Split Reductive Group Research Articles

Chow groups of products of Severi-Brauer varieties and invariants of degree 3

We study the semi-decomposable invariants of a split semisimple group and their extension to a split reductive group by using the torsion in the codimension 2 2 Chow groups of a product of Severi-Brauer varieties. In particular, for any n ≥ 2 n\geq 2 we completely determine the degree 3 3 invariants of a split semisimple group, the quotient of ( S L 2 ) n (\mathbf {SL}_{2})^{n} by its maximal central subgroup, as well as of the corresponding split reductive group. We also provide an example illustrating that a modification of our method can be applied to find the semi-decomposable invariants of a split semisimple group of type A.

Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups

AbstractThe representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the questions of whether a given complex representation is symplectic or orthogonal have been solved since at least the 1950s. Similar results for Weyl modules of split reductive groups over fields of characteristic different from z hold by using similar proofs. This paper considers analogues of these results for simple, induced, and tilting modules of split reductive groups over fields of prime characteristic as well as a complete answer for Weyl modules over fields of characteristic 2.

Dimensions of some locally analytic representations

Let G G be the group of points of a split reductive group over a finite extension of Q p \mathbb {Q}_p . In this paper, we compute the dimensions of certain classes of locally analytic G G -representations. This includes principal series representations and certain representations coming from homogeneous line bundles on p p -adic symmetric spaces. As an application, we compute the dimensions of the unitary GL 2 ( Q p ) \textrm {GL}_2(\mathbb {Q}_p) -representations appearing in Colmez’ p p -adic local Langlands correspondence.

From pro-p Iwahori–Hecke modules to (φ,Γ)-modules, I

Let ${\mathfrak o}$ be the ring of integers in a finite extension $K$ of ${\mathbb Q}_p$, let $k$ be its residue field. Let $G$ be a split reductive group over ${\mathbb Q}_p$, let $T$ be a maximal split torus in $G$. Let ${\mathcal H}(G,I_0)$ be the pro-$p$-Iwahori Hecke ${\mathfrak o}$-algebra. Given a semiinfinite reduced chamber gallery (alcove walk) $C^{({\bullet})}$ in the $T$-stable apartment, a period $\phi\in N(T)$ of $C^{({\bullet})}$ of length $r$ and a homomorphism $\tau:{\mathbb Z}_p^{\times}\to T$ compatible with $\phi$, we construct a functor from the category ${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$ of finite length ${\mathcal H}(G,I_0)$-modules to \'{e}tale $(\varphi^r,\Gamma)$-modules over Fontaine's ring ${\mathcal O}_{\mathcal E}$. If $G={\rm GL}_{d+1}({\mathbb Q}_p)$ there are essentially two choices of ($C^{({\bullet})}$, $\phi$, $\tau$) with $r=1$, both leading to a functor from ${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$ to \'{e}tale $(\varphi,\Gamma)$-modules and hence to ${\rm Gal}_{{\mathbb Q}_p}$-representations. Both induce a bijection between the set of absolutely simple supersingular ${\mathcal H}(G,I_0)\otimes_{\mathfrak o} k$-modules of dimension $d+1$ and the set of irreducible representations of ${\rm Gal}_{{\mathbb Q}_p}$ over $k$ of dimension $d+1$. We also compute these functors on modular reductions of tamely ramified locally unitary principal series representations of $G$ over $K$. For $d=1$ we recover Colmez' functor (when restricted to ${\mathfrak o}$-torsion ${\rm GL}_{2}({\mathbb Q}_p)$-representations generated by their pro-$p$-Iwahori invariants)

Eisenstein congruences for split reductive groups

We present a general conjecture on congruences between Hecke eigenvalues of parabolically induced and cuspidal automorphic representations of split reductive groups, modulo divisors of critical values of certain L-functions. We examine the consequences in several special cases and use the Bloch–Kato conjecture to further motivate a belief in the congruences.

Compactifications of reductive groups as moduli stacks of bundles

Let$G$be a split reductive group. We introduce the moduli problem ofbundle chainsparametrizing framed principal$G$-bundles on chains of lines. Any fan supported in a Weyl chamber determines a stability condition on bundle chains. Its moduli stack provides an equivariant toroidal compactification of$G$. All toric orbifolds may be thus obtained. Moreover, we get a canonical compactification of any semisimple$G$, which agrees with the wonderful compactification in the adjoint case, but which in other cases is an orbifold. Finally, we describe the connections with Losev–Manin’s spaces of weighted pointed curves and with Kausz’s compactification of$GL_{n}$.

Whittaker Coefficients of Metaplectic Eisenstein Series

We study Whittaker coefficients for maximal parabolic Eisenstein series on metaplectic covers of split reductive groups. By the theory of Eisenstein series these coefficients have meromorphic continuation and functional equation. However they are not Eulerian and the standard methods to compute them in the reductive case do not apply to covers. For “cominuscule” maximal parabolics, we give an explicit description of the coefficients as Dirichlet series whose arithmetic content is expressed in an exponential sum. The exponential sum is then shown to satisfy a twisted multiplicativity, reducing its determination to prime power contributions. These, in turn, are connected to Lusztig data for canonical bases on the dual group using a result of Kamnitzer. The exponential sum at prime powers is shown to simplify for generic Lusztig data. At the remaining degenerate cases, the exponential sum seems best expressed in terms of Gauss sums depending on string data for canonical bases, as shown in a detailed example in GL 4. Thus we demonstrate that the arithmetic part of metaplectic Whittaker coefficients is intimately connected to the relations between these two expressions for canonical bases.

Geometric Satake, Springer correspondence, and small representations II

For a split reductive group scheme G ˇ \check G over a commutative ring k \Bbbk with Weyl group W W , there is an important functor R e p ( G ˇ , k ) → R e p ( W , k ) {\mathsf {Rep}}(\check G,\Bbbk )\to {\mathsf {Rep}}(W,\Bbbk ) defined by taking the zero weight space. We prove that the restriction of this functor to the subcategory of small representations has an alternative geometric description, in terms of the affine Grassmannian and the nilpotent cone of the Langlands dual group G G . The translation from representation theory to geometry is via the Satake equivalence and the Springer correspondence. This generalizes the result for the k = C \Bbbk =\mathbb {C} case proved by the first two authors, and also provides a better explanation than in the earlier paper, since the current proof is uniform across all types.

Geometry of canonical bases and mirror symmetry

A decorated surface S is an oriented surface with boundary and a finite, possibly empty, set of special points on the boundary, considered modulo isotopy. Let G be a split reductive group over Q. A pair (G, S) gives rise to a moduli spaceAG,S , closely related to the moduli space of G-local systems on S. It is equipped with a positive structure (Fock and Goncharov, Publ Math IHES 103:1–212, 2006). So a set AG,S(Z ) of its integral tropical points is defined. We introduce a rational positive function W on the space AG,S , called the potential. Its tropicalisation is a function W t : AG,S(Z ) → Z. The condition W t ≥ 0 defines a subset of positive integral tropical points AG,S(Zt ). For G = SL2, we recover the set of positive integralA-laminations on S from Fock and Goncharov (Publ Math IHES 103:1–212, 2006). We prove that when S is a disc with n special points on the boundary, the set AG,S(Zt ) parametrises top dimensional components of the fibers of the convolution maps. Therefore, via the geometric Satake correspondence (Lusztig, Asterisque 101–102:208– 229, 1983; Ginzburg,1995; Mirkovic and Vilonen, Ann Math (2) 166(1):95– 143, 2007; Beilinson and Drinfeld, Chiral algebras. American Mathematical Society Colloquium Publications, vol. 51, 2004) they provide a canonical basis in the tensor product invariants of irreducible modules of the Langlands dual group GL : (Vλ1 ⊗ . . .⊗ Vλn )G L . (1) A. Goncharov (B) · L. Shen Mathematics Department, Yale University, New Haven, CT 06520, USA e-mail: alexander.goncharov@yale.edu

Geometrization of principal series representations of reductive groups

In geometric representation theory, one often wishes to describe representations realized on spaces of invariant functions as trace functions of equivariant perverse sheaves. In the case of principal series representations of a connected split reductive group G over a local field, there is a description of families of these representations realized on spaces of functions on G invariant under the translation action of the Iwahori subgroup, or a suitable smaller compact open subgroup, studied by Howe, Bushnell and Kutzko, Roche, and others. In this paper, we construct categories of perverse sheaves whose traces recover the families associated to regular characters of T(F[[t]]), and prove conjectures of Drinfeld on their structure. We also propose conjectures on the geometrization of families associated to more general characters.

Dirac cohomology of one-$W$-type representations

The smooth hermitian representations of a split reductive p-adic group whose restriction to a maximal hyperspecial compact subgroup contain a single K-type with Iwahori fixed vectors have been studied in [D. Barbasch, A. Moy, Classification of one K-type representations, Trans. Amer. Math. Soc. 351 (1999), no. 10, 4245-4261] in the more general setting of modules for graded affine Hecke algebras with parameters. We show that every such one K-type module has nonzero Dirac cohomology (in the sense of [D. Barbasch, D. Ciubotaru, P. Trapa, The Dirac operator for graded affine Hecke algebras, arXiv:1006.3822]), and use Dirac operator techniques to determine the semisimple part of the Langlands parameter for these modules, thus completing their classification.

Vanishing of certain equivariant distributions on spherical spaces

We prove vanishing of distributions on a split real reductive group which change according to a non-degenerate character under the left action of the unipotent radical of the Borel subgroup, and are equivariant under the right action of a spherical subgroup, and eigen with respect to the center of the universal enveloping algebra of the Lie algebra of G. This is a generalization of a result by Shalika, that concerned the group case. Shalika's result was crucial in the proof of his multiplicity one theorem. We view our result as a step in the study of multiplicities of quasi-regular representations on spherical varieties. As an application we prove non-vanishing of spherical Bessel functions.

Compatibility between Satake and Bernstein isomorphisms in characteristicp

We study the center of the pro-p Iwahori-Hecke ring H of a connected split p-adic reductive group G. For k an algebraically closed field with characteristic p, we prove that the center of the k-algebra H_k:= H\otimes_Z k contains an affine semigroup algebra which is naturally isomorphic to the Hecke algebra attached to any irreducible smooth k-representation of a given hyperspecial maximal compact subgroup of G. This isomorphism is obtained using the inverse Satake isomorphism constructed in arXiv:1207.5557. We apply this to classify the simple supersingular H_k-modules, study the supersingular block in the category of finite length H_k-modules, and relate the latter to supersingular representations of G.

Witt Groups of Algebraic Groups

Let G_k be a split reductive group over a subfield k of \mathbb C corresponding to a Lie group G . Let T be a maximal torus of G . We show the isomorphism W^*(G_k)\cong W^*(G_k/T_k) of Balmer–Witt groups. When k is algebraically closed, we prove that W^*(G_k) is isomorphic to the topological K -theory KO^{2*-1}(G/T) of the flag manifold G/T . Then we compute it explicitly by using the fact that W^*(G_k) is a Hopf algebra.

Pro- Iwahori–Hecke algebras are Gorenstein

AbstractLet$\mathfrak{F}$be a locally compact nonarchimedean field with residue characteristic$p$, and let$\mathrm{G} $be the group of$\mathfrak{F}$-rational points of a connected split reductive group over$\mathfrak{F}$. For$k$an arbitrary field of any characteristic, we study the homological properties of the Iwahori–Hecke$k$-algebra${\mathrm{H} }^{\prime } $and of the pro-$p$Iwahori–Hecke$k$-algebra$\mathrm{H} $of$\mathrm{G} $. We prove that both of these algebras are Gorenstein rings with self-injective dimension bounded above by the rank of$\mathrm{G} $. If$\mathrm{G} $is semisimple, we also show that this upper bound is sharp, that both$\mathrm{H} $and${\mathrm{H} }^{\prime } $are Auslander–Gorenstein, and that there is a duality functor on the finite length modules of$\mathrm{H} $(respectively${\mathrm{H} }^{\prime } $). We obtain the analogous Gorenstein and Auslander–Gorenstein properties for the graded rings associated to$\mathrm{H} $and${\mathrm{H} }^{\prime } $.When$k$has characteristic$p$, we prove that in ‘most’ cases$\mathrm{H} $and${\mathrm{H} }^{\prime } $have infinite global dimension. In particular, we deduce that the category of smooth$k$-representations of$\mathrm{G} = {\mathrm{PGL} }_{2} ({ \mathbb{Q} }_{p} )$generated by their invariant vectors under the pro-$p$Iwahori subgroup has infinite global dimension (at least if$k$is algebraically closed).

On a classification of irreducible admissible modulo representations of a -adic split reductive group

AbstractWe give a classification of irreducible admissible modulo $p$ representations of a split$p$-adic reductive group in terms of supersingular representations. This is a generalization of a theorem of Herzig.

Spherical functions on spherical varieties

Let ${\bf X}={\bf H}\backslash{\bf G}$ be a homogeneous spherical variety for a split reductive group ${\bf G}$ over the integers $\frak o$ of a $p$-adic field $k$, and $K={\bf G}(\frak o)$ a hyperspecial maximal compact subgroup of $G={\bf G}(k)$. We compute eigenfunctions (``spherical functions'') on $X={\bf X}(k)$ under the action of the unramified (or spherical) Hecke algebra of $G$, generalizing many classical results of ``Casselman-Shalika'' type. Under some additional assumptions on ${\bf X}$ we also prove a variant of the formula which involves a certain quotient of $L$-values, and we present several applications such as: (1) a statement on ``good test vectors'' in the multiplicity-free case (namely, that an $H$-invariant functional on an irreducible unramified representation $\pi$ is non-zero on $\pi^K$), (2) the unramified Plancherel formula for $X$, including a formula for the ``Tamagawa measure'' of ${\bf X}(\frak o)$, and (3) a computation of the most continuous part of ${\bf H}$-period integrals of principal Eisenstein series.

An integrality theorem of Grosshans over arbitrary base ring

We revisit a theorem of Grosshans and show that it holds over arbitrary commutative base ring k. One considers a split reductive group scheme G acting on a k-algebra A and leaving invariant a subalgebra R. Let U be the unipotent radical of a split Borel subgroup scheme. If R U = A U then the conclusion is that A is integral over R.

On locally analytic Beilinson–Bernstein localization and the canonical dimension

Let $$\mathbf{G}$$ be a connected split reductive group over a $$p$$ -adic field. In the first part of the paper we prove, under certain assumptions on $$\mathbf{G}$$ and the prime $$p$$ , a localization theorem of Beilinson–Bernstein type for admissible locally analytic representations of principal congruence subgroups in the rational points of $$\mathbf{G}$$ . In doing so we take up and extend some recent methods and results of Ardakov–Wadsley on completed universal enveloping algebras (Ardakov and Wadsley, Ann. Math., 2013) to a locally analytic setting. As an application we prove, in the second part of the paper, a locally analytic version of Smith’s theorem on the canonical dimension.

The subrepresentation theorem for automorphic representations

We prove that every irreducible subrepresentation in the space of automorphic forms on G(A), where G is a split reductive group over a number field k, and A is the related ring of adeles, is a subrepresentation of the representation induced from a cuspidal automorphic representation of a Levi subgroup.