A generalization of Kloosterman sums to a simply connected Chevalley group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is discussed. These sums are parameterized …
A generalization of Kloosterman sums to a simply connected Chevalley group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is discussed. These sums are parameterized by pairs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis w comma t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(w,t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w"> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding="application/x-tex">w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an element of the Weyl group of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an element of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf {Q}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-split torus in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L left-parenthesis 2 comma bold upper Q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">SL(2,{\mathbf {Q}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Kloosterman sums coincide with the classical Kloosterman sums and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L left-parenthesis r comma bold upper Q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">SL(r,{\mathbf {Q}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Kloosterman sums, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">r \geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, coincide with the sums introduced in [B-F-G,F,S]. Algebraic properties of the sums are proved by root system methods. In particular an explicit decomposition of a general Kloosterman sum over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf {Q}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into the product of local <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 factors is obtained. Using this factorization one can show that the Kloosterman sums corresponding to a toral element, which acts trivially on the highest weight space of a fundamental irreducible representation, splits into a product of Kloosterman sums for Chevalley groups of lower rank.
Let $\mathfrak{g}$ be a semi-simple Lie algebra over the field $C$ of complex numbers.Then there is a basis of $\mathfrak{g}$ such that every structure constant of $\mathfrak{g}$ with respect to …
Let $\mathfrak{g}$ be a semi-simple Lie algebra over the field $C$ of complex numbers.Then there is a basis of $\mathfrak{g}$ such that every structure constant of $\mathfrak{g}$ with respect to this basis is an integer.This basis generates a Lie algebra $\mathfrak{g}_{Z}$
Introduction Gaussian binomial coefficients The quantized enveloping algebra $U_q(\mathfrak s \mathfrak {1}_2)$ Representations of $U_q(\mathfrak{sl}_2)$ Tensor products or: $U_q(\mathfrak{sl}_2)$ as a Hopf algebra The quantized enveloping algebra $U_q(\mathfrak g)$ Representations …
Introduction Gaussian binomial coefficients The quantized enveloping algebra $U_q(\mathfrak s \mathfrak {1}_2)$ Representations of $U_q(\mathfrak{sl}_2)$ Tensor products or: $U_q(\mathfrak{sl}_2)$ as a Hopf algebra The quantized enveloping algebra $U_q(\mathfrak g)$ Representations of $U_q(\mathfrak g)$ Examples of representations The center and bilinear forms $R$-matrices and $k_q[G]$ Braid group actions and PBW type basis Proof of proposition 8.28 Crystal bases I Crystal bases II Crystal bases III References Notations Index.
Revised second edition. The text covers the material presented for a graduate-level course on quantum groups at Harvard University. Covered topics include: Poisson algebras and quantization, Poisson-Lie groups, coboundary Lie …
Revised second edition. The text covers the material presented for a graduate-level course on quantum groups at Harvard University. Covered topics include: Poisson algebras and quantization, Poisson-Lie groups, coboundary Lie bialgebras, Drinfelds double construction, Belavin-Drinfeld classification, Infinite dimensional Lie bialgebras, Hopf algebras, Quantized universal enveloping algebras, formal groups and h-formal groups, infinite dimensional quantum groups, the quantum double, tensor categories and quasi Hopf-algebras, braided tensor categories, KZ equations and the Drinfeld Category, Quasi-Hpf enveloping algebras, Lie associators, Fiber functors and Tannaka-Driein duality, Quantization of finite Lie bialgebras, Universal constructions, Universal quantization, Dequantization and the equivalence theorem, KZ associator and multiple zeta functions, and Mondoromy of trigonometric KZ equations. Probems are given with each subject and an answer key is included. New paperback re-issue of the revised second edition.
These are expanded lecture notes from lectures given at the Workshop on higher structures at MATRIX Melbourne. These notes give an introduction to Feynman categories and their applications. Feynman categories …
These are expanded lecture notes from lectures given at the Workshop on higher structures at MATRIX Melbourne. These notes give an introduction to Feynman categories and their applications. Feynman categories give a universal categorical way to encode operations and relations. This includes the aspects of operad--like theories such as PROPs, modular operads, twisted (modular) operads, properads, hyperoperads and their colored versions. There is more depth to the general theory as it applies as well to algebras over operads and an abundance of other related structures, such as crossed simplicial groups, the augmented simplicial category or FI--modules. Through decorations and transformations the theory is also related to the geometry of moduli spaces. Furthermore the morphisms in a Feynman category give rise to Hopf-- and bi--algebras with examples coming from topology, number theory and quantum field theory. All these aspects are covered.
Let H be a group acting on a building ∆.We analyze three transitivity properties that this action could have, namely strong, Weyl and weak transitivity.We present and analyze a collection …
Let H be a group acting on a building ∆.We analyze three transitivity properties that this action could have, namely strong, Weyl and weak transitivity.We present and analyze a collection of groups H and buildings ∆ for which the action is not weakly transitive but may nonetheless be Weyl transitive.In these examples, the failure to be weakly transitive is in some sense precisely determined, and in some cases is shown to be extremely severe.The first situation we consider is Chevalley groups.Let K be a local field and G = g(K) a Chevalley group.Let (B, N ) be the standard spherical BN -pair and W = N/B ∩ N the Weyl group.We precisely characterize which elements w of W admit only finite-order representatives in N .In particular for such a w of order m, all representatives of w in N have the same order, and that order is either m or 2m.Using this we can find a variety of subgroups H of G, in particular if H is dense and torsionfree, such that H acts Weyl transitively but not weakly transitively on the affine building arising from G.Next we consider the case of division algebras, where the failure to be weakly transitive can be more precisely characterized and shown to be very extreme.Let D be a finite-dimensional F -division algebra of degree d > 2, and let H be either D × or SL 1 (D).For any splitting field K, H admits an action on the buildings associated to G = GL d (K) or G = SL d (K).It is easy to show that this action is not weakly transitive, and in the present context we can show that it even fails "dramatically" ii to be weakly transitive.If F is a global field we can construct examples where the action of H on the affine building of G is nonetheless Weyl transitive.In the global case, for "most" D we can even show that SL 1 (D) acts on the fundamental affine apartment only by translations -the most extreme possible situation.
In this paper we investigate the image of the $l$-adic representation attached to the Tate module of an abelian variety defined over a number field. We consider simple abelian varieties …
In this paper we investigate the image of the $l$-adic representation attached to the Tate module of an abelian variety defined over a number field. We consider simple abelian varieties of type III in the Albert classification. We compute the image of the $l$-adic and mod $l$ Galois representations and we prove the Mumford-Tate and Lang conjectures for a wide class of simple abelian varieties of type III.
We study the variety of actions of a fixed (Chevalley) group on arbitrary geodesic, Gromov hyperbolic spaces.In high rank we obtain a complete classification.In rank one, we obtain some partial …
We study the variety of actions of a fixed (Chevalley) group on arbitrary geodesic, Gromov hyperbolic spaces.In high rank we obtain a complete classification.In rank one, we obtain some partial results and give a conjectural picture.
Abstract Let G be a simple algebraic group over an algebraically closed field K of characteristic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {p>0} . We consider connected reductive …
Abstract Let G be a simple algebraic group over an algebraically closed field K of characteristic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {p>0} . We consider connected reductive subgroups X of G that contain a given distinguished unipotent element u of G . A result of Testerman and Zalesski [D. Testerman and A. Zalesski, Irreducibility in algebraic groups and regular unipotent elements, Proc. Amer. Math. Soc. 141 2013, 1, 13–28] shows that if u is a regular unipotent element, then X cannot be contained in a proper parabolic subgroup of G . We generalize their result and show that if u has order p , then except for two known examples which occur in the case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi>p</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>C</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {(G,p)=(C_{2},2)} , the subgroup X cannot be contained in a proper parabolic subgroup of G . In the case where u has order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi /> <m:mo>></m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> {>p} , we also present further examples arising from indecomposable tilting modules with quasi-minuscule highest weight.
Let $p$ be a prime integer and $\mathbb{Z}_p$ be the ring of $p$-adic integers. By a purely computational approach we prove that each nonzero normal element of a completed group …
Let $p$ be a prime integer and $\mathbb{Z}_p$ be the ring of $p$-adic integers. By a purely computational approach we prove that each nonzero normal element of a completed group algebra over the special linear group ${\rm SL}_3(\mathbb{Z}_p)$ is a unit. This give a positive answer to an open question in \cite{WeiBian2} and make up for an earlier mistake in \cite{WeiBian1} simultaneously.
Let [Formula: see text] be a group and [Formula: see text] be an automorphism of [Formula: see text]. Two elements [Formula: see text] are said to be [Formula: see text]-twisted …
Let [Formula: see text] be a group and [Formula: see text] be an automorphism of [Formula: see text]. Two elements [Formula: see text] are said to be [Formula: see text]-twisted conjugate if [Formula: see text] for some [Formula: see text]. We say that a group [Formula: see text] has the [Formula: see text]-property if the number of [Formula: see text]-twisted conjugacy classes is infinite for every automorphism [Formula: see text] of [Formula: see text]. In this paper, we prove that twisted Chevalley groups over a field [Formula: see text] of characteristic zero have the [Formula: see text]-property as well as the [Formula: see text]-property if [Formula: see text] has finite transcendence degree over [Formula: see text] or [Formula: see text] is periodic.
After some general remarks about characters of finite groups (possibly twisted by an automorphism), this chapter focuses on the generalised characters $R(T,\theta)$ which where introduced by Deligne and Lustzig using …
After some general remarks about characters of finite groups (possibly twisted by an automorphism), this chapter focuses on the generalised characters $R(T,\theta)$ which where introduced by Deligne and Lustzig using cohomological methods. We refer to the books by Carter and Digne-Michel for proofs of some fundamental properties, like orthogonality relations and degree formulae. Based on these results, we develop in some detail the basic formalism of Lusztig's book, which leads to a classification of the irreducible characters of finite groups of Lie type in terms of a fundamental Jordan decomposition. Using the general theory about regular embeddings in Chapter 1, we state and discuss that Jordan decomposition in complete generality, that is, without any assumption on the center of the underlying algebraic group. The final two sections give an introduction to the problems of computing Green functions and characteristic functions of character sheaves.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
After collectiong some properties of irreducible representations of finite Coxeter groups we state and explain Lusztig's result on the decomposition of Deligne-Lusztig characters and then give a detailed exposition of …
After collectiong some properties of irreducible representations of finite Coxeter groups we state and explain Lusztig's result on the decomposition of Deligne-Lusztig characters and then give a detailed exposition of the parametrisation and the properties of unipotent characters of finite reductive groups and related data like Fourier matrices and eigenvalues of Frobenius. We then describe the decomposition of Lusztig induction and collect the most recent results on its commutation with Jordan decomposition. We end the chapter with a survey of the character theory of finite disconnected reductive groups.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Certain classical generating functions for elements of reflection groups can be expressed using fundamental invariants called <italic>exponents</italic>. We give new analogues of such generating functions that accommodate orbits of reflecting …
Certain classical generating functions for elements of reflection groups can be expressed using fundamental invariants called <italic>exponents</italic>. We give new analogues of such generating functions that accommodate orbits of reflecting hyperplanes using similar invariants we call <italic>reflexponents</italic>. Our verifications are case-by-case.
Communicated by Academician Vyacheslav I. Yanchevskii In a number of cases the minimal polynomials of the images of unipotent elements of non-prime order in irreducible representations of the exceptional algebraic …
Communicated by Academician Vyacheslav I. Yanchevskii In a number of cases the minimal polynomials of the images of unipotent elements of non-prime order in irreducible representations of the exceptional algebraic groups in good characteristics are found. It is proved that if p > 5 for a group of type E 8 and p > 3 for other exceptional algebraic groups, then for irreducible representations of these groups in characteristic p with large highest weights with respect to p, the degree of the minimal polynomial of the image of a unipotent element is equal to the order of this element.
Abstract This is the last in a series of articles where we are concerned with normal elements of noncommutative Iwasawa algebras over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> …
Abstract This is the last in a series of articles where we are concerned with normal elements of noncommutative Iwasawa algebras over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>ℤ</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{SL}_{n}(\mathbb{Z}_{p})} . Our goal in this portion is to give a positive answer to an open question in [D. Han and F. Wei, Normal elements of noncommutative Iwasawa algebras over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mn>3</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>ℤ</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{SL}_{3}(\mathbb{Z}_{p}) , Forum Math. 31 2019, 1, 111–147] and make up for an earlier mistake in [F. Wei and D. Bian, Normal elements of completed group algebras over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>ℤ</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{SL}_{n}(\mathbb{Z}_{p}) , Internat. J. Algebra Comput. 20 2010, 8, 1021–1039] simultaneously. Let n ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {n\geq 2} ) be a positive integer. Let p ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {p>2} ) be a prime integer, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>ℤ</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {\mathbb{Z}_{p}} the ring of p -adic integers and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝔽</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {\mathbb{F}_{p}} the finite filed of p elements. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>Γ</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>ℤ</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {G=\Gamma_{1}(\mathrm{SL}_{n}(\mathbb{Z}_{p}))} be the first congruence subgroup of the special linear group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>ℤ</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{SL}_{n}(\mathbb{Z}_{p})} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>Ω</m:mi> <m:mi>G</m:mi> </m:msub> </m:math> {\Omega_{G}} the mod- p Iwasawa algebra of G defined over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝔽</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {\mathbb{F}_{p}} . By a purely computational approach, for each nonzero element <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>W</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mi>Ω</m:mi> <m:mi>G</m:mi> </m:msub> </m:mrow> </m:math> {W\in\Omega_{G}} , we prove that W is a normal element if and only if W contains constant terms. In this case, W is a unit. Also, the main result has been already proved under “nice prime” condition by Ardakov, Wei and Zhang [Non-existence of reflexive ideals in Iwasawa algebras of Chevalley type, J. Algebra 320 2008, 1, 259–275; Reflexive ideals in Iwasawa algebras, Adv. Math. 218 2008, 3, 865–901]. This paper currently provides a new proof without the “nice prime” condition. As a consequence of the above-mentioned main result, we observe that the center of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>Ω</m:mi> <m:mi>G</m:mi> </m:msub> </m:math> {\Omega_{G}} is trivial.
Let $\mathfrak {F}$ be a semisimple Jordan algebra over an algebraically closed field $\Phi$ of characteristic zero. Let $G$ be the automorphism group of $\mathfrak {F}$ and $\Gamma$ the structure …
Let $\mathfrak {F}$ be a semisimple Jordan algebra over an algebraically closed field $\Phi$ of characteristic zero. Let $G$ be the automorphism group of $\mathfrak {F}$ and $\Gamma$ the structure groups of $\mathfrak {F}$. General results on $G$ and $\Gamma$ are given, the proofs of which do not involve the use of the classification theory of simple Jordan algebras over $\Phi$. Specifically, the algebraic components of the linear algebraic groups $G$ and $\Gamma$ are determined, and a formula for the number of components in each case is given. In the course of this investigation, certain Lie algebras and root spaces associated with $\mathfrak {F}$ are studied. For each component ${G_i}$ of $G$, the index of $G$ is defined to be the minimum dimension of the $1$-eigenspace of the automorphisms belonging to ${G_i}$. It is shown that the index of ${G_i}$ is also the minimum dimension of the fixed-point spaces of automorphisms in ${G_i}$. An element of $G$ is called regular if the dimension of its $1$-eigenspace is equal to the index of the component to which it belongs. It is proven that an automorphism is regular if and only if its $1$-eigenspace is an associative subalgebra of $\mathfrak {F}$. A formula for the index of each component ${G_i}$ is given. In the Appendix, a new proof is given of the fact that the set of primitive idempotents of a simple Jordan algebra over $\Phi$ is an irreducible algebraic set.
A polarized abelian variety (X,\lambda) of dimension g over a local field K determines an admissible representation of GSpin_{2g+1}(K). We show that the restriction of this representation to Spin_{2g+1}(K) is …
A polarized abelian variety (X,\lambda) of dimension g over a local field K determines an admissible representation of GSpin_{2g+1}(K). We show that the restriction of this representation to Spin_{2g+1}(K) is reducible if and only if X is isogenous to its twist by the quadratic unramified extension of K. When g=1 and K = Q_p, we recover the well-known fact that the admissible GL_2(K) representation attached to an elliptic curve E is reducible upon restriction to SL_2(K) if and only if E has supersingular reduction.