La theorie generale des groupes algebriques semi-simples sur un corps K quelconque (racines, groupe de Weyl, sous-groupes paraboliques, BN-paire associee a un sous-groupe parabolique minimal, etc) est maintenant bien connue …
La theorie generale des groupes algebriques semi-simples sur un corps K quelconque (racines, groupe de Weyl, sous-groupes paraboliques, BN-paire associee a un sous-groupe parabolique minimal, etc) est maintenant bien connue (cf. [2] et [12]). Notre but est d’exposer une theorie analogue lorsque K est un corps local de corps residuel k. Un groupe algebrique simple simplement connexe G defini sur K apparait alors comme une sorte de «groupe algebrique de dimension infinie» sur le corps residuel, plus precisement comme une limite inductive de limites projectives de varietes algebriques sur k. En particulier, on obtient dans G K une BN-paire (B, N) de groupe de Weyl en general infini (isomorphe au groupe de Weyl affiine d’un systeme de racines), qui est caracterisee (lorsque G n’est pas anisotrope sur K) par la propriete suivante: un sous-groupe de G K est borne (au sens de la valuation de K) si et seulement si il est contenu dans la reunion d’un nombre fini de doubles classes modulo B. Ceci permet la classification (a automorphismes interieurs pres) des sousgroupes bornes maximaux (c’est-a-dire des sous-groupes compacts maximaux lorsque K est localement compact) de G K : on trouve qu’il y en a exactement l+1 classes, ou l est le rang relatif de G sur K.
We develop a Deligne-Lusztig theory for thé complex characters of a non-connected reductive group over a finite field.RÉSUMÉ.-Nous développons une théorie de Deligne-Lustzig pour les caractères d'un groupe réductif non …
We develop a Deligne-Lusztig theory for thé complex characters of a non-connected reductive group over a finite field.RÉSUMÉ.-Nous développons une théorie de Deligne-Lustzig pour les caractères d'un groupe réductif non connexe sur un corps fini.PROPOSITION 1.6.-Soit a un automorphisme quasi-semi-simple d'un groupe réductif connexe, et soit L C P un sous-groupe de Levi d'un sous-groupe parabolique de G. Alors, si la G-orbite (pour l'action par conjugaison) du couple L C P est à-stable, elle contient un couple à-stable.Preuve.-L'hypothèse dit qu'il existe g ç G tel que ^(L, P) = (L, P).Soit T C B un couple formé d'un tore maximal de L et d'un sous-groupe de Borel de P le contenant.Deux tels couples sont conjugués par L, donc il existe l G L tel que ^(T, B) = ^(T, B).D'autre part, a étant quasi-semi-simple, il existe un couple formé d'un tore maximal cr-stable de G et d'un sous-groupe de Borel à-stable le contenant.Ce couple est conjugué à T C B par un élément h G G. On a donc:d'où h~1 0 ' hg~11 e T, donc "hg^ l et h ont même action sur (L, P) ; on en déduit que le couple ^(L, P) est à-stable, ce qui démontre la proposition, car: ^(L, P)) = '^(L, P)) = '^(L, P) = ^-1 \L^ P) = \L^ P).•Nous étudions maintenant les points fixes d'un automorphisme quasi-semi-simple, ce qui généralise l'étude du centraliseur d'un élément semi-simple.Notons que le (iii) de la proposition suivante montre que si un élément d'un groupe réductif est quasi-semi-simple, il l'est dans tout « Borel » le contenant.
Soient G un groupe algébrique réductif connexe, K le sous-groupe des points fixes d'un automorphisme involutif de G, le corps de base étant algébriquement clos et de caractéristique nulle.Via l'étude …
Soient G un groupe algébrique réductif connexe, K le sous-groupe des points fixes d'un automorphisme involutif de G, le corps de base étant algébriquement clos et de caractéristique nulle.Via l'étude de l'opération de K dans certains cônes presque homogènes pour G, on détermine l'ensemble des types de G-modules rationnels irréductibles de dimension finie possédant des invariants non nuls par K.
We prove the existence of weak integral canonical models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we solve a conjecture of …
We prove the existence of weak integral canonical models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we solve a conjecture of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence of integral canonical models of Shimura varieties of preabelian (resp. of abelian) type in mixed characteristic $(0,p)$ with $p\Ge 3$ (resp. with $p=2$) and with respect to hyperspecial subgroups; if $p=3$ (resp. if $p=2$) we restrict in this part I either to the $A_n$, $C_n$, $D_n^{\dbH}$ (resp. $A_n$ and $C_n$) types or to the $B_n$ and $D_n^{\dbR}$ (resp. $B_n$, $D_n^{\dbH}$ and $D_n^{\dbR}$) types which have compact factors over $\dbR$ (resp. which have compact factors over $\dbR$ in some $p$-compact sense). Though the second application is new just for $p\Le 3$, a great part of its proof is new even for $p\Ge 5$ and corrects [Va1, 6.4.11] in most of the cases. The second application forms progress towards the proof of a conjecture of Milne. It also provides in arbitrary mixed characteristic the very first examples of general nature of projective varieties over number fields which are not embeddable into abelian varieties and which have Néron models over certain local rings of rings of integers of number fields.
This paper was motivated by a question of Vilonen, and the main results have been used by Mirković and Vilonen to give a geometric interpretation of the dual group (as …
This paper was motivated by a question of Vilonen, and the main results have been used by Mirković and Vilonen to give a geometric interpretation of the dual group (as a Chevalley group over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Z})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a reductive group. We define a quasi-reductive group over a discrete valuation ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to be an affine flat group scheme over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that (i) the fibers are of finite type and of the same dimension; (ii) the generic fiber is smooth and connected, and (iii) the identity component of the reduced special fiber is a reductive group. We show that such a group scheme is of finite type over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the generic fiber is a reductive group, the special fiber is connected, and the group scheme is smooth over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in most cases, for example when the residue characteristic is not 2, or when the generic fiber and reduced special fiber are of the same type as reductive groups. We also obtain results about group schemes over a Dedekind scheme or a Noetherian scheme. We show that in residue characteristic 2 there are non-smooth quasi-reductive group schemes with generic fiber <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper O Subscript 2 n plus 1"> <mml:semantics> <mml:msub> <mml:mi>SO</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\operatorname {SO}_{2n+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and they can be classified when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is strictly Henselian.
Schneider–Stuhler and Vignéras have used cosheaves on the affine Bruhat–Tits building to construct natural projective resolutions of finite type for admissible representations of reductive p-adic groups in characteristic not equal …
Schneider–Stuhler and Vignéras have used cosheaves on the affine Bruhat–Tits building to construct natural projective resolutions of finite type for admissible representations of reductive p-adic groups in characteristic not equal to p. We use a system of idempotent endomorphisms of a representation with certain properties to construct a cosheaf and a sheaf on the building and to establish that these are acyclic and compute homology and cohomology with these coefficients. This implies Bernstein's result that certain subcategories of the category of representations are Serre subcategories. Furthermore, we also get results for convex subcomplexes of the building. Following work of Korman, this leads to trace formulas for admissible representations.
In this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence …
In this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them, analogue to a differentiability property. As an application of this convergence, we prove that the Hausdorff dimension of the limit set of a hyperconvex representation is equal to a suitably chosen critical exponent.
We obtain the optimal global upper and lower bounds for the transition density $p_n(x,y)$ of a finite range isotropic random walk on affine buildings. We present also sharp estimates for …
We obtain the optimal global upper and lower bounds for the transition density $p_n(x,y)$ of a finite range isotropic random walk on affine buildings. We present also sharp estimates for the corresponding Green function.
We consider affine buildings with refined chamber structure. For each vertex <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we construct a contraction, based at <inline-formula …
We consider affine buildings with refined chamber structure. For each vertex <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we construct a contraction, based at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, that is used to prove exactness of Schneider-Stuhler resolutions of arbitrary depth.
In this paper, we establish strong and <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>Δ</mi> </math> convergence results for mappings satisfying condition <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mfenced open="(" close=")"> <mrow> <msub> <mrow> <mi>B</mi> </mrow> <mrow> …
In this paper, we establish strong and <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>Δ</mi> </math> convergence results for mappings satisfying condition <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mfenced open="(" close=")"> <mrow> <msub> <mrow> <mi>B</mi> </mrow> <mrow> <mi>γ</mi> <mo>,</mo> <mi>μ</mi> </mrow> </msub> </mrow> </mfenced> </math> through a newly introduced iterative process called JA iteration process. A nonlinear Hadamard space is used the ground space for establishing our main results. A novel example is provided for the support of our main results and claims. The presented results are the good extension of the corresponding results present in the literature.
We consider norms on a complex separable Hilbert space such that $\langle a\xi ,\xi \rangle \leq \|\xi \|^2\leq \langle b\xi ,\xi \rangle$ for positive invertible operators $a$ and $b$ that …
We consider norms on a complex separable Hilbert space such that $\langle a\xi ,\xi \rangle \leq \|\xi \|^2\leq \langle b\xi ,\xi \rangle$ for positive invertible operators $a$ and $b$ that differ by an operator in the Schatten class. We prove that these norms have unitarizable isometry groups. As a result, if their isometry groups do not leave any finite dimensional subspace invariant, then the norms must be Hilbertian. The approach involves metric geometric arguments related to the canonical action on the non-positively curved space of positive invertible Schatten perturbations of the identity. Our proof of the main result uses a generalization of a unitarization theorem which follows from the Bruhat-Tits fixed point theorem.
We give a quite general construction of irreducible supercuspidal representations and supercuspidal types (in the sense of Bushnell and Kutzko) of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> …
We give a quite general construction of irreducible supercuspidal representations and supercuspidal types (in the sense of Bushnell and Kutzko) of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic groups. In the tame case, the construction should include all known constructions, and it is expected that this gives all supercuspidal representations. We also give a conjectural Hecke algebra isomorphism, which can be used to analyze arbitrary irreducible admissible representations, following the ideas of Howe and Moy.
This paper concerns the dimensions of certain affine Deligne-Lusztig varieties, both in the affine Grassmannian and in the affine flag manifold.Rapoport conjectured a formula for the dimensions of the varieties …
This paper concerns the dimensions of certain affine Deligne-Lusztig varieties, both in the affine Grassmannian and in the affine flag manifold.Rapoport conjectured a formula for the dimensions of the varieties Xμ(b) in the affine Grassmannian.We prove his conjecture for b in the split torus; we find that these varieties are equidimensional; and we reduce the general conjecture to the case of superbasic b.In the affine flag manifold, we prove a formula that reduces the dimension question for Xx(b) with b in the split torus to computations of dimensions of intersections of Iwahori orbits with orbits of the unipotent radical.Calculations using this formula allow us to verify a conjecture of Reuman in many new cases, and to make progress toward a generalization of his conjecture.
We compare precisely and explicitly Bushnell-Kutzko and Yu's constructions of supercuspidal representations. At the end, we draw conclusions and ask a natural question about the existence of a general construction.
We compare precisely and explicitly Bushnell-Kutzko and Yu's constructions of supercuspidal representations. At the end, we draw conclusions and ask a natural question about the existence of a general construction.
Let G be a reductive group over a non-Archimedean local field. Then the canonical functor from the derived category of smooth tempered representations of G to the derived category of …
Let G be a reductive group over a non-Archimedean local field. Then the canonical functor from the derived category of smooth tempered representations of G to the derived category of all smooth representations of G is fully faithful. Here we consider representations on bornological vector spaces. As a consequence, if V and W are two tempered irreducible representations and if V or W is square-integrable, then Ext_G^n(V,W) vanishes for all n>0. We use this to prove in full generality a formula for the formal dimension of square-integrable representations due to Schneider and Stuhler.
A masure (a.k.a affine ordered hovel) I is a generalization of the Bruhat-Tits building that is associated to a split Kac-Moody group G over a non-archimedean local field. This is …
A masure (a.k.a affine ordered hovel) I is a generalization of the Bruhat-Tits building that is associated to a split Kac-Moody group G over a non-archimedean local field. This is a union of affine spaces called apartments. When G is a reductive group, I is a building and there is a G-invariant distance inducing a norm on each apartment. In this paper, we study distances on I inducing the affine topology on each apartment. We show that some properties (completeness, local compactness, ...) cannot be satisfyed when G is not reductive. Nevertheless, we construct distances such that each element of G is a continuous automorphism of I.
Abstract Sarnak’s density conjecture is an explicit bound on the multiplicities of nontempered representations in a sequence of cocompact congruence arithmetic lattices in a semisimple Lie group, which is motivated …
Abstract Sarnak’s density conjecture is an explicit bound on the multiplicities of nontempered representations in a sequence of cocompact congruence arithmetic lattices in a semisimple Lie group, which is motivated by the work of Sarnak and Xue ([58]). The goal of this work is to discuss similar hypotheses, their interrelation and their applications. We mainly focus on two properties – the spectral spherical density hypothesis and the geometric Weak injective radius property. Our results are strongest in the p -adic case, where we show that the two properties are equivalent, and both imply Sarnak’s general density hypothesis. One possible application is that either the spherical density hypothesis or the Weak injective radius property imply Sarnak’s optimal lifting property ([57]). Conjecturally, all those properties should hold in great generality. We hope that this work will motivate their proofs in new cases.
This paper is a continuation of the paper [14], henceforth cited as I. Let k be a field of characteristic $ and ks a separable closure of k. Let G/k …
This paper is a continuation of the paper [14], henceforth cited as I. Let k be a field of characteristic $ and ks a separable closure of k. Let G/k be a semisimple simply connected algebraic group which is assumed absolutely almost k-simple, i.e. G × kks is almost simple. Let $ be an element of order n. The goal of the paper is to define and study an invariant $ with value in K2(k)/nK2(k), where K2(k) denotes the second Milnor K-group of k. This invariant is characteristic-free and therefore it permits us to work in an arithmetical set-up. The case when n = p = char(k) > 0 and G is split is especially interesting, and an element of order p is then unipotent. In [28], Tits constructed explicit examples of anisotropic unipotent elements, i.e. unipotent elements which do not belong to any proper k-parabolic subgroup of G, and a large part of this paper is devoted to the computation of the invariant for such elements. For the split groups of type G2. (resp. F4, E8) and n =p = 2 (resp. 3, 5), we show that this invariant classifies conjugacy classes of anisotropic elements of order p, and it turns out that any such element is conjugate to one of the anisotropic elements constructed by Tits. In particular, we prove the remaining case of Theorem 1.3, i.e. that any anisotropic unipotent element of order 5 in the split group E8/k normalizes a maximal k-split torus.
We show some of the conjectures of Pappas and Rapoport concerning the moduli stack $${{\rm Bun}_\mathcal {G}}$$ of $${\mathcal {G}}$$ -torsors on a curve C, where $${\mathcal {G}}$$ is a …
We show some of the conjectures of Pappas and Rapoport concerning the moduli stack $${{\rm Bun}_\mathcal {G}}$$ of $${\mathcal {G}}$$ -torsors on a curve C, where $${\mathcal {G}}$$ is a semisimple Bruhat-Tits group scheme on C. In particular we prove the analog of the uniformization theorem of Drinfeld-Simpson in this setting. Furthermore we apply this to compute the connected components of these moduli stacks and to calculate the Picard group of $${{\rm Bun}_\mathcal {G}}$$ in case $${\mathcal {G}}$$ is simply connected.
We introduce a conjugation invariant normalized height h(F ) on finite subsets of matrices F in GL d (Q) and describe its properties.In particular, we prove an analogue of the …
We introduce a conjugation invariant normalized height h(F ) on finite subsets of matrices F in GL d (Q) and describe its properties.In particular, we prove an analogue of the Lehmer problem for this height by showing that h(F ) > ε whenever F generates a nonvirtually solvable subgroup of GL d (Q), where ε = ε(d) > 0 is an absolute constant.This can be seen as a global adelic analog of the classical Margulis Lemma from hyperbolic geometry.As an application we prove a uniform version of the classical Burnside-Schur theorem on torsion linear groups.In a companion paper we will apply these results to prove a strong uniform version of the Tits alternative.
We call a non-discrete Euclidean building a {\it Bruhat-Tits space} if its automorphism group contains a subgroup that induces the subgroup generated by all the root groups of a root …
We call a non-discrete Euclidean building a {\it Bruhat-Tits space} if its automorphism group contains a subgroup that induces the subgroup generated by all the root groups of a root datum of the building at infinity. This is the class of non-discrete Euclidean buildings introduced and studied by Bruhat and Tits. We give the complete classification of Bruhat-Tits spaces whose building at infinity is the fixed point set of a polarity of an ambient building of type $\hbox{\sf B}_2$, $\hbox{\sf F}_4$ or $\hbox{\sf G}_2$ associated with a Ree or Suzuki group endowed with the usual root datum. (In the $\hbox{\sf B}_2$ and $\hbox{\sf G}_2$ cases, this fixed point set is a building of rank one; in the $\hbox{\sf F}_4$ case, it is a generalized octagon whose Weyl group is not crystallographic.) We also show that each of these Bruhat-Tits spaces has a natural embedding in the unique Bruhat-Tits space whose building at infinity is the corresponding ambient building.
We apply G. Prasad's volume formula for the arithmetic quotients of semi-simple groups and Bruhat-Tits theory to study the covolumes of arithmetic subgroups of SO(1,n). As a result we prove …
We apply G. Prasad's volume formula for the arithmetic quotients of semi-simple groups and Bruhat-Tits theory to study the covolumes of arithmetic subgroups of SO(1,n). As a result we prove that for any even dimension n there exists a unique compact arithmetic hyperbolic n-orbifold of the smallest volume. We give a formula for the Euler-Poincare characteristic of the orbifolds and present an explicit description of their fundamental groups as the stabilizers of certain lattices in quadratic spaces. We also study hyperbolic 4-manifolds defined arithmetically and obtain a number theoretical characterization of the smallest compact arithmetic 4-manifold.
The main purpose of this paper is to prove a group-theoretic generalization of a theorem of Katz on isocrystals. Along the way we reprove the group-theoretic generalization of Mazur's inequality …
The main purpose of this paper is to prove a group-theoretic generalization of a theorem of Katz on isocrystals. Along the way we reprove the group-theoretic generalization of Mazur's inequality for isocrystals due to Rapoport-Richartz, and generalize from split groups to unramified groups a result of Kottwitz-Rapoport which determines when an affine Deligne-Lusztig subset of the affine Grassmannian is non-empty.
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 …
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 |$(x, r, \beta)$|, where |$x$| is a point in the Bruhat-Tits building of the reductive group |$G$|, |$r$| is a nonnegative real number, and |$\beta$| is a semistable functional on the degree |$r$| associated graded piece of the Moy–Prasad filtration corresponding to |$x$|. Recent work on the wild ramification case of the geometric Langlands conjectures suggests that fundamental strata also play a role in the geometric setting. In this paper, we develop a theory of minimal |$K$|-types for formal flat |$G$|-bundles. We show that any formal flat |$G$|-bundle contains a fundamental stratum; moreover, all such strata have the same rational depth. We thus obtain a new invariant of a flat |$G$|-bundle called the slope, generalizing the classical definition for flat vector bundles. The slope can also be realized as the minimum depth of a stratum contained in the flat |$G$|-bundle, and in the case of positive slope, all such minimal depth strata are fundamental. Finally, we show that a flat |$G$|-bundle is irregular singular if and only if it has positive slope.
Let $G$ be a connected reductive algebraic group over a non-Archimedean local field $K$, and let $g$ be its Lie algebra. By a theorem of Harish-Chandra, if $K$ has characteristic …
Let $G$ be a connected reductive algebraic group over a non-Archimedean local field $K$, and let $g$ be its Lie algebra. By a theorem of Harish-Chandra, if $K$ has characteristic zero, the Fourier transforms of orbital integrals are represented on the set of regular elements in $g(K)$ by locally constant which, extended by zero to all of $g(K)$, are locally integrable. In this paper, we prove that these functions are in fact specializations of constructible motivic exponential functions. Combining this with the Principle for integrability [R. Cluckers, J. Gordon, I. Halupczok, Transfer principles for integrability and boundedness conditions for motivic exponential functions, preprint arXiv:1111.4405], we obtain that Harish-Chandra's theorem holds also when $K$ is a non-Archimedean local field of sufficiently large positive characteristic. Under the hypothesis on the existence of the mock exponential map, this also implies local integrability of Harish-Chandra characters of admissible representations of $G(K)$, where $K$ is an equicharacteristic field of sufficiently large (depending on the root datum of $G$) characteristic.
This paper is a development of [4], and gives a more detailed treatment of the topic named in the title.It includes in particular the birational equivalence with affine space, over …
This paper is a development of [4], and gives a more detailed treatment of the topic named in the title.It includes in particular the birational equivalence with affine space, over the groundfield, of the variety of Cartan subgroups of a k-group G, the splitting of G over a separable extension of k if G is reductive, some results on unipotent groups operated upon by tori, and on the existence of subgroups of G whose Lie algebra contains a given nilpotent element of the Lie algebra g of G.Discussing as it does a number of known results (due mostly to Rosenlicht and Grothendieck), this paper is to be viewed as partly expository.In fact, besides proving some new results, our main goal is to provide a rather comprehensive, albeit not exhaustive, account of our topic, from the point of view sketched in [4],Our basic tools are some rationality properties of transversal intersections and of separable mappings, the Jordan decomposition in g, and purely inseparable isogenies of height one.They are reviewed or discussed in section 1.13, §3 and §5 respectively.Thus Lie algebras of algebraic groups play an important role in this paper and, for the sake of completeness, we have collected in §1 a number of definitions and facts pertaining to them.§2 reproves a result of Grothendieck ([12], Exp.XIV) stating that g is the union of the subalgebras of its Borel subgroups.Its main use for us is to reduce to Lie algebras of solvable groups the existence proof of the Jordan decomposition, §4 discusses subalgebras § of g consisting of semi-simple elements, to be called "toral subalgebras" of g.They are tangent to maximal tori, and have several properties similar to that of tori in G, in particular: the centralizer Z( 8)={g*G 9 Adg(X) = X(X € 8)} of jg in G is defined over k if 3 is, (see 4.3 for Z(3)°, 6.14 for Z(β)\ its Lie algebra is 3(8) = {X z g, [8, X] = 0}.If § is spanned by one element X, the conjugacy class of X is isomorphic to G/Z(β).This paragraph also gives some conditions under which a subalgebra of g is algebraic, and reproves some results of Chevalley [8] in characteristic zero.§6 introduces regular elements, Cartan subalgebras in g, and the subgroups of type (C) of ([12], Exp.XΠI) in G.By definition here, a Cartan subalgebra
Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact …
Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces On the classification of symmetric spaces Functions on symmetric spaces Bibliography List of notational conventions Symbols frequently used Author index Subject index Reviews for the first edition.
Article Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque. was published on January 1, 1971 in the journal Journal für die reine und angewandte Mathematik (volume 1971, issue …
Article Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque. was published on January 1, 1971 in the journal Journal für die reine und angewandte Mathematik (volume 1971, issue 247).
By A. Borel: pp. x, 398; Cloth $12.50, Paper $4.95. (W. A. Benjamin, Inc., New York, 1969).
By A. Borel: pp. x, 398; Cloth $12.50, Paper $4.95. (W. A. Benjamin, Inc., New York, 1969).