characteristic.
Abstract: We compute the Hodge and the de Rham cohomology of the classifying space BG (defined as etale cohomology on the algebraic stack BG) for reductive groups G over many fields, including fields of small characteristic. These calculations have a direct relation with representation theory, yielding new results there. The calculations are closely analogous to, but not always the same as, the cohomology of classifying spaces in topology.
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TL;DR: In this article, a soergel theory for Bruhat-constructible perverse sheaves on the flag variety G/B of a complex reductive group G, with coefficients in an arbitrary field k, was developed.
Abstract: We develop a "Soergel theory" for Bruhat-constructible perverse sheaves on the flag variety G/B of a complex reductive group G, with coefficients in an arbitrary field k. Namely, we describe the endomorphisms of the projective cover of the skyscraper sheaf in terms of a "multiplicative" coinvariant algebra, and then establish an equivalence of categories between projective (or tilting) objects in this category and a certain category of "Soergel modules" over this algebra. We also obtain a description of the derived category of T-monodromic k-sheaves on G/U (where U,T ⊂ B are the unipotent radical and the maximal torus), as a monoidal category.
TL;DR: In this article, a general family of partition functions attached to the complex L-group of the unramified p-adic group G was introduced, which specialize to Kostant's q-partition function for complex connected groups and also specialize to the Langlands L-function of a spherical representation.
Abstract: This article gives a proof of the Langlands-Shelstad fundamental lemma for the spherical Hecke algebra for every unramified p-adic reductive group G in large positive characteristic. The proof is based on the transfer principle for constructible motivic integration. To carry this out, we introduce a general family of partition functions attached to the complex L-group of the unramified p-adic group G. Our partition functions specialize to Kostant's q-partition function for complex connected groups and also specialize to the Langlands L-function of a spherical representation. These partition functions are used to extend numerous results that were previously known only when the L-group is connected (that is, when the p-adic group is split). We give explicit formulas for branching rules, the inverse of the weight multiplicity matrix, the Kato-Lusztig formula for the inverse Satake transform, the Plancherel measure, and Macdonald's formula for the spherical Hecke algebra on a non-connected complex group (that is, non-split unramified p-adic group).
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TL;DR: In this paper, the exactness of parabolic restriction and induction functors for conjugation equivariant sheaves on a reductive group generalizing a well known result of Lusztig who established this property for character sheaves was proved.
Abstract: We prove exactness of parabolic restriction and induction functors for conjugation equivariant sheaves on a reductive group generalizing a well known result of Lusztig who established this property for character sheaves. We propose a conjectural (but known for character sheaves) t-exactness property of the Harish-Chandra transform and provide an evidence for that conjecture. We also present two applications generalizing some results of Gabber and Loeser on perverse sheaves on an algebraic torus to an arbitrary reductive group.
TL;DR: In this article, a progenerator for the category of representations of a finite reductive group coming from generalised Gelfand-Graev representations was constructed, based on the conjecture of Geck and Malle.
Abstract: We prove a long-standing conjecture of Geck which predicts that cuspidal unipotent characters remain irreducible after $$\ell $$
-reduction To this end, we construct a progenerator for the category of representations of a finite reductive group coming from generalised Gelfand–Graev representations This is achieved by showing that cuspidal representations appear in the head of generalised Gelfand–Graev representations attached to cuspidal unipotent classes, as defined and studied in Geck and Malle (J Lond Math Soc 2(53):63–78, 1996)
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TL;DR: In this paper, the Grothendieck-Serre conjecture was generalized to the case of arbitrary reductive group schemes, and it was shown that a simple simply connected affine semi-local scheme over a field is trivial if it is trivial over the generic fiber of the projection.
Abstract: Let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. Assuming that $G$ has an appropriate parabolic subgroup scheme, we prove the following statement. Given an affine $k$-scheme $W$, a principal $G$-bundle over $W\times_kU$ is trivial if it is trivial over the generic fiber of the projection $W\times_kU\to U$. We also simplify the proof of the Grothendieck-Serre conjecture: let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. A principal $G$-bundle over $U$ is trivial if it is trivial over the generic point of $U$. We generalize some other related results from the simple simply-connected case to the case of arbitrary reductive group schemes.
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TL;DR: In this paper, the authors give a new proof of Geometric Stabilisation Theorem, which implies the Fundamental Lemma, based on results for $p$-adic integration on coarse moduli spaces of Deligne-Mumford stacks.
Abstract: In this article we give a new proof of Ng\^o's Geometric Stabilisation Theorem, which implies the Fundamental Lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme $G$ to the cohomology of Hitchin fibres for the endoscopy groups $H_{\kappa}$. Our proof avoids the Decomposition and Support Theorem, instead the argument is based on results for $p$-adic integration on coarse moduli spaces of Deligne-Mumford stacks. Along the way we establish a description of the inertia stack of the (anisotropic) moduli stack of $G$-Higgs bundles in terms of endoscopic data, and extend duality for generic Hitchin fibres of Langlands dual group schemes to the quasi-split case.
TL;DR: In this article, a bilinear form B on the space of automorphic forms of a split reductive group G over a function field is defined, which is related to miraculous duality in the geometric Langlands program through the functions-sheaves dictionary.
Abstract: This article concerns the study of a new invariant bilinear form B on the space of automorphic forms of a split reductive group G over a function field. We define B using the asymptotics maps from recent work of Bezrukavnikov, Kazhdan, Sakellaridis, and Venkatesh, which involve the geometry of the wonderful compactification of G. We show that B is naturally related to miraculous duality in the geometric Langlands program through the functions-sheaves dictionary. In the proof, we highlight the connection between the classical non-Archimedean Gindikin–Karpelevich formula and certain factorization algebras acting on geometric Eisenstein series. We then give another definition of B using the constant term operator and the inverse of the standard intertwining operator. The form B defines an invertible operator L from the space of compactly supported automorphic forms to a new space of pseudocompactly supported automorphic forms. We give a formula for L−1 in terms of pseudo-Eisenstein series and constant term operators which suggests that L−1 is an analogue of the Aubert–Zelevinsky involution.
TL;DR: For a reductive group G$ defined over an algebraically closed field of positive characteristic, it was shown in this article that the Frobenius contraction functor of G$-modules is right adjoint to the Frobius twist of the modules tensored with the Steinberg module twice.
Abstract: For a reductive group $G$ defined over an algebraically closed field of positive characteristic, we show that the Frobenius contraction functor of $Gs-modules is right adjoint to the Frobenius twist of the modules tensored with the Steinberg module twice. It follows that the Frobenius contraction functor preserves injectivity, good filtrations, but not semi-simplicity.
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TL;DR: In this paper, the Lusztig-Vogan bijection is shown to be independent of the characteristic of the algebraically closed field k. This allows us to extend all of its known properties from the characteristic-0 setting to the general case, which is a step towards a proof of the Humphreys conjecture on support varieties of tilting modules for reductive groups in positive characteristic.
Abstract: Let G be a reductive group over an algebraically closed field k of very good characteristic. The Lusztig-Vogan bijection is a bijection between the set of dominant weights for G and the set of irreducible G-equivariant vector bundles on nilpotent orbits, conjectured by Lusztig and Vogan independently , and constructed in full generality by Bezrukavnikov. In characteristic 0, this bijection is related to the theory of 2-sided cells in the affine Weyl group, and plays a key role in the proof of the Humphreys conjecture on support varieties of tilting modules for quantum groups at a root of unity. In this paper, we prove that the Lusztig-Vogan bijection is (in a way made precise in the body of the paper) independent of the characteristic of k. This allows us to extend all of its known properties from the characteristic-0 setting to the general case. We also expect this result to be a step towards a proof of the Humphreys conjecture on support varieties of tilting modules for reductive groups in positive characteristic.
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TL;DR: In this paper, a block decomposition of the dg category of character sheaves on a simple and simply-connected complex reductive group was given, similar to the one in generalized Springer correspondence.
Abstract: We give a block decomposition of the dg category of character sheaves on a simple and simply-connected complex reductive group $G$, similar to the one in generalized Springer correspondence. As a corollary, we identify the category of character sheaves on $G$ as the category of quasi-coherent sheaves on an explicitly defined derived stack $\widehat{G}$.
TL;DR: In this paper, a lifting n of a maximal torus T in a split reductive group over F_q was constructed explicitly such that the image of n under the Frobenius map is equal to the inverse of n.
Abstract: Let N be the normalizer of a maximal torus T in a split reductive group over F_q and let w be an involution in the Weyl group N/T. We construct explicitly a lifting n of w in N such that the image of n under the Frobenius map is equal to the inverse of n.
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TL;DR: In this article, a necessary and sufficient condition for the existence of a G-0-equivariant k_0-form of G/H is given, where G is a quasi-split k-0 form of G and H is a spherical subgroup defined over k.
Abstract: Let k_0 be a field of characteristic 0, k its algebraic closure, G a connected reductive group defined over k. Let H\subset G be a spherical subgroup. We assume that k_0 is a large field, for example, k_0 is either the field R of real numbers or a p-adic field. Let G_0 be a quasi-split k_0-form of G. We show that if H has self-normalizing normalizer, and Gal(k/k_0) preserves the combinatorial invariants of G/H, then H is conjugate to a subgroup defined over k_0, and hence, the G-variety G/H admits a G_0-equivariant k_0-form. In the case when G_0 is not assumed to be quasi-split, we give a necessary and sufficient Galois-cohomological condition for the existence of a G_0-equivariant k_0-form of G/H.
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TL;DR: In this article, a modified definition of the tensor category of displays and variants is proposed, which is similar to the Frobenius gauges of Fontaine-Jannsen.
Abstract: Deformations of ordinary varieties of K3 type can be described in terms of displays by recent work of Langer-Zink. We extend this to the general (non-ordinary) case using displays with $G$-structure for a reductive group $G$. As a basis we suggest a modified definition of the tensor category of displays and variants which is similar to the Frobenius gauges of Fontaine-Jannsen.
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TL;DR: In this paper, the space of vector-valued conjugate invariant functions on a connected reductive group was studied, and the authors studied vector-value invariant function space of two-sided invariants.
Abstract: We study the space of vector-valued (twisted) conjugate invariant functions on a connected reductive group
TL;DR: The theory of minimal K-types for p-adic reductive groups was developed in part to classify irreducible admissible representations with wild ramification as mentioned in this paper, and it was shown that minimal k-types associated to such representations correspond to fundamental strata.
Abstract: The theory of minimal K-types for p-adic reductive groups was developed in part to classify irreducible admissible representations with wild ramification. An important observation was that minimal K-types associated to such representations correspond to fundamental strata. These latter objects are triples (x, r, β), where x is a point in the Bruhat-Tits building of the reductive group G, r is a nonnegative real number, and β is a semistable functional on the degree r associated graded piece of the Moy-Prasad filtration
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TL;DR: The Mumford cone is the polyhedral cone generated by the T-weights of the polynomial functions on H which are semi-invariant under the Borel subgroup as mentioned in this paper.
Abstract: Let G be a complex reductive group acting on a finite-dimensional complex vector space H. Let B be a Borel subgroup of G and let T be the associated torus. The Mumford cone is the polyhedral cone generated by the T-weights of the polynomial functions on H which are semi-invariant under the Borel subgroup. In this article, we determine the inequalities of the Mumford cone in the case of the linear representation associated to a quiver and a dimension vector n=(n_x). We give these inequalities in terms of filtered dimension vectors, and we directly adapt Schofield's argument to inductively determine the dimension vectors of general subrepresentations in the filtered context. In particular, this gives one further proof of the Horn inequalities for tensor products.
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TL;DR: In this paper, the authors investigated the flexibility of affine varieties with an action of a linear algebraic group and proved that a normal affine complexity zero horospherical variety is flexible.
Abstract: We investigate flexibility of affine varieties with an action of a linear algebraic group. Flexibility of a smooth affine variety with only constant invertible functions and a locally transitive action of a reductive group is proved. Also we show that a normal affine complexity-zero horospherical variety is flexible.
TL;DR: In this article, a new algebraic object called an FFG-algebra has been defined, which is a function from a group to a ring with polynomial relations that make it "look like" the character of a representation.
Abstract: A $GL_d$-pseudocharacter is a function from a group $\Gamma$ to a ring $k$ satisfying polynomial relations which make it "look like" the character of a representation. When $k$ is an algebraically closed field, Taylor proved that $GL_d$-pseudocharacters of $\Gamma$ are the same as degree-$d$ characters of $\Gamma$ with values in $k$, hence are in bijection with equivalence classes of semisimple representations $\Gamma \rightarrow GL_d(k)$. Recently, V. Lafforgue generalized this result by showing that, for any connected reductive group $H$ over an algebraically closed field $k$ of characteristic 0 and for any group $\Gamma$, there exists an infinite collection of functions and relations which are naturally in bijection with $H^0(k)$-conjugacy classes of semisimple representations $\Gamma \rightarrow H(k)$. In this paper, we reformulate Lafforgue's result in terms of a new algebraic object called an FFG-algebra. We then define generating sets and generating relations for these objects and show that, for all $H$ as above, the corresponding FFG-algebra is finitely presented. Hence we can always define $H$-pseudocharacters consisting of finitely many functions satisfying finitely many relations. Next, we use invariant theory to give explicit finite presentations of the FFG-algebras for (general) orthogonal groups, (general) symplectic groups, and special orthogonal groups. Finally, we use our pseudocharacters to answer questions about conjugacy vs. element-conjugacy of representations, following Larsen.