Classical Algebraic Groups Research Articles

On abelian unitary involutions of crossed products

In the theory of classical linear algebraic groups, of importance are special unitary groups of non-commuted involution crossed products with division. The description of these groups largely depends on the involution type of these products. The class of Abelian involution crossed products is considered and the criterion for their existence is set provided that unitary bases (with respect to these involutions) are present in these products.

Minimum codimension of eigenspaces in irreducible representations of simple classical linear algebraic groups

Let k be an algebraically closed field of characteristic p ≥ 0 , let G be a simple simply connected classical linear algebraic group of rank l and let T be a maximal torus in G with rational character group X ( T ) . For a nonzero p-restricted dominant weight λ ∈ X ( T ) , let V be the associated irreducible kG-module. We define ν G ( V ) as the minimum codimension of any eigenspace on V for any non-central element of G. In this paper, we determine lower-bounds for ν G ( V ) for G of type A l and dim ( V ) ≤ l 3 2 , and for G of type B l , C l , or D l and dim ( V ) ≤ 4 l 3 . Moreover, we give the exact value of ν G ( V ) for G of type A l with l ≥ 15 ; for G of type B l or C l with l ≥ 14 ; and for G of type D l with l ≥ 16 .

Strongly invertible knots, equivariant slice genera, and an equivariant algebraic concordance group

AbstractWe use the Blanchfield form to obtain a lower bound on the equivariant slice genus of a strongly invertible knot. For our main application, let be a strongly invertible genus one slice knot with nontrivial Alexander polynomial. We show that the equivariant slice genus of an equivariant connected sum is at least . We also formulate an equivariant algebraic concordance group, and show that the kernel of the forgetful map to the classical algebraic concordance group is infinite rank.

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MULTIPLE FLAG IND-VARIETIES WITH FINITELY MANY ORBITS

Let G be one of the ind-groups GL(∞), O(∞), Sp(∞), and let P1, ..., Pℓ be an arbitrary set of ℓ splitting parabolic subgroups of G. We determine all such sets with the property that G acts with finitely many orbits on the ind-variety X1 × × Xℓ where Xi = G/Pi. In the case of a finite-dimensional classical linear algebraic group G, the analogous problem has been solved in a sequence of papers of Littelmann, Magyar–Weyman–Zelevinsky and Matsuki. An essential difference from the finite-dimensional case is that already for ℓ = 2, the condition that G acts on X1 × X2 with finitely many orbits is a rather restrictive condition on the pair P1, P2. We describe this condition explicitly. Using the description we tackle the most interesting case where ℓ = 3, and present the answer in the form of a table. For ℓ ≥ 4 there always are infinitely many G-orbits on X1 × × Xℓ.

Regular functions on spherical nilpotent orbits in complex symmetric pairs: Classical Hermitian cases

Given a classical semisimple complex algebraic group G and a symmetric pair (G,K) of Hermitian type, we study the closures of the spherical nilpotent K-orbits in the isotropy representation of K. We show that all such orbit closures are normal and describe the K-module structure of their ring of regular functions.

Noetherianity up to conjugation of locally diagonal inverse limits

We prove that the inverse limit of the sequence dual to a sequence of Lie algebras is Noetherian up to the action of the direct limit of the corresponding sequence of classical algebraic groups when the sequence of groups consists of diagonal embeddings. We also classify all conjugation-stable closed subsets of the space of N×N matrices.

INTERSECTION MULTIPLICITY ONE FOR CLASSICAL GROUPS

In this note we show that when G is a classical semi-simple algebraic group, B ⊂ G a Borel subgroup, and X = G/B, then the structure coefficients of the Belkale–Kumar product ⨀0 on H*(X, Z) are all either 0 or 1.

A new family of irreducible subgroups of the orthogonal algebraic groups

Let n ≥ 3 , n\geq 3, and let Y Y be a simply connected, simple algebraic group of type D n + 1 D_{n+1} over an algebraically closed field K . K. Also let X X be the subgroup of type B n B_n of Y , Y, embedded in the usual way. In this paper, we correct an error in a proof of a theorem of Seitz (Mem. Amer. Math. Soc. 67 (1987), no. 365), resulting in the discovery of a new family of triples ( X , Y , V ) , (X,Y,V), where V V denotes a finite-dimensional, irreducible, rational K Y KY -module, on which X X acts irreducibly. We go on to investigate the impact of the existence of the new examples on the classification of the maximal closed connected subgroups of the classical algebraic groups.

Outer automorphisms of classical algebraic groups

The so-called Tits class, associated to an adjoint absolutely almost simple algebraic group, provides a cohomological obstruction for this group to admit an outer automorphism. If the group has inner type, this obstruction is the only one. In this paper, we prove this is not the case for classical groups of outer type, except for groups of type $^2\mathsf{A}_n$ with $n$ even, or $n=5$. More precisely, we prove a descent theorem for exponent $2$ and degree $6$ algebras with unitary involution, which shows that their automorphism groups have outer automorphisms. In all other relevant classical types, namely $^2\mathsf{A}_n$ with $n$ odd, $n\geq3$ and $^2\mathsf{D}_n$, we provide explicit examples where the Tits class obstruction vanishes, and yet the group does not have outer automorphism. As a crucial tool, we use generic sums of algebras with involution.

Regular functions on spherical nilpotent orbits in complex symmetric pairs: Classical non-Hermitian cases

Given a classical semisimple complex algebraic group G and a symmetric pair (G,K) of non-Hermitian type, we study the closures of the spherical nilpotent K-orbits in the isotropy representation of K. For all such orbit closures, we study the normality, and we describe the K-module structure of the ring of regular functions of the normalizations.

Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block

Abstract Let G be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field F of characteristic p ≥ 0 {p\geq 0} , and let u ∈ G {u\in G} be a nonidentity unipotent element. Let ϕ be a non-trivial irreducible representation of G. Then the Jordan normal form of ϕ ⁢ ( u ) {\phi(u)} contains at most one non-trivial block if and only if G is of type G 2 {G_{2}} , u is a regular unipotent element and dim ⁡ ϕ ≤ 7 {\dim\phi\leq 7} . Note that the irreducible representations of the simple classical algebraic groups in which a non-trivial unipotent element is represented by a matrix whose Jordan form has a single non-trivial block were determined by I. D. Suprunenko [21].

Regular Unipotent Elements from Subsystem Subgroups of Type A 2 in Representations of the Special Linear Groups

For p > 2, odd Jordan block sizes of the images of regular unipotent elements from subsystem subgroups of type A2 in irreducible p-restricted representations for groups of type Ar over the field of characteristic p, the weights of which are locally small with respect to p, are found. The weight is called locally small if the double sum of its two neighboring coefficients is less than p. This result is part of a more general program investigating the behavior of unipotent elements in representations of the classical algebraic groups. It can be used to solve recognition problems for representations or linear groups by the presence of certain elements.

Irreducible geometric subgroups of classical algebraic groups

Let G be a simple classical algebraic group over an algebraically closed eld K of characteristic p 0 with natural module W . Let H be a closed subgroup of G and let V be a nontrivial irreducible tensor-indecomposable p-restricted rational KG-module such that the restriction of V to H is irreducible. In this paper we classify all such triples (G;H;V ), where H is a maximal closed disconnected positive-dimensional subgroup of G and H preserves a natural geometric structure on W .

Irreducible almost simple subgroups of classical algebraic groups

Let G be a simple classical algebraic group over an algebraically closed field K of characteristic p ? 0 with natural module W. Let H be a closed subgroup of G and let V be a nontrivial irreducible tensor indecomposable -restricted rational KG-module such that the restriction of V to H is irreducible. In this paper we classify the triples (G,H,V ) of this form, where H is a closed disconnected almost simple positive-dimensional subgroup of G acting irreducibly on W. Moreover, by combining this result with earlier work, we complete the classifcation of the irreducible triples (G,H,V ) where G is a simple algebraic group over K, and H is a maximal closed subgroup of positive dimension.

Unipotent Elements of Nonprime Order in Representations of the Classical Algebraic Groups: Two Big Jordan Blocks

For irreducible rational representations of the classical algebraic groups in characteristic p > 2, which are not equivalent to the composition of a group morphism and the standard representation, it is proved that usually the image of a unipotent element of order ps+1 > p has at least two Jordan blocks of size > ps; all exceptions are indicated explicitly. As a corollary, irreducible rational representations of these groups whose images contain unipotent elements with just one Jordan block of size > 1 are classified.

Classification theorems for hermitian forms, the Rost kernel and Hasse principle over fields with [formula omitted

In this paper we prove classification theorems for hermitian forms over some central simple algebras with involution over a field k with cd2(k)⩽3. We apply these results to show the triviality of the kernel of the Rost invariant for the classical algebraic groups associated to such hermitian forms over k. We also deduce a Hasse principle for algebraic groups defined over function fields of curves over p-adic fields thus proving a conjecture due to Colliot-Thélène–Parimala–Suresh for a large class of groups.

Centralizer of the elementary subgroup of an isotropic reductive group

Let R be a commutative ring with 1, and let G be an isotropic reductive algebraic group over R. In [5] Victor Petrov and the second author gave a definition of an elementary subgroup E(R) of the group of points G(R). This definition of E(R) generalized the well known definition of the elementary subgroup of a Chevalley group, as well as several other definitions of the elementary subgroup of isotropic classical groups and simple algebraic groups over fields, see the references in [5]. The definition is as follows. Assume that G is isotropic in the following strong sense: it possesses a par abolic subgroup that intersects properly any semisimple normal subgroup of G. Such a parabolic subgroup P is called strictly proper. Denote by EP(R) the subgroup of G(R) generated by the R points of the unipotent radicals of P and of an opposite parabolic subgroup P–. The main theorem of [5] states that EP(R) does not depend on the choice of P, as soon as for any maximal ideal M of R all irreducible components of the rel ative root system of (see [2, Exp. XXVI, §7] for the definition) are of rank ≥2. Under this assumption,

Support varieties for Frobenius kernels of classical groups

Let G be a classical simple algebraic group over an algebraically closed field k of characteristic p>0, and denote by G(r) the r-th Frobenius kernel of G. We show that for p large enough, the support variety of a simple G-module over G(r) can be described in terms of support varieties of simple G-modules over G(1). We use this, together with the computation of the varieties VG(1)(H0(λ)), given by Jantzen (1987) in [8] and by Nakano et al. (2002) in [10], to explicitly compute the support variety of a block of Dist(G(r)).

A Bott–Borel–Weil Theorem for Diagonal Ind-groups

AbstractA diagonal ind-group is a direct limit of classical affine algebraic groups of growing rank under a class of inclusions that contains the inclusionas a typical special case. If G is a diagonal ind-group and B ⊂ G is a Borel ind-subgroup, we consider the ind-variety G/B and compute the cohomology H𝓁(G/B,𝒪−λ) of any G-equivariant line bundle 𝒪−λ on G/B. It has been known that, for a generic λ, all cohomology groups of 𝒪−λ vanish, and that a non-generic equivariant line bundle 𝒪−λ has at most one nonzero cohomology group. The new result of this paper is a precise description of when Hj (G/B,𝒪−λ) is nonzero and the proof of the fact that, whenever nonzero, Hj (G/B,𝒪−λ) is a G-module dual to a highest weight module. The main difficulty is in defining an appropriate analog WB of the Weyl group, so that the action of WB on weights of G is compatible with the analog of the Demazure “action” of the Weyl group on the cohomology of line bundles. The highest weight corresponding to Hj (G/B,𝒪−λ) is then computed by a procedure similar to that in the classical Bott–Borel–Weil theorem.