Split Reductive Group Research Articles

Colored vertex models and Iwahori Whittaker functions

We give a recursive method for computing all values of a basis of Whittaker functions for unramified principal series invariant under an Iwahori or parahoric subgroup of a split reductive group G over a nonarchimedean local field F. Structures in the proof have surprising analogies to features of certain solvable lattice models. In the case G=GLr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G=\ extrm{GL}_r$$\\end{document} we show that there exist solvable lattice models whose partition functions give precisely all of these values. Here ‘solvable’ means that the models have a family of Yang–Baxter equations which imply, among other things, that their partition functions satisfy the same recursions as those for Iwahori or parahoric Whittaker functions. The R-matrices for these Yang–Baxter equations come from a Drinfeld twist of the quantum group Uq(gl^(r|1))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_q(\\widehat{\\mathfrak {gl}}(r|1))$$\\end{document}, which we then connect to the standard intertwining operators on the unramified principal series. We use our results to connect Iwahori and parahoric Whittaker functions to variations of Macdonald polynomials.

Genuine pro-$p$ Iwahori–Hecke algebras, Gelfand–Graev representations, and some applications

We study the Iwahori-component of the Gelfand–Graev representation of a central cover of a split linear reductive group and utilize our results for three applications. In fact, it is advantageous to begin at the pro- p level. Thus to begin we study the structure of a genuine pro- p Iwahori–Hecke algebra, establishing a Bernstein presentation. With this structure theory we first describe the pro- p part of the Gelfand–Graev representation and then the more subtle Iwahori part. For the first application we relate the Gelfand–Graev representation to the metaplectic representation of Sahi–Stokman–Venkateswaran, which conceptually realizes the Chinta–Gunnells action from the theory of Weyl group multiple Dirichlet series. For the second we compute the Whittaker dimension of the constituents of regular unramified principal series representations; for the third we do the same for unitary unramified principal series representations.

G-torsors and universal torsors over nonsplit del Pezzo surfaces

Let S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of SL in PicSL, or a form of it containing the Néron–Severi torus. Let G be the G-torsor over SL obtained by extension of structure group from a universal torsor T over SL. We prove that G does not descend to S unless T does. This is in contrast to a result of Friedman and Morgan that such G always descend to singular del Pezzo surfaces over C from their desingularizations.

Cellular [formula omitted]-homology and the motivic version of Matsumoto's theorem

We define a new version of A1-homology, called cellular A1-homology, for smooth schemes over a field that admit an increasing filtration by open subschemes with cohomologically trivial closed strata. We provide several explicit computations of cellular A1-homology and use them to determine the A1-fundamental group of a split reductive group over an arbitrary field, thereby obtaining the motivic version of Matsumoto's theorem on universal central extensions of split, semisimple, simply connected algebraic groups. As applications, we uniformly explain and generalize results due to Brylinski-Deligne and Esnault-Kahn-Levine-Viehweg, determine the isomorphism classes of central extensions of such an algebraic group by an arbitrary strictly A1-invariant sheaf and also reprove classical results of E. Cartan on homotopy groups of complex Lie groups.

G-valued crystalline deformation rings in the Fontaine–Laffaille range

Let$G$be a split reductive group over the ring of integers in a$p$-adic field with residue field$\mathbf {F}$. Fix a representation$\overline {\rho }$of the absolute Galois group of an unramified extension of$\mathbf {Q}_p$, valued in$G(\mathbf {F})$. We study the crystalline deformation ring for$\overline {\rho }$with a fixed$p$-adic Hodge type that satisfies an analog of the Fontaine–Laffaille condition for$G$-valued representations. In particular, we give a root theoretic condition on the$p$-adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups.

Additivity of the motivic trace and the motivic Euler-characteristic

In this paper, we settle an open conjecture regarding the assertion that the Euler-characteristic of G/NG(T) for a split reductive group scheme G and the normalizer of a split maximal torus NG(T) over a field is 1 in the Grothendieck-Witt ring with the characteristic exponent of the field inverted, under the assumption that the base field contains a −1. Numerous applications of this to splittings in the motivic stable homotopy category and to Algebraic K-Theory are worked out in several related papers by Gunnar Carlsson and the authors.

On the derived category of the Iwahori–Hecke algebra

We state a conjecture that relates the derived category of smooth representations of a $p$-adic split reductive group with the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We investigate the conjecture in the case of the principal block of ${\rm GL}_n$ by showing that the functor should be given by the derived tensor product with the family of representations interpolating the modified Langlands correspondence over the stack of L-parameters that is suggested by the work of Helm and of Emerton and Helm.

Motivic cohomology and infinitesimal group schemes

For $k$ a perfect field of characteristic $p>0$ and $G/k$ a split reductive group with $p$ a non-torsion prime for $G,$ we compute the mod $p$ motivic cohomology of the geometric classifying space $BG_{(r)}$, where $G_{(r)}$ is the $r$th Frobenius kernel of $G.$ Our main tool is a motivic version of the Eilenberg-Moore spectral sequence, due to Krishna. For a flat affine group scheme $G/k$ of finite type, we define a cycle class map from the mod $p$ motivic cohomology of the classifying space $BG$ to the mod $p$ \'etale motivic cohomology of the classifying stack $\mathcal{B}G.$ This also gives a cycle class map into the Hodge cohomology of $\mathcal{B}G.$ We study the cycle class map for some examples, including Frobenius kernels.

A note on Galois representations valued in reductive groups with open image

Let G be a split reductive group with dim⁡Z(G)≤1. We show that for any prime p that is large enough relative to G, there is a finitely ramified Galois representation ρ:ΓQ→G(Zp) with open image. We also show that for any given integer e, if the index of irregularity of p is at most e and if p is large enough relative to G and e, then there is a Galois representation ρ:ΓQ→G(Zp) ramified only at p with open image, generalizing a theorem of Ray [8]. The first type of Galois representation is constructed by lifting a suitable Galois representation into G(Fp) using a lifting theorem of Fakhruddin–Khare–Patrikis [4], and the second type of Galois representation is constructed using a variant of the argument in Ray's work [8].

Affine representability of quadrics revisited

The quadric Q2n is the Z-scheme defined by the equation ∑i=1nxiyi=z(1−z). We show that Q2n is a homogeneous space for the split reductive group scheme SO2n+1 over Z. The quadric Q2n is known to have the A1-homotopy type of a motivic sphere and the identification as a homogeneous space allows us to give a characteristic independent affine representability statement for motivic spheres. This last observation allows us to give characteristic independent comparison results between Chow–Witt groups, motivic stable cohomotopy groups and Euler class groups.

Lifting involutions in a Weyl group to the normalizer of the torus

Let N N be the normalizer of a maximal torus T T in a split reductive group over F q \mathbb {F}_q and let w w be an involution in the Weyl group N / T N/T . We construct a section of W W satisfying the braid relations, such that the image of the lift n n of w w under the Frobenius map is equal to the inverse of n n .

Localized operations on T-equivariant oriented cohomology of projective homogeneous varieties

In the present paper we provide a general algorithm to compute multiplicative cohomological operations on algebraic oriented cohomology of projective homogeneous G -varieties, where G is a split reductive algebraic group over a field k of characteristic 0. More precisely, we extend such operations to the respective T -equivariant ( T is a maximal split torus of G ) oriented theories, and then compute them using equivariant Schubert calculus techniques. This generalizes an approach suggested by Garibaldi-Petrov-Semenov for Steenrod operations. As an application we establish the Riemann-Roch type formula for the Hecke action on theories of additive type.

Bessel F-isocrystals for reductive groups

We construct the Frobenius structure on a rigid connection $\mathrm{Be}_{\check{G}}$ on $\mathbb{G}_m$ for a split reductive group $\check{G}$ introduced by Frenkel-Gross. These data form a $\check{G}$-valued overconvergent $F$-isocrystal $\mathrm{Be}_{\check{G}}^{\dagger}$ on $\mathbb{G}_{m,\mathbb{F}_p}$, which is the $p$-adic companion of the Kloosterman $\check{G}$-local system $\mathrm{Kl}_{\check{G}}$ constructed by Heinloth-Ng\^o-Yun. By exploring the structure of the underlying differential equation, we calculate the monodromy group of $\mathrm{Be}_{\check{G}}^{\dagger}$ when $\check{G}$ is almost simple (which recovers the calculation of monodromy group of $\mathrm{Kl}_{\check{G}}$ due to Katz and Heinloth-Ng\^o-Yun), and establish functoriality between different Kloosterman $\check{G}$-local systems as conjectured by Heinloth-Ng\^o-Yun. We show that the Frobenius Newton polygons of $\mathrm{Kl}_{\check{G}}$ are generically ordinary for every $\check{G}$ and are everywhere ordinary on $|\mathbb{G}_{m,\mathbb{F}_p}|$ when $\check{G}$ is classical or $G_2$.

A reduction principle for Fourier coefficients of automorphic forms

We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group mathbf {G}(mathbb {A}_mathbb {K}), which we refer to as Fourier coefficients associated to the data of a ‘Whittaker pair’. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are ‘Levi-distinguished’ Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a {mathbb K}-distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients.

Gelfand’s Trick for the Spherical Derived Hecke Algebra

Abstract Gelfand’s trick shows that the spherical Hecke algebra of a $p$-adic split reductive group is commutative. We adapt this strategy in order to show that the spherical derived Hecke algebra is graded commutative under mild assumptions on the coefficient ring.

On the asymptotics of Hecke operators for reductive groups

In this paper, we study the asymptotic behavior of the traces of Hecke operators for spherical discrete automorphic representations of fixed level on general split reductive groups over \(\mathbb {Q}\). Under a condition on the analytic behavior of intertwining operators, which is known for the classical groups and the exceptional group \(G_2\), we obtain the expected asymptotics in terms of the spherical Plancherel measure and an explicit estimate for the remainder.

Rodier type theorem for generalized principal series

Given a regular supercuspidal representation rho of the Levi subgroup M of a standard parabolic subgroup P=MN in a connected reductive group G defined over a non-archimedean local field F, we serve you a Rodier type structure theorem which provides us a geometrical parametrization of the set JH(Ind^G_P(rho )) of Jordan–Hölder constituents of the Harish-Chandra parabolic induction representation Ind^G_P(rho ), vastly generalizing Rodier structure theorem for P=B=TU Borel subgroup of a connected split reductive group about 40 years ago. Our novel contribution is to overcome the essential difficulty that the relative Weyl group W_M=N_G(M)/M is not a coxeter group in general, as opposed to the well-known fact that the Weyl group W_T=N_G(T)/T is a coxeter group. Along the way, we sort out all regular discrete series/tempered/generic representations for arbitrary G, generalizing Tadić’s work on regular discrete series representation for split (G)Sp_{2n} and SO_{2n+1}, and also providing a new simple proof of Casselman–Shahidi’s theorem on generalized injectivity conjecture for regular generalized principal series. Indeed, such a beautiful structure theorem also holds for finite central covering groups.

Cuspidal cohomology of stacks of shtukas

Let$G$be a connected split reductive group over a finite field$\mathbb{F}_{q}$and$X$a smooth projective geometrically connected curve over$\mathbb{F}_{q}$. The$\ell$-adic cohomology of stacks of$G$-shtukas is a generalization of the space of automorphic forms with compact support over the function field of$X$. In this paper, we construct a constant term morphism on the cohomology of stacks of shtukas which is a generalization of the constant term morphism for automorphic forms. We also define the cuspidal cohomology which generalizes the space of cuspidal automorphic forms. Then we show that the cuspidal cohomology has finite dimension and that it is equal to the (rationally) Hecke-finite cohomology defined by V. Lafforgue.

Homogeneous vector bundles over abelian varieties via representation theory

Let A A be an abelian variety over a field. The homogeneous (or translation-invariant) vector bundles over A A form an abelian category HVec A \textrm {HVec}_A ; the Fourier-Mukai transform yields an equivalence of HVec A \textrm {HVec}_A with the category of coherent sheaves with finite support on the dual abelian variety. In this paper, we develop an alternative approach to homogeneous vector bundles, based on the equivalence of HVec A \textrm {HVec}_A with the category of finite-dimensional representations of a commutative affine group scheme (the “affine fundamental group” of A A ). This displays remarkable analogies between homogeneous vector bundles over abelian varieties and representations of split reductive algebraic groups.

THE INTERSECTION MOTIVE OF THE MODULI STACK OF SHTUKAS

For a split reductive group$G$over a finite field, we show that the intersection (cohomology) motive of the moduli stack of iterated$G$-shtukas with bounded modification and level structure is defined independently of the standard conjectures on motivic$t$-structures on triangulated categories of motives. This is in accordance with general expectations on the independence of$\ell$in the Langlands correspondence for function fields.