In this paper, Steinberg’s concept of a regular element in a semisimple algebraic group defined over an algebraically closed field is generalized to the concept of a <inline-formula content-type="math/mathml"> <mml:math …
In this paper, Steinberg’s concept of a regular element in a semisimple algebraic group defined over an algebraically closed field is generalized to the concept of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-regular element in a semisimple algebraic group defined over an arbitrary field of characteristic zero. The existence of semisimple and unipotent <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-regular elements in a semisimple algebraic group defined over a field of characteristic zero is proved. The structure of all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-regular unipotent elements is given. The number of minimal parabolic subgroups containing a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-regular element is given. The number of conjugacy classes of <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>-regular unipotent elements is given, where <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 the real field. The number of conjugacy classes of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q Subscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{Q_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-regular unipotent elements is shown to be finite, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q Subscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{Q_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the field 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 numbers.
Given a connected linear algebraic group $G$, we descrive the subgroup of $G$ generated by all semisimple elements.
Given a connected linear algebraic group $G$, we descrive the subgroup of $G$ generated by all semisimple elements.
Let A be a C-algebra with 1. Let ξ=s λ1e1 + … +λ nen ε A where each λi C and e1,… en are orthogonal idempotents in A. We give …
Let A be a C-algebra with 1. Let ξ=s λ1e1 + … +λ nen ε A where each λi C and e1,… en are orthogonal idempotents in A. We give conditions on A to guarantee that a certain function of ξ is bounded in terms of the λi's. Examples of such algebra are the complex group ring C[G] and the ring Mm (C[G]) of matrices over C[G], for arbitrary groups G. Thus this paper generalizes the result of [PP]: on C[G] (with out using C *-algebra theory), and that of [LP] on Mm (C[G]) (without assuming that G is finite.
The projection theorem for weights of a representation of a semisimple group G on restriction to a semisimple subgroup is derived, and the existence of a subgroup corresponding to a …
The projection theorem for weights of a representation of a semisimple group G on restriction to a semisimple subgroup is derived, and the existence of a subgroup corresponding to a given projection is discussed. Dynkin's definition of the index of a simple subgroup is extended to the case of G being only semisimple, and the geometrical meaning of the index is given. A method is developed for finding branching rules for both regular and nonregular subgroups. Explicit general formulas for the branching multiplicities are obtained for all cases when G is of rank 2 and for B3(R7) → G2. Applications to the construction of weight diagrams and the ``state-labeling'' problems for B2 and G2 are mentioned.
Abstract Let G be a simple linear algebraic group defined over an algebraically closed field of characteristic p ≥ 0 and let ϕ be a nontrivial p -restricted irreducible representation …
Abstract Let G be a simple linear algebraic group defined over an algebraically closed field of characteristic p ≥ 0 and let ϕ be a nontrivial p -restricted irreducible representation of G . Let T be a maximal torus of G and s ϵ T . We say that s is Ad - regular if α ( s ) ≠ β ( s ) for all distinct T -roots α and β of G . Our main result states that if all but one of the eigenvalues of ϕ ( s ) are of multiplicity 1 then, with a few specified exceptions, s is Ad-regular. This can be viewed as an extension of our earlier work, in which we show that, under the same hypotheses, either s is regular or G is a classical group and ϕ is “essentially” (a twist of) the natural representation of G .
Abstract We deal here with the structure of semisimple groups and the definition of the generalized Fitting subgroup F* (G) in our context. This is a notion which is used …
Abstract We deal here with the structure of semisimple groups and the definition of the generalized Fitting subgroup F* (G) in our context. This is a notion which is used constantly in the analysis of finite simple groups and may be expected to play a parallel role in the context of connected simple groups of finite Morley rank, notably in connection with the solution of Problem C of Chapter B. As the most concrete applications of the general theory given here are to rather small groups, in Chapter 14, we do not yet have solid evidence that the arguments parallel to the finite case go through here; but barring major surprises this notion should turn out to be a workhorse for the further development of the theory. In any case both the Fitting subgroup and the socle in the definable versions used here have proved their utility already, and from that point it is a small step to the generalized Fitting subgroup.
It is shown that the classification theorems for semisimple algebraic groups in characteristic zero can be derived quite simply and naturally from the corresponding theorems for Lie algebras by using …
It is shown that the classification theorems for semisimple algebraic groups in characteristic zero can be derived quite simply and naturally from the corresponding theorems for Lie algebras by using a little of the theory of tensor categories. This article is extracted from Milne 2007.
In this chapter, we discuss the representation theory of semisimple algebraic groups. We also sketch the construction of finite dimensional irreducible representations of semisimple algebraic groups. We further discuss the …
In this chapter, we discuss the representation theory of semisimple algebraic groups. We also sketch the construction of finite dimensional irreducible representations of semisimple algebraic groups. We further discuss the geometric realization of finite dimensional irreducible representations of a semisimple algebraic group (over ℂ).
In this chapter, we discuss the representation theory of semisimple algebraic groups. We also sketch the construction of finite dimensional irreducible representations of semisimple algebraic groups. We further discuss the …
In this chapter, we discuss the representation theory of semisimple algebraic groups. We also sketch the construction of finite dimensional irreducible representations of semisimple algebraic groups. We further discuss the geometric realization of finite dimensional irreducible representations of a semisimple algebraic group (over ℂ).
Every complex semisimple Lie algebra has a compact real form, as a consequence of a particular normalization of root vectors whose construction uses the Isomorphism Theorem of Chapter II. If …
Every complex semisimple Lie algebra has a compact real form, as a consequence of a particular normalization of root vectors whose construction uses the Isomorphism Theorem of Chapter II. If go is a real semisimple Lie algebra, then the use of a compact real form of (g0)ℂ leads to the construction of a "Cartan involution" θ of go. This involution has the property that if go = t 0 ⊕ p0 is the corresponding eigenspace decomposition or "Cartan decomposition", then to ⊕ ipo is a compact real form of (g0)ℂ. Any two Cartan involutions of go are conjugate by an inner automorphism. The Cartan decomposition generalizes the decomposition of a classical matrix Lie algebra into its skew-Hermitian and Hermitian parts. If G is a semisimple Lie group, then a Cartan decomposition g0 = t0 ⊕ p0 of its Lie algebra leads to a global decomposition G = K exp p0, where K is the analytic subgroup of G with Lie algebra go. This global decomposition generalizes the polar decomposition of matrices. The group K contains the center of G and, if the center of G is finite, is a maximal compact subgroup of G. The Iwasawa decomposition G = K AN exhibits closed subgroups A and N of G such that A is simply connected abelian, N is simply connected nilpotent, A normalizes N, and multiplication from K × A × N to G is a diffeomorphism onto. This decomposition generalizes the Gram-Schmidt orthogonalization process. Any two Iwasawa decompositions of G are conjugate. The Lie algebra a0 of A may be taken to be any maximal abelian subspace of p0, and the Lie algebra of N is defined from a kind of root-space decomposition of g0 with respect to a0. The simultaneous eigenspaces are called "restricted roots", and the restricted roots form an abstract root system. The Weyl group of this system coincides with the quotient of normalizer by centralizer of a0 in K. A Cartan subalgebra of g0 is a subalgebra whose complexification is a Cartan sub-algebra of (g0)ℂ. One Cartan subalgebra of g0 is obtained by adjoining to the above ao a maximal abelian subspace of the centralizer of a0 in t0. This Cartan subalgebra is θ stable. Any Cartan subalgebra of g0 is conjugate by an inner automorphism to a θ stable one, and the subalgebra built from a0 as above is maximally noncompact among all θ stable Cartan subalgebras. Any two maximally noncompact Cartan subalgebras are conjugate, and so are any two maximally compact ones. Cayley transforms allow one to pass between any two θ stable Cartan subalgebras, up to conjugacy. A Vogan diagram of g0 superimposes certain information about the real form g0 on the Dynkin diagram of (go)ℂ. The extra information involves a maximally compact θ stable Cartan subalgebra and an allowable choice of a positive system off roots. The effect of θ on simple roots is labeled, and imaginary simple roots are painted if they are "noncompact," left unpainted if they are "compact", Such a diagram is not unique for g0, but it determines go up to isomorphism. Every diagram that looks formally like a Vogan diagram arises from some g0. Vogan diagrams lead quicakly to a classification of all simple real Lie algebras, the only diffcuylty being eliminating the redundancy in the chosice of postitigve system of roots. This difficylty is resolved by the Borel and de Siebenthal Theotem. Using a succession of Catkey transforms to pass form a maximally compact Cartan subalgebre to a maximally non compact Cartan subalgebre, on e readily identifies the restricted roots for each simple real Lie alebra.
Introduction. Let A be a finite-dimensional Hopf algebra over a field k. As in [S], we let A, c, and S denote the coalgebra operations and the antipode (which is …
Introduction. Let A be a finite-dimensional Hopf algebra over a field k. As in [S], we let A, c, and S denote the coalgebra operations and the antipode (which is uniquely determined and bijective). The linear dual A* is again a Hopf algebra. One says that A is semisimple if it is a semisimple k-algebra; it is cosemisimple when A* is semisimple. In a recent paper [R], Radford proved in most cases that when A is semisimple and cosemisimple, its automorphism group is finite. When char(k) = p > 0, his proof required the extra assumptions that p > dim(A) and S2 = J. In this paper, I shall combine Radford's ideas with the theory of affine group schemes to show that the theorem is true without those assumptions. In addition, the conclusion will be a bit stronger in characteristic p:
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We shall investigate a flipping contraction g : X -> Y from a 4-fold X with at most isolated complete intersection singularities. If Y has an anti-bi-canonical divisor (=bi-elephant) with …
We shall investigate a flipping contraction g : X -> Y from a 4-fold X with at most isolated complete intersection singularities. If Y has an anti-bi-canonical divisor (=bi-elephant) with only rational singularities, then g carries an inductive structure chained up by blow-ups (La Torre Pendente), and in particular the flip exists. This naturally contains Miles Reid's `Pagoda' as an anti-canonical divisor (=elephant) and its proper transforms.
According to a theorem of Brieskorn and Slodowy, the intersection of the nilpotent cone of a simple Lie algebra with a transverse slice to the subregular nilpotent orbit is a …
According to a theorem of Brieskorn and Slodowy, the intersection of the nilpotent cone of a simple Lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface singularity. At the opposite extremity of the poset of nilpotent orbits, the closure of the minimal nilpotent orbit is also an isolated symplectic singularity, called a minimal singularity. For classical Lie algebras, Kraft and Procesi showed that these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures: specifically, any such singularity is either a simple surface singularity, a minimal singularity, or a union of two simple surface singularities of type A2k−1. In the present paper, we complete the picture by determining the generic singularities of all nilpotent orbit closures in exceptional Lie algebras (up to normalization in a few cases). We summarize the results in some graphs at the end of the paper. In most cases, we also obtain simple surface singularities or minimal singularities, though often with more complicated branching than occurs in the classical types. There are, however, six singularities that do not occur in the classical types. Three of these are unibranch non-normal singularities: an SL2(C)-variety whose normalization is A2, an Sp4(C)-variety whose normalization is A4, and a two-dimensional variety whose normalization is the simple surface singularity A3. In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, extending Slodowy's work for the singularity of the nilpotent cone at a point in the subregular orbit.
This is a brief introduction to the study of growth in groups of Lie type, with $SL_2(\mathbb{F}_q)$ and some of its subgroups as the key examples. They are an edited …
This is a brief introduction to the study of growth in groups of Lie type, with $SL_2(\mathbb{F}_q)$ and some of its subgroups as the key examples. They are an edited version of the notes I distributed at the Arizona Winter School in 2016. Those notes were, in turn, based in part on my survey in Bull. Am. Math. Soc. (2015) and in part on the notes for courses I gave on the subject in Cusco (AGRA) and Gottingen.
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an affine spherical variety, possibly singular, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper L Superscript …
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an affine spherical variety, possibly singular, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper L Superscript plus Baseline upper X"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">L</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf L^+X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> its arc space. The intersection complex of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper L Superscript plus Baseline upper X"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">L</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf L^+X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions. Such relationships were previously established in Braverman–Finkelberg–Gaitsgory–Mirković for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in Bouthier–Ngô–Sakellaridis for toric varieties and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-monoids. In this paper, we compute this intersection complex for the large class of those spherical <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>-varieties whose dual group is equal to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper G With ˇ"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>G</mml:mi> <mml:mo stretchy="false">ˇ<!-- ˇ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\check G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the stalks of its nearby cycles on the horospherical degeneration of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We formulate the answer in terms of a Kashiwara crystal, which conjecturally corresponds to a finite-dimensional <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper G With ˇ"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>G</mml:mi> <mml:mo stretchy="false">ˇ<!-- ˇ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\check G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-representation determined by the set of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant valuations on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove the latter conjecture in many cases. Under the sheaf–function dictionary, our calculations give a formula for the Plancherel density of the IC function of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper L Superscript plus Baseline upper X"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">L</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf L^+X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as a ratio of local <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-values for a large class of spherical varieties.
A bstract The singularity structure of the Coulomb and Higgs branches of good 3 d $$ \mathcal{N}=4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:math> circular quiver gauge theories (CQGTs) with …
A bstract The singularity structure of the Coulomb and Higgs branches of good 3 d $$ \mathcal{N}=4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:math> circular quiver gauge theories (CQGTs) with unitary gauge groups is studied. The central method employed is the Kraft-Procesi transition . CQGTs are described as a generalisation of a class of linear quivers. This class degenerates into the familiar class T ρ σ ( SU ( N )) in the linear case, however the circular case does not have the degeneracy and so the class of CQGTs contains many more theories and much more structure. We describe a collection of good, unitary, CQGTs from which the entire class can be found using Kraft-Procesi transitions. The singularity structure of a general member of this collection is fully determined, encompassing the singularity structure of a generic CQGT. Higher-level Hasse diagrams are introduced in order to write the results compactly. In higher-level Hasse diagrams, single nodes represent lattices of nilpotent orbit Hasse diagrams and edges represent traversing structure between lattices. The results generalise the case of linear quiver moduli spaces which are known to be nilpotent varieties of $$ \mathfrak{s}{\mathfrak{l}}_n $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>s</mml:mi> <mml:msub> <mml:mi>l</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> .
Every conic symplectic singularity admits a universal Poisson deformation and a universal filtered quantization, thanks to the work of Losev and Namikawa. We begin this paper by showing that every …
Every conic symplectic singularity admits a universal Poisson deformation and a universal filtered quantization, thanks to the work of Losev and Namikawa. We begin this paper by showing that every such variety admits a universal equivariant Poisson deformation and a universal equivariant quantization with respect to a reductive group acting on it by \mathbb{C}^\times -equivariant Poisson automorphisms. We go on to study these definitions in the context of nilpotent Slodowy slices. First, we give a complete description of the cases in which the finite W -algebra is a universal filtered quantization of the slice, building on the work of Lehn–Namikawa–Sorger. This leads to a near-complete classification of the filtered quantizations of nilpotent Slodowy slices. The subregular slices in non-simply laced Lie algebras are especially interesting: with some minor restrictions on Dynkin type, we prove that the finite W -algebra is a universal equivariant quantization with respect to the Dynkin automorphisms coming from the unfolding of the Dynkin diagram. This can be seen as a non-commutative analogue of Slodowy's theorem. Finally, we apply this result to give a presentation of the subregular finite W -algebra of type \mathsf{B} as a quotient of a shifted Yangian.
The purpose of this paper is to give a classification of the orbits in a real reductive Lie algebra under the adjoint action of a corresponding connected Lie group. The …
The purpose of this paper is to give a classification of the orbits in a real reductive Lie algebra under the adjoint action of a corresponding connected Lie group. The classification is obtained by examining the intersection of the Lie algebra with the orbits in its complexification. An algebraic characterization of the minimal points in the closed orbits is also given.
Let $Ω$ be a finite symmetric subset of GL$_n(\mathbb{Z}[1/q_0])$, and $Γ:=\langle Ω\rangle$. Then the family of Cayley graphs $\{{\rm Cay}(π_m(Γ),π_m(Ω))\}_m$ is a family of expanders as $m$ ranges over fixed …
Let $Ω$ be a finite symmetric subset of GL$_n(\mathbb{Z}[1/q_0])$, and $Γ:=\langle Ω\rangle$. Then the family of Cayley graphs $\{{\rm Cay}(π_m(Γ),π_m(Ω))\}_m$ is a family of expanders as $m$ ranges over fixed powers of square-free integers and powers of primes that are coprime to $q_0$ if and only if the connected component of the Zariski-closure of $Γ$ is perfect. Some of the immediate applications, e.g. orbit equivalence rigidity, {\em largeness} of certain $\ell$-adic Galois representations, are also discussed.
It is shown that if G is a finite Chevalley group or twisted type over a field of characteristic p and U is a maximal p-subgroup of G then any …
It is shown that if G is a finite Chevalley group or twisted type over a field of characteristic p and U is a maximal p-subgroup of G then any nonlinear irreducible character of U vanishes on regular elements. For groups of adjoint type the linear content of the restriction to U of a discrete series character J of G is calculated and it is deduced that J takes the value 0 or ${( - 1)^s}$ on regular elements of U $(s = {\text {rank}}\;G)$.
In this paper we define the chromatic number of a lattice: It is the least number of colors one needs to color the interiors of the cells of the Voronoi …
In this paper we define the chromatic number of a lattice: It is the least number of colors one needs to color the interiors of the cells of the Voronoi tessellation of a lattice so that no two cells sharing a facet are of the same color. We compute the chromatic number of the root lattices, their duals, and of the Leech lattice, we consider the chromatic number of lattices of Voronoi's first kind, and we investigate the asymptotic behaviour of the chromatic number of lattices when the dimension tends to infinity. We introduce a spectral lower bound for the chromatic number of lattices in spirit of Hoffman's bound for finite graphs. We compute this bound for the root lattices and relate it to the character theory of the corresponding Lie groups.
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.
The moduli space of cubic threefolds in CP4, with some minor birational modifications, is the Baily-Borel compactification of the quotient of the complex 10-ball by a discrete group. We describe …
The moduli space of cubic threefolds in CP4, with some minor birational modifications, is the Baily-Borel compactification of the quotient of the complex 10-ball by a discrete group. We describe both the birational modifications and the discrete group explicitly.
Abstract We study basic geometric properties of Kottwitz–Viehmann varieties, which are certain generalizations of affine Springer fibers that encode orbital integrals of spherical Hecke functions. Based on the previous work …
Abstract We study basic geometric properties of Kottwitz–Viehmann varieties, which are certain generalizations of affine Springer fibers that encode orbital integrals of spherical Hecke functions. Based on the previous work of A. Bouthier and the author, we show that these varieties are equidimensional and give a precise formula for their dimension. Also we give a conjectural description of their number of irreducible components in terms of certain weight multiplicities of the Langlands dual group and we prove the conjecture in the case of unramified conjugacy class.
In this note, we formulate an observation that “almost all” irreducible ordinary characters of finite groups of Lie type remain irreducible when restricted to the derived subgroups. To see this, …
In this note, we formulate an observation that “almost all” irreducible ordinary characters of finite groups of Lie type remain irreducible when restricted to the derived subgroups. To see this, key ingredients are some asymptotic results for conjugacy classes of finite groups of Lie type and strongly regular semisimple elements in dual groups.
Using equivariant geometry, we find a universal formula that computes the number of times a general cubic surface arises in a family. As applications, we show that the PGL(4) orbit …
Using equivariant geometry, we find a universal formula that computes the number of times a general cubic surface arises in a family. As applications, we show that the PGL(4) orbit closure of a generic cubic surface has degree 96120, and that a general cubic surface arises 42120 times as a hyperplane section of a general cubic 3-fold.
We study the contribution of multiple covers of an irreducible rational curve C in a Calabi-Yau threefold Y to the genus 0 Gromov-Witten invariants in the following cases. (1) If …
We study the contribution of multiple covers of an irreducible rational curve C in a Calabi-Yau threefold Y to the genus 0 Gromov-Witten invariants in the following cases. (1) If the curve C has one node and satisfies a certain genericity condition, we prove that the contribution of multiple covers of degree d is given by the sum of all 1/n^3 where n divides d. (2) For a smoothly embedded contractable curve C in Y we define schemes C_i for i=1,...,l where C_i is supported on C and has multiplicity i, and the integer l (0l). In the latter case we also get a formula for arbitrary genus. These results show that the curve C contributes an integer amount to the so-called instanton numbers that are defined recursively in terms of the Gromov-Witten invariants and are conjectured to be integers.
The aim of this second part is to compute explicitly the Lusztig restriction of the characteristic function of a regular unipotent class of a finite reductive group, improving slightly a …
The aim of this second part is to compute explicitly the Lusztig restriction of the characteristic function of a regular unipotent class of a finite reductive group, improving slightly a theorem of Digne, Lehrer and Michel. We follow their proof but introduce new information, namely the computation of morphisms
This is the first part of a guide to deformations and moduli, especially viewed from the perspective of algebraic surfaces (the simplest higher dimensional varieties). It contains also new results, …
This is the first part of a guide to deformations and moduli, especially viewed from the perspective of algebraic surfaces (the simplest higher dimensional varieties). It contains also new results, regarding the question of local homeomorphism between Kuranishi and Teichmueller space, and a survey of new results with Ingrid Bauer, concerning the discrepancy between the deformation of the action of a group G on a minimal models S, respectively the deformation of the action of G on the canonical model X. Here Def(S) maps properly onto Def(X), but the same does not hold for pairs: Def(S,G) does not map properly onto Def(X,G). Indeed the connected components of Def(S), in the case of tertiary Burniat surfaces, only map to locally closed sets. The last section contains anew result on some surfaces whise Albanese map has generic degree equal to 2.
Let $G$ be a simple complex factorizable Poisson Lie algebraic group. Let $\U_\hbar(\g)$ be the corresponding quantum group. We study $\U_\hbar(\g)$-equivariant quantization $\C_\hbar[G]$ of the affine coordinate ring $\C[G]$ along …
Let $G$ be a simple complex factorizable Poisson Lie algebraic group. Let $\U_\hbar(\g)$ be the corresponding quantum group. We study $\U_\hbar(\g)$-equivariant quantization $\C_\hbar[G]$ of the affine coordinate ring $\C[G]$ along the Semenov-Tian-Shansky bracket. For a simply connected group $G$ we prove an analog of the Kostant-Richardson theorem stating that $\C_\hbar[G]$ is a free module over its center.
We show that Nichols algebras of most simple Yetter–Drinfeld modules over the projective symplectic linear group over a finite field, corresponding to unipotent orbits, have infinite dimension. We give a …
We show that Nichols algebras of most simple Yetter–Drinfeld modules over the projective symplectic linear group over a finite field, corresponding to unipotent orbits, have infinite dimension. We give a criterion to deal with unipotent classes of general finite simple groups of Lie type and apply it to regular classes in Chevalley and Steinberg groups.
Let $G$ be a semisimple, simply connected, algebraic group over an algebraically closed field $k$ with Lie algebra $\mathfrak {g}$. We study the spaces of parahoric subalgebras of a given …
Let $G$ be a semisimple, simply connected, algebraic group over an algebraically closed field $k$ with Lie algebra $\mathfrak {g}$. We study the spaces of parahoric subalgebras of a given type containing a fixed nil-elliptic element of $\mathfrak {g}\otimes k((\pi ))$, i.e. fixed point varieties on affine flag manifolds. We define a natural class of $k^*$-actions on affine flag manifolds, generalizing actions introduced by Lusztig and Smelt. We formulate a condition on a pair $(N,f)$ consisting of $N\in \mathfrak {g}\otimes k((\pi ))$ and a $k^*$-action $f$ of the specified type which guarantees that $f$ induces an action on the variety of parahoric subalgebras containing $N$. For the special linear and symplectic groups, we characterize all regular semisimple and nil-elliptic conjugacy classes containing a representative whose fixed point variety admits such an action. We then use these actions to find simple formulas for the Euler characteristics of those varieties for which the $k^*$-fixed points are finite. We also obtain a combinatorial description of the Euler characteristics of the spaces of parabolic subalgebras containing a given element of certain nilpotent conjugacy classes of $\mathfrak {g}$.
These are the expanded lecture notes from the author's mini-course during the graduate summer school of the Park City Math Institute in 2015. The main topics covered are: geometry of …
These are the expanded lecture notes from the author's mini-course during the graduate summer school of the Park City Math Institute in 2015. The main topics covered are: geometry of Springer fibers, affine Springer fibers and Hitchin fibers; representations of (affine) Weyl groups arising from these objects; relation between affine Springer fibers and orbital integrals.
Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of …
Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of corresponding finite simple groups. (Each algebraic group is assumed to be defined over a universal field which is algebraically closed and of infinite degree of transcendence over the prime field, and all of its representations are assumed to take place on vector spaces over this field.)
Ce travail apporte quelques renseignements sur la torsion du groupe de cohomologie entiere d'un groupe de Lie compact connexe G, que nous appellerons, suivant Γusage, la torsion de G, sur …
Ce travail apporte quelques renseignements sur la torsion du groupe de cohomologie entiere d'un groupe de Lie compact connexe G, que nous appellerons, suivant Γusage, la torsion de G, sur les sous-groupes commutatifs de G, et met ces deux questions en relation.Nous nous interesserons en particulier aux rapports qui existent entre la ^-torsion (p nombre premier) et les sous-groupes commutatifs de type (p, ,p), que nous appellerons ici les [^>]-sous-groupes.Ce travail a ete resume dans une Note au Bull.Amer.Math.Soc.66 (I960),.pp.285-288.En nous appuyant sur quelques remarques concernant les ί/-espaces dont la cohomologie entiere est de type fini, faites dans le §1, et sur le Theoreme V de [14], on verra que le groupe simplement connexe exceptionnel E έ n'a pas de ^-torsion lorsque p = 5, i = 6, 7 et p = 7, i = 7, 8, et Ton determinera aussi if* (E 6 ; Z 3 ), H* (E 8 ; Z 5 ), (Theor.2.2,2.3).Compte tenu de resultats connus [5,8,14], on pourra alors indiquer les nombres premiers intervenant dans les coefficients de torsion de tous les groupes simples et simplement connexes (2.5) et Γon verifiera (2.6) le theoreme suivant, conjecture dans [7]: THEOREME A. Supposons G, simple et simplement connexe, et soit p un nombre premier ne divisant pas les coefficients de la plus grande racine de G, exprimee comme somme de racines simples.Alors G na pas de p-torsion.Dans un groupe de Lie compact connexe G il existe des [/>]-sous-groupes evidents, les elements d'ordre p d'un tore maximal, mais il peut y en avoir d'autres, (pour p = 2, les matrices diagonales de SO (n), (n > 3) par exemple).Cependant, d'apres [9, XII,5,3, 5.4], si G n'a pas de p-torsion, tout [/>]-sousgroupe // fait partie d'un tore, ce qui precise un resultat de [11] affirmant que,.sous Γhypothese faite, le rang de H est au plus egal a la dimension des tores maximaux de G. Cette derniere condition est evidemment necessaire pour que H fasse partie d'un tore de G, mais elle n'est pas suffisante en general.En effet, on demontrera: (i) Pour que le groupe fundamental TΓ^G) de G soit sans p-torsion, il faut et il suffit que tout [/>]-sous-groupe de rang deux soit contenu dans un
We recall that if an algebraic group G operates regularly on a variety V, by a quotient variety is meant a pair (V/G, t), where V/G is a variety and …
We recall that if an algebraic group G operates regularly on a variety V, by a quotient variety is meant a pair (V/G, t), where V/G is a variety and t: V-*V/G is a rational map, everywhere defined and surjective, such that two points of V have the same image under t if and only if they have the same orbit on V, and such that, for any xE V, any rational function on V that is G-invariant (i.e., constant on orbits) and defined at x is actually (under the natural injection of function fields Q,(V/G)-*Q(V), ß denoting the universal domain) a rational function on V/G that is defined at tx (cf.[l, exposé 8]).Q,(V/G) must therefore consist precisely of all G-invariant elements of ß(V), so t is separable.A quotient variety need not exist (obvious necessary condition: all orbits on V must be closed), but when it exists it is clearly unique to within an isomorphism; in this case, for any open subset UQV/G, r~lU/G exists and equals U.
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.
0.Introduction. 1.Let G be a group of linear transformations on a finite dimensional real or complex vector space X.Assume X is completely reducible as a G-module.Let 5 be the ring …
0.Introduction. 1.Let G be a group of linear transformations on a finite dimensional real or complex vector space X.Assume X is completely reducible as a G-module.Let 5 be the ring of all complexvalued polynomials on X, regarded as a G-module in the obvious way, and let JC5 be the subring of all G-invariant polynomials on X.Now let J + be the set of all ƒ £ J having zero constant term and let HQS be any graded subspace such that S=J + S+H is a G-module direct sum.It is then easy to see that