Computation of SK 1(D)

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.

In [3] some constructions of exceptional Jordan algebras due to H. Freudenthal, T. A. Springer, and J. Tits were carried over to quadratic Jordan algebras (as in [4]) of arbitrary characteristic. The question was left open whether the Tits Constructions yielded all exceptional finite-dimensional central simple algebras in characteristic 2 (it was known that they do for characteristics #2). In this paper we settle this question in the affirmative. This result completes the structure theory for finite-dimensional quadratic Jordan algebras. The Tits Constructions as given in [3] involve the construction of a norm form. To prove that all the exceptional algebras arise from these constructions we need to show that all such algebras have a suitable norm form. This necessitates a slight detour in ??1 and 2 to verify that Jordan algebras in characteristic 2 have generic norms with the same properties as in the other characteristics. In ?3 we define the centroid for quadratic Jordan algebras and the corresponding notion of central simple algebras. In the next section we establish certain conditions under which a central simple Jordan algebra remains simple upon extension of the base field. In ?5 we apply these results to show that every exceptional finite-dimensional central simple algebra is a form of the 27-dimensional exceptional algebra ?((i3) (i.e. becomes ( upon suitable extension of the base field). In the final section we show that the Tits Constructions yield precisely all the exceptional finite-dimensional central simple Jordan algebras. Our proofs will be valid for all characteristics.

The notions of the rank or principal polynomial of an associative algebra and the corresponding notions of trace and norm are classical. These notions have been generalized recently by the author ([13,I]) to apply to strictly power associative algebras, and we have renamed these concepts the generic minimal polynomial, trace and norm, since this terminology appears to be more in keeping with present day usage in analogous situations. In our paper we investigated the groups of linear transformations which preserve the norms in special central simple Jordan algebras. This applies to central simple associative algebras as a special case. In a later paper ([13, III]) we studied the norm preserving groups of exceptional central simple Jordan algebras. The groups obtained in this way are generalizations of the complex Lie group E 6 and include certain geometrically defined subgroups of the collineation groups of Cayley planes.

In this paper, we will develop a theory of C-functions with characters in a division algebra. The ordinary C-function of a division algebra was introduced by K. Hey [4], and generalized by M. Eichler [1] to L-functions with abelian characters. The first attempt to generalize these theories to C-functions with non-abelian characters is due to H. Maass [5]. Later, R. Godement [2] gave a method to get the most general formulations on these matters. In this note, we will define a type of C-functions of a division algebra over an algebraic number field which are included in Godement's work as a special case, and for which one can develop the theory of Euler products. The latter theory has its own meaning as an application of the theory of spherical functions on p-adic algebraic groups. Here we have a generalization of Hecke's theory of so-called Heckeoperators. (Theorem 1-6). One can expect that there exist similar theories for other simple algebraic groups defined over p-adic number fields, and that there will be applications of these theories to non-commutative number theory. The author wishes to express his thanks to Professor Shimura for some valuable suggestions about the first part of this paper.

Superstability and central extensions of algebraic groups

Introduction. Let G1 and G be connected semisimple algebraic groups defined over a field K of characteristic zero and assume that there is an isomorphism f of G1 onto G which is defined over R, the algebraic closure of K. If p: G -? GL(V) is an absolutely irreducible (finite-dimensional) representation of G defined over K, then p of is an absolutely irreducible representation of G1 defined over K. Satake [7, p. 230] has shown that there is a field K1 which is a finite extension of K, a (unique) central simple division algebra K# defined over K1, a finite-dimensional right vector space V1 over K#, and a K1-homomorphism pi: G1 -GL(V1/K#) (the group of all nonsingular K#-linear endomorphisms of V1) such that (p of)(g) = 01(p1(g)) for all g E G1 where 01 is a unique absolutely irreducible representation of End (V1/K#) (the algebra of all K#-linear endomorphisms of V1) onto End (V). In this paper we are interested in the case where K= K1 and where there are invariant forms on V and V1. More precisely, we state the following two problems. PROBLEM 1. Assume that K#=K and that there are invariant bilinear forms B on V and B1 on V1 which are defined over K. What is the relationship between these two forms over K? Of course, if B is alternating, so is B1 and both are determined by dim V=dim V1. Hence, we shall always take B and B1 to be symmetric. PROBLEM 2. Assume that K# is a nontrivial division algebra over K (i.e., K# AK) and that there is an invariant bilinear form B on V and an invariant ?hermitian form F (c = + 1 or 1) on V1 both of which are defined over K. What is the relationship between these two forms over K? We are especially interested in the case K= Q,, a p-adic field. (In a future paper, we shall discuss the case K=R.) Here, some simplifications are immediately available. In Problem 2, it can be shown [7, p. 232] that K# has an involution of the first kind; but over Q,, it is known that the only such division algebra is the quaternion division algebra. Furthermore, it is known that a hermitian form on a finite-dimensional vector space over a quaternion division algebra defined over Q, is determined only by the dimension of the vector space. Therefore, in Problem 2 we shall always take F to be skew-hermitian; in the case where K# is a quaterion division algebra, this means that the form B is symmetric [7, p. 233]. If W is a vector space defined over K and if S is a symmetric form on W which is also defined over K, then three invariants can be associated with the pair (W, S),

Given r, n two relatively prime integers, and m be the minimal integer such that r m ≡ 1 (mod n). We construct, for each s = r t , (t, m) = 1 a central division algebra A S of dimension m 2 over a field of rational function K(X) in a commutative indeterminate X, and K a finite extension of the rationals Q. These algebras have peculiar properties: The ring of rational functions A s (ξ) over a commutative indeterminate ξ-are isomorphic as algebras, but belong to different Brauer classes over their center K(X, ξ) (theorem 4.1). (b) The division algebras A s are non-isomorphic (as rings) except that A s ≅A v if sv ≡ l(mod m), yet for m≠ 2-they belong to different Brauer classes, (c) Nevertheless, there exist a chain of ring isomorphism maps into: (but not onto!) A r→A r v1 →…→A r Vl →A r where {v i } is a set of integers relatively prime to m. (Corollary 4.4). (d) By Tsen’s theorem, the algebras A S have splitting fields of the form L(X) where L/K is an algebraic extension. (Theorem 5.1) For all these algebras there exists an algebraic extension F⊃K such that F(X) is a common maximal subfield of all A S , and L(X) splits A s if and only if L⊇F. Further relations between these divisions algebras are discussed (§5).

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. One of the major theorems in the area is Borovik's trichotomy theorem. The ‘trichotomy’ here is a case division of the generic minimal counterexamples within odd type, that is, groups with a large and divisible Sylow° 2-subgroup. The so-called ‘uniqueness case’ in the trichotomy theorem is the existence of a proper 2-generated core. It is our aim to drive the presence of a proper 2-generated core to a contradiction, and hence bind the complexity of the Sylow° 2-subgroup of a minimal counterexample to the Cherlin–Zilber conjecture. This paper shows that the group in question is a minimal connected simple group and has a strongly embedded subgroup, a far stronger uniqueness case. As a corollary, a tame counterexample to the Cherlin–Zilber conjecture has Prüfer rank at most two.

Let be either a simple linear algebraic group over an algebraically closed field of characteristic or a quantum group at an ‐th root of unity. We define a tensor ideal of singular ‐ modules in the category of finite‐dimensional ‐modules and study the associated quotient category , called the regular quotient . For , the Coxeter number of , we establish a ‘linkage principle’ and a ‘translation principle’ for tensor products: Let be the essential image in of the principal block of . We first show that is closed under tensor products in . Then we prove that the monoidal structure of is governed to a large extent by the monoidal structure of . These results can be combined to give an external tensor product decomposition , where denotes the Verlinde category of .

A theorem of Cassidy states that any Zariski closed differential algebraic subgroup of a simple linear algebraic group, defined over a differential field, is either the whole group or is conjugate to the subgroup of constant matrices. An arithmetic analogue of this theorem is proved in which usual derivations on fields are replaced by certain nonlinear operators on p -adic rings, called " p -derivations."

We show that if H is a reductive group, then nth roots of conjugacy classes are a finite union of conjugacy classes, and that if G is an algebraic overgroup of H, then the intersection of H with a conjugacy class of G is a finite union of H-conjugacy classes. These results follow from results on finiteness of unipotent classes in an almost simple algebraic group.

Chapter 1. A classification of Curtis-Tits amalgams.- Chapter 2. The use of valuations for classifying point-line geometries.- Chapter 3. An outline of polar spaces: basics and advances.- Chapter 4. Embeddings of Line-grassmannians of Polar Spaces in Grassmann Varieties.- Chapter 5. Generation of Lie Incidence Geometries: A Survey.- Chapter 6. Witt-Type Theorems for Subspaces of Lie Geometries: A Survey.- Chapter 7. Embeddings of cotriangular spaces.- Chapter 8. Unipotent Overgroups in Simple Algebraic Groups.- Chapter 9. The axes of a Majorana representation of A12.- Chapter 10. GIT related problems of the flag variety for the action of a maximal torus.- Chapter 11. Characterizations of trialities of type Iid in buildings of type D4.- Chapter 12. On the isomorphism problem for Coxeter groups and related topics.- Chapter 13. Lectures on Artin groups and the K(pi 1) conjecture. Chapter 14. Algebraic codes and geometry of some classical generalized polygons.- Chapter 15. Some Weyl modules of the algebraic groups of type E_6.- Problem Set.

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski, and Y. Peterzil.

Level Zero Types and Hecke Algebras for Local Central Simple Algebras

Let A be an algebra with involution ∗ over a field F. We recall that A is said to be normal if xx∗ = x∗x for every x ∈ A. Over the decades, normal algebras with involutions have been extensively investigated on their own (see e.g. [1], [3], [4], [5], [7], [8], [9], [10], [11]) and, moreover, they have several applications in linear algebra and functional analysis as well (see e.g. [2], [6], [11], [12], [13]). It is well-known that any normal algebra with involution satisfies the standard polynomial identity of degree 4 (see [7, §5]). Moreover, in [11] Maxwell determined the structure of a normal simple algebra of matrices with entries in a field with involution. He also proved that a division algebra D with involution is normal if and only if D is either a field or a generalized quaternion algebra over its center. Furthermore, a characterization of group algebras which are normal under the standard involution was established by Bovdi, Gudivok, and Semirot in [3]. Subsequently, such a result has been extended to twisted group algebras in [4] and to group algebras under a Novikov involution in [5]. On the other hand, it seems that the rather natural problems of characterizing ordinary and restricted enveloping algebras which are normal under their canonical involutions have not been settled yet. The present paper is just devoted to answering these questions. For an arbitrary Lie algebra L we denote by U(L) the universal enveloping algebra of L. Moreover, if L is restricted with a p-map [p] over a field F of characteristic p > 0, then we denote by u(L) the restricted enveloping algebra of L. We consider U(L) and u(L) with the principal involution ∗, namely, the unique F-antiautomorphism such that x∗ = −x for every x in L. Note that ∗ is just the antipode of the F-Hopf algebras U(L) or u(L). We use the symbols Z(L) and L′ for the center and the derived subalgebra of L, respectively. If S ⊆ L then we denote by 〈S〉F and 〈S〉 the F-vector space and the subalgebra generated by S. Also, if L is restricted then 〈S〉p denotes the restricted subalgebra generated by S and we put S = {x |x ∈ S}. In our first main result we completely settle the restricted case: