Showing papers on "Reductive group published in 2019"

Total Positivity in Reductive Groups

Abstract: An invertible n×n matrix with real entries is said to be totally ≥0 (resp. totally >0) if all its minors are ≥0 (resp. >0). This definition appears in Schoenberg’s 1930 paper [S] and in the 1935 note [GK] of Gantmacher and Krein. (For a recent survey of totally positive matrices, see [A].)

$\hat{G}$-local systems on smooth projective curves are potentially automorphic

Abstract: Let $X$ be a smooth, projective, geometrically connected curve over a finite field $\mathbb{F}_q$, and let $G$ be a split semisimple algebraic group over $\mathbb{F}_q$. Its dual group $\widehat{G}$ is a split reductive group over $\mathbb{Z}$. Conjecturally, any $l$-adic $\widehat{G}$-local system on $X$ (equivalently, any conjugacy class of continuous homomorphisms $\pi_1(X) \to \widehat{G}(\overline{\mathbb{Q}}_l)$) should be associated with an everywhere unramified automorphic representation of the group $G$. We show that for any homomorphism $\pi_1(X) \to \widehat{G}(\overline{\mathbb{Q}}_l)$ of Zariski dense image, there exists a finite Galois cover $Y \to X$ over which the associated local system becomes automorphic.

$G$-valued local deformation rings and global lifts

Abstract: We study G-valued Galois deformation rings with prescribed properties, where G is an arbitrary (not necessarily connected) reductive group over an extension of ℤl for some prime l. In particular, for the Galois groups of p-adic local fields (with p possibly equal to l) we prove that these rings are generically regular, compute their dimensions, and show that functorial operations on Galois representations give rise to well-defined maps between the sets of irreducible components of the corresponding deformation rings. We use these local results to prove lower bounds on the dimension of global deformation rings with prescribed local properties. Applying our results to unitary groups, we improve results in the literature on the existence of lifts of mod l Galois representations, and on the weight part of Serre’s conjecture.

Dimensions of affine Deligne-Lusztig varieties: a new approach via labeled folded alcove walks and root operators

Abstract: Let G be a reductive group over the field F=k((t)), where k is an algebraic closure of a finite field, and let W be the (extended) affine Weyl group of G. The associated affine Deligne-Lusztig varieties $X_x(b)$, which are indexed by elements b in G(F) and x in W, were introduced by Rapoport. Basic questions about the varieties $X_x(b)$ which have remained largely open include when they are nonempty, and if nonempty, their dimension. We use techniques inspired by geometric group theory and representation theory to address these questions in the case that b is a pure translation, and so prove much of a sharpened version of Conjecture 9.5.1 of Gortz, Haines, Kottwitz, and Reuman. Our approach is constructive and type-free, sheds new light on the reasons for existing results in the case that b is basic, and reveals new patterns. Since we work only in the standard apartment of the building for G(F), our results also hold in the p-adic context, where we formulate a definition of the dimension of a p-adic Deligne-Lusztig set. We present two immediate consequences of our main results, to class polynomials of affine Hecke algebras and to affine reflection length.

Colored Vertex Models and Iwahori Whittaker Functions

Abstract: 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 $G=\mathrm{GL}_r$ 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 $U_q(\widehat{\mathfrak{gl}}(r|1))$, 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.

-Nice triples and the Grothendieck–Serre conjecture concerning principal G-bundles over reductive group schemes

Abstract: The main result of this article is to reduce a proof of the conjecture to a statement about principal bundles on affine line over a regular local scheme. This reduction is obtained via a theory of nice triples, which goes back to the ideas of Voevodsky. As an application, an unpublished result due to Gabber is proved.

Relative deformation theory and lifting irreducible Galois representations

Abstract: We study irreducible odd mod $p$ Galois representations $\bar{\rho} \colon \mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_p)$, for $F$ a totally real number field and $G$ a general reductive group. For $p \gg_{G, F} 0$, we show that any $\bar{\rho}$ that lifts locally, and at places above $p$ to de Rham and Hodge-Tate regular representations, has a geometric $p$-adic lift. We also prove non-geometric lifting results without any oddness assumption.

Motivic euler characteristics and witt-valued characteristic classes

Abstract: This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker–Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.

Cohomologically induced distinguished representations and cohomological test vectors

Abstract: Let G be a real reductive group, and let χ be a character of a reductive subgroup H of G. We construct χ-invariant linear functionals on certain cohomologically induced representations of G, and we show that these linear functionals do not vanish on the bottom layers. Applying this construction, we prove two Archimedean nonvanishing hypotheses which are vital to the arithmetic study of special values of certain L-functions via modular symbols.

On the construction of tame supercuspidal representations

Abstract: Let F be a non-archimedean local field of odd residual characteristic. Let G be a (connected) reductive group over F that splits over a tamely ramified field extension of F. We revisit Yu's construction of smooth complex representations of G(F) from a slightly different perspective and provide a proof that the resulting representations are supercuspidal. We also provide a counterexample to Proposition 14.1 and Theorem 14.2 in [Yu01], whose proofs relied on a typo in a reference.

Endoscopy for Hecke categories, character sheaves and representations

Abstract: For a split reductive group $G$ over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group $H$ with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on $G$ with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group $H$, after passing to asymptotic versions. The other is a similar relationship between representations of $G(\mathbb{F}_q)$ with a fixed semisimple parameter and unipotent representations of $H(\mathbb{F}_{q})$.

Combinatorial characterization of the weight monoids of smooth affine spherical varieties

Abstract: Let G be a connected complex reductive group. A well known theorem of I. Losev's says that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper, we use the combinatorial theory of spherical varieties and a smoothness criterion of R. Camus to characterize the weight monoids of smooth affine spherical varieties.

Higher Orbit Integrals, Cyclic Cocyles, and K-theory of Reduced Group C*-algebra

Abstract: Let G be a connected real reductive group. Orbit integrals define traces on the group algebra of G. We introduce a construction of higher orbit integrals in the direction of higher cyclic cocycles on the Harish-Chandra Schwartz algebra of G. We analyze these higher orbit integrals via Fourier transform by expressing them as integrals on the tempered dual of G. We obtain explicit formulas for the pairing between the higher orbit integrals and the K-theory of the reduced group C*-algebra, and discuss their applications to representation theory and K-theory.

On the extended Whittaker category

Abstract: Let G be a connected reductive group with connected center and X a smooth complete curve, both defined over an algebraically closed field of characteristic zero. Let BunG denote the stack of G-bundles on X. In analogy with the classical theory of Whittaker coefficients for automorphic functions, we construct a “Fourier transform” functor coeffG,ext from the DG category of D -modules on BunG to a certain DG category Wh(G,ext) , called the extended Whittaker category. This construction allows to formulate the compatibility of the Langlands duality functor LG:IndCohN(LocSysGˇ)→D(BunG) with the Whittaker model. For G=GLn and G=PGLn , we prove that coeffG,ext is fully faithful. This result guarantees that, for those groups, LG is unique (if it exists) and necessarily fully faithful.

The intersection motive of the moduli stack of shtukas

Abstract: 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 l in the Langlands correspondence for function fields.

Relative deformation theory, relative Selmer groups, and lifting irreducible Galois representations

Abstract: We study irreducible odd mod $p$ Galois representations $\bar{\rho} \colon \mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_p)$, for $F$ a totally real number field and $G$ a general reductive group. For $p \gg_{G, F} 0$, we show that any $\bar{\rho}$ that lifts locally, and at places above $p$ to de Rham and Hodge-Tate regular representations, has a geometric $p$-adic lift. We also prove non-geometric lifting results without any oddness assumption.

Local coefficients and gamma factors for principal series of covering groups

Abstract: We consider an $n$-fold Brylinski-Deligne cover of a reductive group over a $p$-adic field. Since the space of Whittaker functionals of an irreducible genuine representation of such a cover is not one-dimensional, one can consider a local coefficients matrix arising from an intertwining operator, which is the natural analogue of the local coefficients in the linear case. In this paper, we concentrate on genuine principal series and establish some fundamental properties of such a local coefficients matrix, including the investigation of its arithmetic invariants. As a consequence, we prove a form of the Casselman-Shalika formula which could be viewed as a natural analogue for linear algebraic groups. We also investigate in some depth the behaviour of the local coefficients matrix with respect to the restriction of genuine principal series from covers of ${\rm GL}_2$ to ${\rm SL}_2$. In particular, some further relations are unveiled between local coefficients matrices and gamma factors or metaplectic-gamma factors.

Supercuspidal L-packets

Abstract: Let F be a non-archimedean local field and let G be a connected reductive group defined over F. We assume that G splits over a tame extension of F and that the residual characteristic p does not divide the order of the Weyl group. To each discrete Langlands parameter of the Weil group of F into the complex L-group of G we associate explicitly a finite set of irreducible supercuspidal representations of G(F), and relate its internal structure to the centralizer of the parameter. We give evidence that this assignment is an explicit realization of the local Langlands correspondence.

Unitriangular Shape of Decomposition Matrices of Unipotent Blocks

Abstract: We show that the decomposition matrix of unipotent $\ell$-blocks of a finite reductive group $\mathbf{G}(\mathbb{F}_q)$ has a unitriangular shape, assuming $q$ is a power of a good prime and $\ell$ is very good for $\mathbf{G}$. This was conjectured by Geck in 1990 as part of his PhD thesis. We establish this result by constructing projective modules using a modification of generalised Gelfand--Graev characters introduced by Kawanaka. We prove that each such character has at most one unipotent constituent which occurs with multiplicity one. This establishes a 30 year old conjecture of Kawanaka.

On the generalized Springer correspondence

Abstract: We close a gap in the explicit determination of the generalized Springer correspondence for a connected reductive group in good characteristic.

Flexibility of normal affine horospherical varieties

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.

Automorphic L-invariants for reductive groups

Abstract: Let $G$ be a reductive group over a number field $F$, which is split at a finite place $\mathfrak{p}$ of $F$, and let $\pi$ be a cuspidal automorphic representation of $G$, which is cohomological with respect to the trivial coefficient system and Steinberg at $\mathfrak{p}$. We use the cohomology of $\mathfrak{p}$-arithmetic subgroups of $G$ to attach automorphic $\mathcal{L}$-invariants to $\pi$. This generalizes a construction of Darmon (respectively Spies), who considered the case $G=GL_2$ over the rationals (respectively over a totally real number field). These $\mathcal{L}$-invariants depend a priori on a choice of degree of cohomology, in which the representation $\pi$ occurs. We show that they are independent of this choice provided that the $\pi$-isotypical part of cohomology is cyclic over Venkatesh's derived Hecke algebra. Further, we show that automorphic $\mathcal{L}$-invariants can be detected by completed cohomology. Combined with a local-global compatibility result of Ding it follows that for certain representations of definite unitary groups the automorphic $\mathcal{L}$-invariants are equal to the Fontaine-Mazur $\mathcal{L}$-invariants of the associated Galois representation.

Local theta correspondence and nilpotent invariants

Abstract: We consider two types of nilpotent invariants associated to smooth representations, namely generalized Whittaker models, and associated characters (in the case of a real reductive group). We survey some recent results on the behavior of these nilpotent invariants under local theta correspondence, and highlight the special role of a certain double fiberation of moment maps.

On loop Deligne--Lusztig varieties of Coxeter-type for inner forms of ${\rm GL}_n$

Abstract: For a reductive group $G$ over a local non-archimedean field $K$ one can mimic the construction from the classical Deligne--Lusztig theory by using the loop space functor. We study this construction in special the case that $G$ is an inner form of ${\rm GL}_n$ and the loop Deligne--Lusztig variety is of Coxeter type. After simplifying the proof of its representability, our main result is that its $\ell$-adic cohomology realizes many irreducible supercuspidal representations of $G$, notably almost all among those whose L-parameter factors through an unramified elliptic maximal torus of $G$. This gives a purely local, purely geometric and -- in a sense -- quite explicit way to realize special cases of the local Langlands and Jacquet--Langlands correspondences.

Tame cuspidal representations in non-defining characteristics

Abstract: Let F be a non-archimedean local field of odd residual characteristic p. Let G be a (connected) reductive group that splits over a tamely ramified field extension of F. We show that a construction analogous to Yu's construction of complex supercuspidal representations yields smooth, irreducible, cuspidal representations over an arbitrary algebraically closed field R of characteristic different from p. Moreover, we prove that this construction provides all smooth, irreducible, cuspidal R-representations if p does not divide the order of the Weyl group of G.