4262:
41:
4269:
Chevalley showed in 1958 that the reductive groups over any algebraically closed field are classified up to isomorphism by root data. In particular, the semisimple groups over an algebraically closed field are classified up to central isogenies by their Dynkin diagram, and the simple groups
4044:
2741:, that is, they are direct sums of irreducible representations. That is the source of the name "reductive". Note, however, that complete reducibility fails for reductive groups in positive characteristic (apart from tori). In more detail: an affine group scheme
3549:
3747:(with some edges directed or multiple). The set of vertices of the Dynkin diagram is the set of simple roots. In short, the Dynkin diagram describes the angles between the simple roots and their relative lengths, with respect to a Weyl group-invariant
3187:
4716:
7472:
5303:
and unitary representations have been developed for real reductive groups in this generality. The main differences between this definition and the definition of a reductive algebraic group have to do with the fact that an algebraic group
2305:
5033:
4919:
5851:
As discussed above, the classification of split reductive groups is the same over any field. By contrast, the classification of arbitrary reductive groups can be hard, depending on the base field. Some examples among the
4216:
2667:, which is a complex reductive algebraic group. In fact, this construction gives a one-to-one correspondence between compact connected Lie groups and complex reductive groups, up to isomorphism. For a compact Lie group
2653:
4326:. It is remarkable that the classification of reductive groups is independent of the characteristic. For comparison, there are many more simple Lie algebras in positive characteristic than in characteristic zero.
3924:
7220:, the Galois action on its Dynkin diagram, and a Galois-invariant subset of the vertices of the Dynkin diagram. Traditionally, the Tits index is drawn by circling the Galois orbits in the given subset.
1569:
2171:
8198:
8004:
5783:) of the Schur module ∇(λ), but it need not be equal to the Schur module. The dimension and character of the Schur module are given by the Weyl character formula (as in characteristic zero), by
5697:
has a unique highest weight vector up to scalars; the corresponding "highest weight" λ is dominant; and every dominant weight λ is the highest weight of a unique irreducible representation
3477:
3626:
3298:
3240:
3122:
6019:
7117:
reduces the classification of central simple algebras over a field to the case of division algebras. Generalizing these results, Tits showed that a reductive group over a field
7630:
7598:
7566:
6120:
5918:
5123:
1691:
1266:
2438:
2347:
2094:
7200:
6588:-point of a maximal torus), a graph automorphism (corresponding to an automorphism of the Dynkin diagram), and a field automorphism (coming from an automorphism of the field
6586:
6155:
5953:
5162:
3588:
3114:
3082:
3054:
3030:
3006:
2966:
2409:
2200:
2127:
2057:
1763:
1659:
1293:
5791:(λ) are in general unknown, although a large body of theory has been developed to analyze these representations. One important result is that the dimension and character of
486:
461:
424:
4540:
1425:. (Some authors call this property "almost simple".) This differs slightly from the terminology for abstract groups, in that a simple algebraic group may have nontrivial
2547:
2503:
1207:
7384:
1383:
2380:
2208:
3847:
are in one-to-one correspondence with the subsets of the set Δ of simple roots (or equivalently, the subsets of the set of vertices of the Dynkin diagram). Let
2458:
1353:
1227:
1167:
1139:
1119:
1088:
7235:
with the given action. (For a quasi-split group, every Galois orbit in the Dynkin diagram is circled.) Moreover, any other simply connected semisimple group
7113:
says that a nondegenerate quadratic form over a field is determined up to isomorphism by its Witt index together with its anisotropic kernel. Likewise, the
4956:
4776:
8222:
Building on the Hasse principle, the classification of semisimple groups over number fields is well understood. For example, there are exactly three
7381:
is nontrivial, then it is encoded in the Galois action on the Dynkin diagram: the index-2 subgroup of the Galois group that acts as the identity is
1477:
scheme. Every reductive group over a field admits a central isogeny from the product of a torus and some simple groups. For example, over any field
6253:
algebraically closed, and they are understood for some other fields such as number fields, but for arbitrary fields there are many open questions.
4146:
2004:
is algebraically closed, any two (nondegenerate) quadratic forms of the same dimension are isomorphic, and so it is reasonable to call this group
8216:
2555:
7109:, which reduces the problem to the case of anisotropic groups. This reduction generalizes several fundamental theorems in algebra. For example,
4039:{\displaystyle \left\{{\begin{bmatrix}*&*&*&*\\*&*&*&*\\0&0&*&*\\0&0&0&*\end{bmatrix}}\right\}}
788:
5288:*. The problem of classifying the real reductive groups largely reduces to classifying the simple Lie groups. These are classified by their
1574:
It is slightly awkward that the definition of a reductive group over a field involves passage to the algebraic closure. For a perfect field
9338:
4322:
in the 1880s and 1890s. In particular, the dimensions, centers, and other properties of the simple algebraic groups can be read from the
4438:
with a given Dynkin diagram, with simple groups corresponding to the connected diagrams. At the other extreme, a semisimple group is of
7223:
There is a full classification of quasi-split groups in these terms. Namely, for each action of the absolute Galois group of a field
2942:
8284:
5272:
In particular, every connected semisimple Lie group (meaning that its Lie algebra is semisimple) is reductive. Also, the Lie group
4221:
For the orthogonal group or the symplectic group, the projective homogeneous varieties have a similar description as varieties of
2821:, as in the theories of complex semisimple Lie algebras or compact Lie groups. Here is the way roots appear for reductive groups.
6354:
8306:
1709:. These kinds of groups are useful because their classification can be described through combinatorical data called root data.
923:
346:
9216:
9048:
9009:
8922:
8850:
3373:; this is a combinatorial structure which can be completely classified. More generally, the roots of a reductive group form a
1487:
9398:
9312:
9242:
9091:
9017:
8930:
8858:
7121:
is determined up to isomorphism by its Tits index together with its anisotropic kernel, an associated anisotropic semisimple
2132:
8116:
7922:
3864:
3456:
9056:
4314:. This result is essentially identical to the classifications of compact Lie groups or complex semisimple Lie algebras, by
2746:
296:
1450:
3918:
above are the groups of invertible matrices with zero entries below a given set of squares along the diagonal, such as:
9132:
8977:
8891:
7869:
7847:
5444:
of non-compact type. In fact, every symmetric space of non-compact type arises this way. These are central examples in
781:
291:
9275:
9167:
8818:
8728:
3665:
means a positive root that is not a sum of two other positive roots. Write Δ for the set of simple roots. The number
9490:
8961:
7359:
7080:
3300:
as the direct sum of the diagonal matrices and the 1-dimensional subspaces indexed by the off-diagonal positions (
7110:
6999:
3544:{\displaystyle {\mathfrak {b}}={\mathfrak {t}}\oplus \bigoplus _{\alpha \in \Phi ^{+}}{\mathfrak {g}}_{\alpha }.}
9206:
9153:
1146:
8802:
7889:
5828:
1938:
707:
9382:
9267:
9198:
9083:
8328:
5244:) (in the classical topology). It is also standard to assume that the image of the adjoint representation Ad(
5176:
are classified by root data. This statement includes the existence of
Chevalley groups as group schemes over
3182:{\displaystyle {\mathfrak {g}}={\mathfrak {t}}\oplus \bigoplus _{\alpha \in \Phi }{\mathfrak {g}}_{\alpha }.}
774:
8075:
of positive characteristic was proved earlier by Harder (1975): for every simply connected semisimple group
9473:
Schémas en groupes (SGA 3), II: Groupes de type multiplicatif, et structure des schémas en groupes généraux
8958:
Schémas en groupes (SGA 3), II: Groupes de type multiplicatif, et structure des schémas en groupes généraux
7114:
3744:
3593:
3265:
3207:
2664:
1335:
Over fields of characteristic zero another equivalent definition of a reductive group is a connected group
3751:
on the weight lattice. The connected Dynkin diagrams (corresponding to simple groups) are pictured below.
9193:
9108:
8301:
7055:
6389:
5990:
3762:, an important point is that a root α determines not just a 1-dimensional subspace of the Lie algebra of
391:
205:
1316:
9116:
8315:
8274:
7603:
7571:
7539:
7529:
6093:
5891:
5775:
are typically not direct sums of irreducibles. For a dominant weight λ, the irreducible representation
5096:
1664:
1239:
993:
893:
838:
123:
5708:
There remains the problem of describing the irreducible representation with given highest weight. For
2414:
2323:
2070:
9188:
7178:
6564:
6133:
5931:
5300:
5293:
5140:
4323:
3569:
3095:
3063:
3035:
3011:
2987:
2947:
2738:
2730:
2385:
2176:
2129:
through the diagonal, and from this representation, their unipotent radical is trivial. For example,
2103:
2035:
1739:
1637:
1271:
5712:
of characteristic zero, there are essentially complete answers. For a dominant weight λ, define the
4711:{\displaystyle q(x_{1},\ldots ,x_{2n+1})=x_{1}x_{2}+x_{3}x_{4}+\cdots +x_{2n-1}x_{2n}+x_{2n+1}^{2};}
1627:
1308:
8754:
8289:
8253:
5434:
4927:
4093:
1355:
admitting a faithful semisimple representation which remains semisimple over its algebraic closure
1234:
911:. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the
857:
589:
323:
200:
88:
8219:, saying that a central simple algebra over a number field is determined by its local invariants.
3463:Φ ⊂ Φ, with the property that Φ is the disjoint union of Φ and −Φ. Explicitly, the Lie algebra of
469:
444:
407:
8319:
7815:
6835:
6515:
5552:, the analog of a metric with nonpositive curvature. The dimension of the affine building is the
5324:
3416:. The Weyl group is in fact a finite group generated by reflections. For example, for the group
2508:
9467:
9037:
8999:
8953:
8912:
8279:
7467:{\displaystyle \operatorname {Gal} (k_{s}/k({\sqrt {d}}))\subset \operatorname {Gal} (k_{s}/k)}
6025:
5812:
5753:
5745:
5509:
4353:
2934:
1595:
1015:
1009:
997:
989:
908:
810:
739:
529:
2472:
1176:
9420:
7137:
5411:
1426:
957:
613:
2300:{\displaystyle (a_{1},a_{2})\mapsto {\begin{bmatrix}a_{1}&0\\0&a_{2}\end{bmatrix}}.}
1358:
9449:
9408:
9359:
9322:
9285:
9230:
9177:
9142:
9101:
9073:
9066:
9027:
8987:
8940:
8901:
8868:
8828:
8783:
8763:
8738:
8323:
8293:
8257:
7866:.) It follows, for example, that every reductive group over a finite field is quasi-split.
6918:
5816:
2688:
2358:
1823:
842:
826:
553:
541:
159:
93:
8:
9495:
8839:
8310:
8250:
are the finite simple groups constructed from simple algebraic groups over finite fields.
8014:
6942:
6826:
simply connected and quasi-split, the
Whitehead group is trivial, and so the whole group
5780:
5771:
of positive characteristic, the situation is far more subtle, because representations of
5574:
5449:
5445:
4225:
flags with respect to a given quadratic form or symplectic form. For any reductive group
3670:
3452:
2794:
2320:
is not reductive since its unipotent radical is itself. This includes the additive group
905:
853:
841:. Reductive groups include some of the most important groups in mathematics, such as the
814:
128:
23:
9234:
8767:
5843:-Kazhdan-Lusztig polynomials, which are even more complex, but at least are computable.
9437:
9363:
9220:
8787:
7715:. Namely, such forms (up to isomorphism) are in one-to-one correspondence with the set
7633:
6558:
6268:-rank greater than 0 (that is, if it contains a nontrivial split torus), and otherwise
5836:
5525:
5492:
4947:
4364:
4222:
4085:
2443:
1942:
1470:
1338:
1212:
1152:
1124:
1104:
1073:
113:
85:
9299:, Progress in Mathematics, vol. 9 (2nd ed.), Boston, MA: Birkhäuser Boston,
9394:
9308:
9271:
9238:
9184:
9163:
9149:
9128:
9087:
9052:
9013:
8973:
8926:
8887:
8854:
8814:
8791:
8752:(1971), "Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I.",
8724:
8366:
8297:
8265:
8247:
7863:
7839:
7256:
7227:
on a Dynkin diagram, there is a unique simply connected semisimple quasi-split group
7162:
5395:
5367:
5354:)) is a real reductive group that cannot be viewed as an algebraic group. Similarly,
5028:{\displaystyle \operatorname {Aut} (G)\cong \operatorname {Out} (G)\ltimes (G/Z)(k),}
4914:{\displaystyle q(x_{1},\ldots ,x_{2n})=x_{1}x_{2}+x_{3}x_{4}+\cdots +x_{2n-1}x_{2n}.}
4330:
1303:. (This is equivalent to the definition of reductive groups in the introduction when
1296:
901:
518:
361:
255:
9367:
7649:
5330:(2) is connected as an algebraic group over any field, but its group of real points
5172:
and
Grothendieck showed that split reductive group schemes over any nonempty scheme
4092:. Thus the classification of parabolic subgroups amounts to a classification of the
2460:
on the diagonal. This is an example of a non-reductive group which is not unipotent.
684:
9429:
9386:
9374:
9347:
9333:
9329:
9300:
9120:
8965:
8884:
Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs
8798:
8771:
8716:
8261:
8227:
6986:
6511:
6187:
5757:
5725:
5465:
5221:
4427:
4368:
2782:
1908:
1874:
1474:
960:
in various contexts. First, one can study the representations of a reductive group
889:
868:
830:
669:
661:
653:
645:
637:
625:
565:
505:
495:
337:
279:
154:
9471:
9003:
8916:
7901:
4430:
in the classical topology.) Chevalley's classification gives that, over any field
9463:
9445:
9404:
9355:
9318:
9281:
9173:
9138:
9097:
9062:
9023:
8995:
8983:
8949:
8936:
8908:
8897:
8875:
8864:
8824:
8810:
8779:
8734:
8712:
7905:
7105:
In seeking to classify reductive groups which need not be split, one step is the
6958:
6930:
5853:
5598:
5441:
5399:
5169:
5090:
5054:
has a simpler description: it is the automorphism group of the Dynkin diagram of
4315:
3433:
2317:
2097:
1413:) if it is semisimple, nontrivial, and every smooth connected normal subgroup of
1098:
1067:
976:-vector spaces. But also, one can study the complex representations of the group
753:
732:
689:
577:
500:
330:
244:
184:
64:
8504:
SGA 3 (2011), v. 3, Théorème XXV.1.1; Conrad (2014), Theorems 6.1.16 and 6.1.17.
7348:, and so its automorphism group is of order 2, switching the two "legs" of the D
5358:(2) is simply connected as an algebraic group over any field, but the Lie group
5276:
is reductive in this sense, since it can be viewed as the identity component of
3879:). As a result, there are exactly 2 conjugacy classes of parabolic subgroups in
9292:
9251:
9159:
8879:
6052:
5820:
5804:
5496:
5414:, that is, the product of a semisimple Lie algebra and an abelian Lie algebra.
5383:
5289:
4415:
3740:
3448:
3388:
2907:
2817:
The classification of reductive algebraic groups is in terms of the associated
1919:
1095:
939:
897:
760:
696:
386:
366:
303:
268:
189:
179:
164:
149:
103:
80:
9304:
8720:
5621:. In particular, this parametrization is independent of the characteristic of
4319:
9484:
8270:
7897:
7641:
6377:
4270:
correspond to the connected diagrams. Thus there are simple groups of types A
4077:
3748:
2700:
1918:) is the subgroup of the general linear group that preserves a nondegenerate
1897:
1051:
1023:
822:
679:
601:
435:
308:
174:
7892:(which has cohomological dimension 2). More generally, for any number field
5839:
conjectured the irreducible characters of a reductive group in terms of the
4211:{\displaystyle 0\subset S_{a_{1}}\subset \cdots \subset S_{a_{i}}\subset V.}
2873:(as an algebraic group) is a direct sum of 1-dimensional representations. A
9415:
9005:
Schémas en groupes (SGA 3), III: Structure des schémas en groupes réductifs
8835:
8749:
8745:
8704:
8069:
6697:
6048:
5784:
5614:
5067:
2648:{\displaystyle B_{n}/(R_{u}(B_{n}))\cong \prod _{i=1}^{n}\mathbb {G} _{m}.}
1901:
1027:
947:
919:
534:
233:
222:
169:
144:
139:
98:
69:
32:
9124:
8918:
Schémas en groupes (SGA 3), I: Propriétés générales des schémas en groupes
1209:. (Some authors do not require reductive groups to be connected.) A group
9113:
Algebraic Groups: The Theory of Group
Schemes of Finite Type over a Field
8711:, Graduate Texts in Mathematics, vol. 126 (2nd ed.), New York:
7661:
6615:) is close to being simple, under mild assumptions. Namely, suppose that
6549:, Steinberg also determined the automorphism group of the abstract group
6534:(the root subgroups), with relations determined by the Dynkin diagram of
6241:
essentially includes the problem of classifying all quadratic forms over
4344:
had been constructed earlier, at least in the form of the abstract group
4261:
2818:
2353:
1838:
912:
802:
9390:
7075:-invariant measure). For example, a discrete subgroup Γ is a lattice if
2757:
if its finite-dimensional representations are completely reducible. For
892:
showed that the classification of reductive groups is the same over any
9441:
9351:
9255:
9077:
8969:
8775:
7244:
7106:
7100:
4525:
4509:
3792:
3392:
3379:
3374:
2032:, although they all have the same base change to the algebraic closure
1000:. The structure theory of reductive groups is used in all these areas.
834:
701:
429:
6842:-simple groups the Whitehead group is trivial. In all known examples,
5201:
3887:. Explicitly, the parabolic subgroup corresponding to a given subset
3116:
together with 1-dimensional subspaces indexed by the set Φ of roots:
522:
9433:
6497:
1988:) can always be defined as the maximal smooth connected subgroup of
1586:
is reductive if and only if every smooth connected unipotent normal
9225:
6707:
The exceptions for fields of order 2 or 3 are well understood. For
5787:. The dimensions and characters of the irreducible representations
4473:
For example, the simply connected split simple groups over a field
3815:
and the positive root subgroups. In fact, a split semisimple group
1429:(although the center must be finite). For example, for any integer
59:
5625:. In more detail, fix a split maximal torus and a Borel subgroup,
4116:, parametrizing sequences of linear subspaces of given dimensions
1594:
is trivial. For an arbitrary field, the latter property defines a
8960:. Lecture Notes in Mathematics. Vol. 152. Berlin; New York:
6701:
6696:) by its center is simple (as an abstract group). The proof uses
4477:
corresponding to the "classical" Dynkin diagrams are as follows:
3032:
corresponding to each root is 1-dimensional, and the subspace of
2923:
401:
315:
7632:
on itself by left translation. A torsor can also be viewed as a
5410:) is not a real reductive group, even though its Lie algebra is
4394:
reductive groups is the same over any field. A semisimple group
3787:. The root subgroup is the unique copy of the additive group in
7818:, which are invariants taking values in Galois cohomology with
7508:
5545:
5320:) is not connected, and likewise for simply connected groups.
2881:
means an isomorphism class of 1-dimensional representations of
1121:
is trivial. More generally, a connected linear algebraic group
896:. In particular, the simple algebraic groups are classified by
40:
8110:, the Hasse principle holds in a weaker form: the natural map
7727:)). For example, (nondegenerate) quadratic forms of dimension
6237:
As a result, the problem of classifying reductive groups over
5601:, which are defined as the intersection of the weight lattice
2658:
6561:, a diagonal automorphism (meaning conjugation by a suitable
6368:
of characteristic zero (such as the real numbers), the group
5752:(λ) is isomorphic to the Schur module ∇(λ). Furthermore, the
5693:). Chevalley showed that every irreducible representation of
5180:, and it says that every split reductive group over a scheme
5133:, the corresponding geometric fiber means the base change of
4938:
is isomorphic to the automorphism group of the root datum of
4387:
over a field of positive characteristic were completely new.
6388:
is reductive and anisotropic. Example: the orthogonal group
4434:, there is a unique simply connected split semisimple group
2765:
is linearly reductive if and only if the identity component
2100:. They are examples of reductive groups since they embed in
817:. One definition is that a connected linear algebraic group
7358:
on the Dynkin diagram is trivial if and only if the signed
5312:
may be connected as an algebraic group while the Lie group
5184:
is isomorphic to the base change of a
Chevalley group from
4442:
if its center is trivial. The split semisimple groups over
4256:
7842:'s "Conjecture I": for a connected linear algebraic group
7153:) acts (continuously) on the "absolute" Dynkin diagram of
7094:
5760:(and in particular the dimension) of this representation.
5338:) has two connected components. The identity component of
4100:(with smooth stabilizer group; that is no restriction for
3566:), then this is the obvious decomposition of the subspace
8531:
Jantzen (2003), Proposition II.4.5 and
Corollary II.5.11.
7862:) = 1. (The case of a finite field was known earlier, as
6917:) is compact in the classical topology. Since it is also
5823:'s conjecture in that case). Their character formula for
4418:, being simply connected in this sense is equivalent to
3451:
containing a given maximal torus, and they are permuted
1564:{\displaystyle GL(n)\cong (G_{m}\times SL(n))/\mu _{n}.}
2166:{\displaystyle \mathbb {G} _{m}\times \mathbb {G} _{m}}
926:
says that most finite simple groups arise as the group
8628:
Tits (1964), Main
Theorem; Gille (2009), Introduction.
8193:{\displaystyle H^{1}(k,G)\to \prod _{v}H^{1}(k_{v},G)}
7999:{\displaystyle H^{1}(k,G)\to \prod _{v}H^{1}(k_{v},G)}
7872:
predicts that for a simply connected semisimple group
6941:) contains infinitely many normal subgroups of finite
6718:, Tits's simplicity theorem remains valid except when
6526:). It is generated by copies of the additive group of
5580:
3937:
3558:
is the Borel subgroup of upper-triangular matrices in
2249:
8119:
7925:
7606:
7574:
7542:
7387:
7310:
be a nondegenerate quadratic form of even dimension 2
7181:
6567:
6136:
6096:
5993:
5971:(the maximum dimension of an isotropic subspace over
5934:
5894:
5143:
5099:
4959:
4779:
4543:
4149:
3927:
3799:
and which has the given Lie algebra. The whole group
3596:
3572:
3480:
3268:
3210:
3125:
3098:
3066:
3038:
3014:
2990:
2950:
2558:
2511:
2475:
2446:
2417:
2388:
2361:
2326:
2211:
2179:
2135:
2106:
2073:
2038:
1822:
under multiplication. Another reductive group is the
1742:
1667:
1640:
1490:
1361:
1341:
1274:
1242:
1215:
1179:
1155:
1127:
1107:
1076:
472:
447:
410:
8994:
8948:
8907:
8477:
Chevalley (2005); Springer (1998), 9.6.2 and 10.1.1.
7799:)). These problems motivate the systematic study of
7354:
diagram. The action of the absolute Galois group of
6380:
in the classical topology (based on the topology of
5398:) are real reductive groups. On the other hand, the
5236:) whose kernel is finite and whose image is open in
4406:
if every central isogeny from a semisimple group to
2976:
means a nonzero weight that occurs in the action of
9477:
Revised and annotated edition of the 1970 original.
9032:
Revised and annotated edition of the 1970 original.
8945:
Revised and annotated edition of the 1970 original.
8807:
Classification des groupes algébriques semi-simples
8102:In the slightly different case of an adjoint group
7876:over a field of cohomological dimension at most 2,
6834:) is simple modulo its center. More generally, the
5831:, which are combinatorially complex. For any prime
8192:
8055:) is trivial for every nonarchimidean local field
7998:
7624:
7592:
7560:
7501:
7466:
7194:
6580:
6149:
6114:
6013:
5947:
5912:
5156:
5117:
5027:
4913:
4710:
4210:
4038:
3693:is semisimple). For example, the simple roots for
3620:
3582:
3543:
3459:). A choice of Borel subgroup determines a set of
3292:
3234:
3181:
3108:
3076:
3048:
3024:
3000:
2960:
2647:
2541:
2497:
2452:
2432:
2403:
2374:
2341:
2299:
2194:
2165:
2121:
2088:
2051:
1757:
1733:A fundamental example of a reductive group is the
1685:
1653:
1563:
1377:
1347:
1287:
1260:
1221:
1201:
1161:
1133:
1113:
1082:
480:
455:
418:
9334:"Regular elements of semisimple algebraic groups"
9148:
8420:Demazure & Gabriel (1970), Théorème IV.3.3.6.
6948:
6498:Structure of semisimple groups as abstract groups
4516:+1) associated to a quadratic form of dimension 2
4446:with given Dynkin diagram are exactly the groups
3807:and the root subgroups, while the Borel subgroup
3739:Root systems are classified by the corresponding
2469:Note that the normality of the unipotent radical
9482:
9462:
9418:(1964), "Algebraic and abstract simple groups",
5597:(as an algebraic group) are parametrized by the
5046:. For a split semisimple simply connected group
2733:zero, all finite-dimensional representations of
1578:, that can be avoided: a linear algebraic group
9204:
8874:
5520:plays the role of the symmetric space. Namely,
4755:) associated to a quadratic form of dimension 2
1169:is trivial. This normal subgroup is called the
988:is a finite field, or the infinite-dimensional
7322:≥ 5. (These restrictions can be avoided.) Let
6541:For a simply connected split semisimple group
6502:For a simply connected split semisimple group
5050:over a field, the outer automorphism group of
3324:for the standard basis for the weight lattice
3262:. Then the root-space decomposition expresses
1046:. Equivalently, a linear algebraic group over
8691:Platonov & Rapinchuk (1994), Theorem 6.4.
8682:Platonov & Rapinchuk (1994), section 6.8.
8673:Platonov & Rapinchuk (1994), Theorem 6.6.
8655:Platonov & Rapinchuk (1994), section 9.1.
8592:Platonov & Rapinchuk (1994), Theorem 3.1.
7687:Torsors arise whenever one seeks to classify
7255:is the group associated to an element of the
6486:| ≤ 1, and it is quasi-split if and only if |
6442:. A split reductive group is quasi-split. If
5846:
5736:associated to λ; this is a representation of
5681:acts on that line through its quotient group
4375:. By contrast, the Chevalley groups of type F
2464:
1896:) that preserves a nondegenerate alternating
1330:
1141:over an algebraically closed field is called
1090:over an algebraically closed field is called
1061:
953:, or as minor variants of that construction.
922:, the classification is well understood. The
782:
8233:, corresponding to the three real forms of E
7083:says, in particular: for a simple Lie group
6291:contains a copy of the multiplicative group
6008:
5994:
5212:such that there is a linear algebraic group
3669:of simple roots is equal to the rank of the
2691:, with respect to the classical topology on
968:as an algebraic group, which are actions of
9381:, University Lecture Series, vol. 66,
8549:Riche & Williamson (2018), section 1.8.
8030:is the corresponding local field (possibly
6873:) can be far from simple. For example, let
6680:) is nontrivial, and even Zariski dense in
6557:). Every automorphism is the product of an
6415:), and so it is anisotropic if and only if
6345:perfect, it is also equivalent to say that
6280:, the following conditions are equivalent:
5645:with a smooth connected unipotent subgroup
2659:Other characterizations of reductive groups
1976:) is in fact connected but not smooth over
9205:Riche, Simon; Williamson, Geordie (2018),
9183:
7908:: for a simply connected semisimple group
7904:and Vladimir Chernousov (1989) proved the
7803:-torsors, especially for reductive groups
7751:)), and central simple algebras of degree
7087:of real rank at least 2, every lattice in
6968:can be extended to an affine group scheme
6743:, or non-split (that is, unitary) of type
6070:means the square root of the dimension of
5685:, by some element λ of the weight lattice
5061:
3906:. For example, the parabolic subgroups of
3851:be the order of Δ, the semisimple rank of
3819:is generated by the root subgroups alone.
1388:
789:
775:
9373:
9328:
9224:
8797:
8744:
8346:SGA 3 (2011), v. 3, Définition XIX.1.6.1.
7684:), in the language of Galois cohomology.
6783:, in order to understand the whole group
6688:is infinite.) Then the quotient group of
5748:says that the irreducible representation
5386:. By definition, all finite coverings of
4068:-subgroup such that the quotient variety
2632:
2420:
2329:
2153:
2138:
2076:
2016:, different quadratic forms of dimension
1856:
1601:
1233:is called semisimple or reductive if the
474:
449:
412:
9291:
9260:Automorphic Forms, Representations, and
9250:
8583:Borel & Tits (1971), Corollaire 3.8.
8437:
8435:
8384:Conrad (2014), after Proposition 5.1.17.
7071:/Γ has finite volume (with respect to a
6976:, and this determines an abstract group
6647:-points of copies of the additive group
6635:) be the subgroup of the abstract group
5779:(λ) is the unique simple submodule (the
5195:
4260:
4257:Classification of split reductive groups
3803:is generated (as an algebraic group) by
3766:, but also a copy of the additive group
3467:is the direct sum of the Lie algebra of
3258:be the subgroup of diagonal matrices in
2828:be a split reductive group over a field
2663:Every compact connected Lie group has a
9072:
8561:
8559:
8557:
8555:
8486:Milne (2017), Theorems 23.25 and 23.55.
7838:). In this direction, Steinberg proved
7095:The Galois action on the Dynkin diagram
6360:For a connected linear algebraic group
3369:The roots of a semisimple group form a
2699:). For example, the inclusion from the
2020:can yield non-isomorphic simple groups
9483:
8834:
6324:contains a copy of the additive group
5795:(λ) are known when the characteristic
5093:and affine, and every geometric fiber
4942:. Moreover, the automorphism group of
4390:More generally, the classification of
3822:
924:classification of finite simple groups
347:Classification of finite simple groups
9107:
9035:
9002:(2011) . Gille, P.; Polo, P. (eds.).
8915:(2011) . Gille, P.; Polo, P. (eds.).
8703:
8432:
8285:Weil's conjecture on Tamagawa numbers
7888:) = 1. The conjecture is known for a
7810:When possible, one hopes to classify
7171:(which is also the Dynkin diagram of
6957:be a linear algebraic group over the
6438:if it contains a Borel subgroup over
5987:has the maximum possible Witt index,
5593:, the irreducible representations of
5417:For a connected real reductive group
3777:with the given Lie algebra, called a
3621:{\displaystyle {{\mathfrak {g}}l}(n)}
3293:{\displaystyle {{\mathfrak {g}}l}(n)}
3235:{\displaystyle {{\mathfrak {g}}l}(n)}
2789:is linearly reductive if and only if
1618:if it contains a split maximal torus
9414:
9339:Publications Mathématiques de l'IHÉS
8552:
7303:, as discussed in the next section.
6877:be a division algebra with center a
6245:or all central simple algebras over
6169:is isomorphic to the matrix algebra
3839:that contain a given Borel subgroup
3835:, the smooth connected subgroups of
2411:has a non-trivial unipotent radical
1469:of reductive groups is a surjective
9079:Representations of Algebraic Groups
8217:Albert–Brauer–Hasse–Noether theorem
8068:matter. The analogous result for a
7010:). (Arithmeticity of a subgroup of
6909:-simple group. As mentioned above,
6893:is finite and greater than 1. Then
6310:contains a parabolic subgroup over
6014:{\displaystyle \lfloor n/2\rfloor }
5860:Every nondegenerate quadratic form
5581:Representations of reductive groups
5491:that is complete with respect to a
5224:) is reductive, and a homomorphism
3600:
3575:
3527:
3493:
3483:
3272:
3214:
3165:
3138:
3128:
3101:
3069:
3041:
3017:
2993:
2953:
1952:) is reductive, in fact simple for
13:
9155:Algebraic Groups and Number Theory
9045:Séminaire Bourbaki. Vol. 2007/2008
7779:-forms of a given algebraic group
7018:) is independent of the choice of
6450:, then any two Borel subgroups of
4466:-subgroup scheme of the center of
4458:is the simply connected group and
4132:contained in a fixed vector space
3914:) that contain the Borel subgroup
3513:
3157:
2440:of upper-triangular matrices with
2096:and products of it are called the
1849:) is a simple algebraic group for
1598:, which is somewhat more general.
1268:is semisimple or reductive, where
992:of a real reductive group, or the
14:
9507:
9456:
8364:
8064:, and so only the real places of
7625:{\displaystyle G_{\overline {k}}}
7593:{\displaystyle G_{\overline {k}}}
7561:{\displaystyle X_{\overline {k}}}
7338:. The absolute Dynkin diagram of
7157:, that is, the Dynkin diagram of
6462:). Example: the orthogonal group
6454:are conjugate by some element of
6115:{\displaystyle G_{\overline {k}}}
5913:{\displaystyle G_{\overline {k}}}
5868:determines a reductive group G =
5720:-vector space of sections of the
5220:whose identity component (in the
5118:{\displaystyle G_{\overline {k}}}
3895:together with the root subgroups
3395:of a maximal torus by the torus,
2885:, or equivalently a homomorphism
2679:into the complex reductive group
1873:An important simple group is the
1686:{\displaystyle G_{\overline {k}}}
1449:is simple, and its center is the
1261:{\displaystyle G_{\overline {k}}}
8540:Jantzen (2003), section II.8.22.
8468:Borel (1991), Proposition 21.12.
8459:Milne (2017), Proposition 17.53.
8355:Milne (2017), Proposition 21.60.
8226:-forms of the exceptional group
6885:. Suppose that the dimension of
5819:, and Wolfgang Soergel (proving
5204:rather than algebraic groups, a
4094:projective homogeneous varieties
3590:of upper-triangular matrices in
3088:. Therefore, the Lie algebra of
2505:implies that the quotient group
2433:{\displaystyle \mathbb {U} _{n}}
2342:{\displaystyle \mathbb {G} _{a}}
2089:{\displaystyle \mathbb {G} _{m}}
1837:, the subgroup of matrices with
1693:). It is equivalent to say that
1145:if the largest smooth connected
39:
8685:
8676:
8667:
8658:
8649:
8640:
8631:
8622:
8613:
8604:
8595:
8586:
8577:
8568:
8543:
8534:
8525:
8516:
8507:
8498:
8489:
8480:
8471:
8462:
8453:
8444:
7502:Torsors and the Hasse principle
7195:{\displaystyle {\overline {k}}}
7063:means a discrete subgroup Γ of
7034:) is an arithmetic subgroup of
6933:(but not finite). As a result,
6761:, the theorem holds except for
6581:{\displaystyle {\overline {k}}}
6256:A reductive group over a field
6150:{\displaystyle {\overline {k}}}
5948:{\displaystyle {\overline {k}}}
5560:. For example, the building of
5292:; or one can just refer to the
5168:.) Extending Chevalley's work,
5157:{\displaystyle {\overline {k}}}
3891:of Δ is the group generated by
3583:{\displaystyle {\mathfrak {b}}}
3109:{\displaystyle {\mathfrak {t}}}
3077:{\displaystyle {\mathfrak {t}}}
3049:{\displaystyle {\mathfrak {g}}}
3025:{\displaystyle {\mathfrak {g}}}
3001:{\displaystyle {\mathfrak {g}}}
2961:{\displaystyle {\mathfrak {g}}}
2918:) isomorphic to the product of
2404:{\displaystyle {\text{GL}}_{n}}
2310:
2195:{\displaystyle {\text{GL}}_{2}}
2122:{\displaystyle {\text{GL}}_{n}}
2052:{\displaystyle {\overline {k}}}
1925:on a vector space over a field
1758:{\displaystyle {\text{GL}}_{n}}
1654:{\displaystyle {\overline {k}}}
1288:{\displaystyle {\overline {k}}}
9470:, Gille, P.; Polo, P. (eds.),
9217:Société Mathématique de France
9049:Société Mathématique de France
9010:Société Mathématique de France
8923:Société Mathématique de France
8851:Société Mathématique de France
8664:Steinberg (1965), Theorem 1.9.
8601:Borel (1991), Theorem 20.9(i).
8495:Milne (2017), Corollary 23.47.
8450:Milne (2017), Corollary 21.12.
8423:
8414:
8411:Milne (2017), Corollary 22.43.
8405:
8396:
8387:
8378:
8358:
8349:
8340:
8187:
8168:
8145:
8142:
8130:
7993:
7974:
7951:
7948:
7936:
7890:totally imaginary number field
7783:(sometimes called "twists" of
7711:over the algebraic closure of
7490:is quasi-split if and only if
7461:
7440:
7428:
7425:
7415:
7394:
7318:of characteristic not 2, with
7206:consists of the root datum of
7081:Margulis arithmeticity theorem
6949:Lattices and arithmetic groups
6249:. These problems are easy for
6058:* (as an algebraic group over
5448:of manifolds with nonpositive
5019:
5013:
5010:
4996:
4990:
4984:
4972:
4966:
4818:
4783:
4588:
4547:
3615:
3609:
3471:and the positive root spaces:
3287:
3281:
3229:
3223:
2603:
2600:
2587:
2574:
2536:
2530:
2492:
2486:
2241:
2238:
2212:
1814:-rational points is the group
1717:
1540:
1537:
1531:
1509:
1503:
1497:
1196:
1190:
1003:
708:Infinite dimensional Lie group
1:
9383:American Mathematical Society
9268:American Mathematical Society
9215:, Astérisque, vol. 397,
9084:American Mathematical Society
9047:, Astérisque, vol. 326,
8847:Autour des schémas en groupes
8697:
8619:Steinberg (2016), Theorem 30.
8513:Springer (1979), section 5.1.
8329:Radical of an algebraic group
8038:). Moreover, the pointed set
6627:has at least 4 elements. Let
6623:, and suppose that the field
6607:says that the abstract group
6035:determines a reductive group
5641:is the semidirect product of
4265:The connected Dynkin diagrams
4104:of characteristic zero). For
3685:(which is simply the rank of
3455:by the Weyl group (acting by
3336:, the roots are the elements
2722:) is a homotopy equivalence.
956:Reductive groups have a rich
9379:Lectures on Chevalley Groups
9152:; Rapinchuk, Andrei (1994),
9038:"Le problème de Kneser–Tits"
8610:Steinberg (2016), Theorem 8.
8441:Milne (2017), Theorem 21.11.
8429:Milne (2017), Theorem 12.12.
8402:Milne (2017), Theorem 22.42.
7691:of a given algebraic object
7616:
7584:
7552:
7486:, the maximum possible, and
7243:with the given action is an
7187:
7111:Witt's decomposition theorem
6573:
6142:
6106:
5940:
5928:) over an algebraic closure
5904:
5763:For a split reductive group
5744:of characteristic zero, the
5585:For a split reductive group
5149:
5109:
3827:For a split reductive group
3754:For a split reductive group
2836:be a split maximal torus in
2737:(as an algebraic group) are
2044:
1677:
1646:
1280:
1252:
1042:, for some positive integer
942:of a simple algebraic group
481:{\displaystyle \mathbb {Z} }
456:{\displaystyle \mathbb {Z} }
419:{\displaystyle \mathbb {Z} }
7:
9194:Encyclopedia of Mathematics
8809:, Collected Works, Vol. 3,
8646:Gille (2009), Théorème 6.1.
8574:Borel (1991), section 23.2.
8565:Borel (1991), section 23.4.
8522:Milne (2017), Theorem 22.2.
8302:geometric Langlands program
8240:
7707:which become isomorphic to
7512:for an affine group scheme
6217:) − 1. So the simple group
5829:Kazhdan–Lusztig polynomials
5480:(2) is hyperbolic 3-space.
5370:isomorphic to the integers
5265:)) (which is automatic for
5125:is reductive. (For a point
4930:of a split reductive group
3242:is the vector space of all
3060:is exactly the Lie algebra
2898:. The weights form a group
1712:
994:automorphic representations
884:semisimple algebraic groups
839:irreducible representations
206:List of group theory topics
10:
9512:
9117:Cambridge University Press
8316:Geometric invariant theory
7664:of isomorphism classes of
7175:over an algebraic closure
7098:
7022:-structure.) For example,
6664:. (By the assumption that
5847:Non-split reductive groups
5301:admissible representations
5296:(up to finite coverings).
4767:, which can be written as:
3432:)), the Weyl group is the
3377:, a slight variation. The
2869:. Every representation of
2549:is reductive. For example,
2542:{\displaystyle G/R_{u}(G)}
2465:Associated reductive group
1701:that is maximal among all
1331:With representation theory
1094:if every smooth connected
1062:With the unipotent radical
1007:
894:algebraically closed field
9305:10.1007/978-0-8176-4840-4
8840:"Reductive group schemes"
8721:10.1007/978-1-4612-0941-6
8637:Tits (1964), section 1.2.
7247:of the quasi-split group
6838:asks for which isotropic
6605:Tits's simplicity theorem
6478:is split if and only if |
6272:. For a semisimple group
6229:is a matrix algebra over
5673:maps the line spanned by
5294:list of simple Lie groups
4356:. For example, the group
4324:list of simple Lie groups
3867:to a subgroup containing
3628:. The positive roots are
2910:of representations, with
1818:* of nonzero elements of
1433:at least 2 and any field
1393:A linear algebraic group
825:is reductive if it has a
9208:Tilting Modules and the
9036:Gille, Philippe (2009),
8755:Inventiones Mathematicae
8334:
8290:Langlands classification
8254:Generalized flag variety
7846:over a perfect field of
7816:cohomological invariants
7494:has Witt index at least
7478:is split if and only if
7115:Artin–Wedderburn theorem
7091:is an arithmetic group.
6791:), one can consider the
6599:-simple algebraic group
6530:indexed by the roots of
5803:is much bigger than the
5435:maximal compact subgroup
5421:, the quotient manifold
5164:of the residue field of
5137:to an algebraic closure
4928:outer automorphism group
4410:is an isomorphism. (For
3447:There are finitely many
2812:
2761:of characteristic zero,
2498:{\displaystyle R_{u}(G)}
1964:of characteristic 2 and
1229:over an arbitrary field
1202:{\displaystyle R_{u}(G)}
858:special orthogonal group
324:Elementary abelian group
201:Glossary of group theory
9491:Linear algebraic groups
9297:Linear Algebraic Groups
8709:Linear Algebraic Groups
8368:Linear Algebraic Groups
8215:), this amounts to the
7848:cohomological dimension
7520:means an affine scheme
7067:such that the manifold
6287:is isotropic (that is,
5975:). So the simple group
5665:to be a nonzero vector
5325:projective linear group
5062:Reductive group schemes
4751:: the spin group Spin(2
4730:: the symplectic group
2062:
2012:). For a general field
1777:, for a natural number
1421:is trivial or equal to
1389:Simple reductive groups
1070:linear algebraic group
990:unitary representations
909:semisimple Lie algebras
880:Simple algebraic groups
8849:, vol. 1, Paris:
8393:Borel (1991), 18.2(i).
8280:Real form (Lie theory)
8275:Deligne–Lusztig theory
8194:
8000:
7626:
7594:
7562:
7468:
7196:
7128:For a reductive group
6990:means any subgroup of
6582:
6518:of the abstract group
6357:element other than 1.
6151:
6116:
6026:central simple algebra
6015:
5949:
5914:
5827:large is based on the
5754:Weyl character formula
5613:with a convex cone (a
5483:For a reductive group
5248:) is contained in Int(
5158:
5119:
5029:
4915:
4712:
4531:, for example the form
4266:
4229:with a Borel subgroup
4212:
4040:
3622:
3584:
3545:
3294:
3236:
3183:
3110:
3078:
3050:
3026:
3002:
2962:
2941:by conjugation on its
2935:adjoint representation
2725:For a reductive group
2671:with complexification
2656:
2649:
2629:
2543:
2499:
2454:
2434:
2405:
2376:
2343:
2308:
2301:
2196:
2167:
2123:
2090:
2053:
1968:odd, the group scheme
1929:. The algebraic group
1797:(1), and so its group
1773:matrices over a field
1759:
1687:
1661:is a maximal torus in
1655:
1602:Split-reductive groups
1596:pseudo-reductive group
1565:
1379:
1378:{\displaystyle k^{al}}
1349:
1289:
1262:
1223:
1203:
1163:
1135:
1115:
1084:
1016:linear algebraic group
1010:Linear algebraic group
998:adelic algebraic group
900:, as in the theory of
811:linear algebraic group
740:Linear algebraic group
482:
457:
420:
16:Concept in mathematics
9421:Annals of Mathematics
9125:10.1017/9781316711736
9074:Jantzen, Jens Carsten
8195:
8099:has no real places).
8001:
7870:Serre's Conjecture II
7627:
7595:
7563:
7469:
7202:). The Tits index of
7197:
7138:absolute Galois group
6865:, the abstract group
6583:
6545:over a perfect field
6152:
6117:
6047:), the kernel of the
6016:
5950:
5915:
5728:on the flag manifold
5705:, up to isomorphism.
5651:highest weight vector
5196:Real reductive groups
5159:
5120:
5030:
4916:
4713:
4264:
4213:
4056:of a reductive group
4041:
3623:
3585:
3546:
3383:of a reductive group
3295:
3237:
3184:
3111:
3079:
3051:
3027:
3003:
2963:
2675:, the inclusion from
2650:
2609:
2551:
2544:
2500:
2455:
2435:
2406:
2377:
2375:{\displaystyle B_{n}}
2344:
2302:
2204:
2197:
2168:
2124:
2091:
2054:
1781:. In particular, the
1760:
1688:
1656:
1634:whose base change to
1566:
1473:with kernel a finite
1380:
1350:
1290:
1263:
1224:
1204:
1164:
1136:
1116:
1085:
958:representation theory
882:and (more generally)
483:
458:
421:
8320:Luna's slice theorem
8294:Langlands dual group
8258:Bruhat decomposition
8117:
8106:over a number field
8095:) is trivial (since
7923:
7787:) are classified by
7640:with respect to the
7604:
7572:
7540:
7385:
7326:be the simple group
7179:
6919:totally disconnected
6905:) is an anisotropic
6565:
6446:is quasi-split over
6134:
6094:
6078:-vector space. Here
5991:
5932:
5892:
5653:in a representation
5346:) (sometimes called
5206:real reductive group
5141:
5097:
4957:
4777:
4541:
4414:semisimple over the
4147:
3925:
3743:, which is a finite
3594:
3570:
3478:
3266:
3208:
3123:
3096:
3064:
3036:
3012:
2988:
2948:
2739:completely reducible
2689:homotopy equivalence
2556:
2509:
2473:
2444:
2415:
2386:
2359:
2324:
2209:
2177:
2133:
2104:
2071:
2036:
1939:connected components
1824:special linear group
1783:multiplicative group
1740:
1735:general linear group
1697:is a split torus in
1665:
1638:
1488:
1359:
1339:
1317:multiplicative group
1272:
1240:
1213:
1177:
1153:
1125:
1105:
1074:
843:general linear group
470:
445:
408:
9235:2015arXiv151208296R
8853:, pp. 93–444,
8768:1971InMat..12...95B
8374:. pp. 381–394.
8311:essential dimension
8203:is injective. For
8009:is bijective. Here
7822:coefficient groups
7600:with the action of
7291:associated to some
7079:/Γ is compact. The
6857:For an anisotropic
6836:Kneser–Tits problem
6364:over a local field
5450:sectional curvature
5446:Riemannian geometry
5299:Useful theories of
4704:
3871:by some element of
3823:Parabolic subgroups
3671:commutator subgroup
3453:simply transitively
3204:), its Lie algebra
2805:has order prime to
2795:multiplicative type
1980:. The simple group
1462:th roots of unity.
1149:normal subgroup of
854:invertible matrices
114:Group homomorphisms
24:Algebraic structure
9352:10.1007/bf02684397
9293:Springer, Tonny A.
9256:"Reductive groups"
9252:Springer, Tonny A.
9150:Platonov, Vladimir
9051:, pp. 39–81,
8970:10.1007/BFb0059005
8776:10.1007/BF01404653
8248:groups of Lie type
8190:
8157:
7996:
7963:
7759:are classified by
7735:are classified by
7699:, meaning objects
7634:principal G-bundle
7622:
7590:
7558:
7464:
7377:*) is trivial. If
7283:. In other words,
7192:
6668:is isotropic over
6619:is isotropic over
6578:
6559:inner automorphism
6426:A reductive group
6407:has real rank min(
6147:
6112:
6090:at least 2, since
6011:
5945:
5910:
5888:at least 3, since
5837:Geordie Williamson
5835:, Simon Riche and
5746:Borel–Weil theorem
5677:into itself. Then
5528:with an action of
5526:simplicial complex
5493:discrete valuation
5200:In the context of
5154:
5115:
5025:
4948:semidirect product
4911:
4708:
4681:
4365:automorphism group
4331:exceptional groups
4267:
4208:
4084:, or equivalently
4051:parabolic subgroup
4036:
4026:
3857:parabolic subgroup
3618:
3580:
3541:
3523:
3290:
3232:
3192:For example, when
3179:
3161:
3106:
3074:
3046:
3022:
3008:. The subspace of
2998:
2958:
2844:is isomorphic to (
2777:of characteristic
2773:is reductive. For
2755:linearly reductive
2645:
2539:
2495:
2450:
2430:
2401:
2372:
2339:
2297:
2288:
2192:
2163:
2119:
2086:
2049:
1943:identity component
1888:, the subgroup of
1755:
1683:
1651:
1606:A reductive group
1561:
1375:
1345:
1285:
1258:
1219:
1199:
1159:
1131:
1111:
1080:
1054:group scheme over
902:compact Lie groups
829:that has a finite
590:Special orthogonal
478:
453:
416:
297:Lagrange's theorem
9400:978-1-4704-3105-1
9391:10.1090/ulect/066
9375:Steinberg, Robert
9330:Steinberg, Robert
9314:978-0-8176-4021-7
9270:, pp. 3–27,
9244:978-2-85629-880-0
9189:"Reductive group"
9093:978-0-8218-3527-2
9019:978-2-85629-324-9
8932:978-2-85629-323-2
8886:, Paris: Masson,
8860:978-2-85629-794-0
8799:Chevalley, Claude
8324:Haboush's theorem
8298:Langlands program
8266:Schubert calculus
8148:
7954:
7619:
7587:
7568:is isomorphic to
7555:
7423:
7279:is the center of
7257:Galois cohomology
7190:
7163:separable closure
6722:is split of type
6576:
6514:gave an explicit
6384:) if and only if
6145:
6122:is isomorphic to
6109:
5943:
5920:is isomorphic to
5907:
5396:metaplectic group
5382:) has nontrivial
5368:fundamental group
5323:For example, the
5152:
5112:
5042:is the center of
4112:), these are the
4049:By definition, a
3501:
3146:
2937:is the action of
2453:{\displaystyle 1}
2393:
2184:
2111:
2047:
1960:at least 3. (For
1747:
1680:
1649:
1348:{\displaystyle G}
1307:is perfect.) Any
1297:algebraic closure
1283:
1255:
1222:{\displaystyle G}
1171:unipotent radical
1162:{\displaystyle G}
1134:{\displaystyle G}
1114:{\displaystyle G}
1083:{\displaystyle G}
799:
798:
374:
373:
256:Alternating group
213:
212:
9503:
9476:
9468:Grothendieck, A.
9452:
9411:
9370:
9325:
9288:
9247:
9228:
9212:-Canonical Basis
9201:
9180:
9145:
9104:
9082:(2nd ed.),
9069:
9058:978-285629-269-3
9042:
9031:
9000:Grothendieck, A.
8991:
8954:Grothendieck, A.
8944:
8913:Grothendieck, A.
8904:
8876:Demazure, Michel
8871:
8844:
8831:
8794:
8741:
8692:
8689:
8683:
8680:
8674:
8671:
8665:
8662:
8656:
8653:
8647:
8644:
8638:
8635:
8629:
8626:
8620:
8617:
8611:
8608:
8602:
8599:
8593:
8590:
8584:
8581:
8575:
8572:
8566:
8563:
8550:
8547:
8541:
8538:
8532:
8529:
8523:
8520:
8514:
8511:
8505:
8502:
8496:
8493:
8487:
8484:
8478:
8475:
8469:
8466:
8460:
8457:
8451:
8448:
8442:
8439:
8430:
8427:
8421:
8418:
8412:
8409:
8403:
8400:
8394:
8391:
8385:
8382:
8376:
8375:
8373:
8362:
8356:
8353:
8347:
8344:
8262:Schubert variety
8199:
8197:
8196:
8191:
8180:
8179:
8167:
8166:
8156:
8129:
8128:
8005:
8003:
8002:
7997:
7986:
7985:
7973:
7972:
7962:
7935:
7934:
7631:
7629:
7628:
7623:
7621:
7620:
7612:
7599:
7597:
7596:
7591:
7589:
7588:
7580:
7567:
7565:
7564:
7559:
7557:
7556:
7548:
7473:
7471:
7470:
7465:
7457:
7452:
7451:
7424:
7419:
7411:
7406:
7405:
7287:is the twist of
7201:
7199:
7198:
7193:
7191:
7183:
7049:For a Lie group
6987:arithmetic group
6959:rational numbers
6700:'s machinery of
6587:
6585:
6584:
6579:
6577:
6569:
6512:Robert Steinberg
6188:division algebra
6156:
6154:
6153:
6148:
6146:
6138:
6121:
6119:
6118:
6113:
6111:
6110:
6102:
6020:
6018:
6017:
6012:
6004:
5963:is equal to the
5954:
5952:
5951:
5946:
5944:
5936:
5919:
5917:
5916:
5911:
5909:
5908:
5900:
5854:classical groups
5813:Henning Andersen
5599:dominant weights
5466:hyperbolic plane
5222:Zariski topology
5163:
5161:
5160:
5155:
5153:
5145:
5124:
5122:
5121:
5116:
5114:
5113:
5105:
5081:if the morphism
5034:
5032:
5031:
5026:
5006:
4920:
4918:
4917:
4912:
4907:
4906:
4894:
4893:
4866:
4865:
4856:
4855:
4843:
4842:
4833:
4832:
4817:
4816:
4795:
4794:
4763:with Witt index
4717:
4715:
4714:
4709:
4703:
4698:
4677:
4676:
4664:
4663:
4636:
4635:
4626:
4625:
4613:
4612:
4603:
4602:
4587:
4586:
4559:
4558:
4428:simply connected
4404:simply connected
4369:octonion algebra
4217:
4215:
4214:
4209:
4198:
4197:
4196:
4195:
4172:
4171:
4170:
4169:
4045:
4043:
4042:
4037:
4035:
4031:
4030:
3811:is generated by
3627:
3625:
3624:
3619:
3608:
3604:
3603:
3589:
3587:
3586:
3581:
3579:
3578:
3554:For example, if
3550:
3548:
3547:
3542:
3537:
3536:
3531:
3530:
3522:
3521:
3520:
3497:
3496:
3487:
3486:
3299:
3297:
3296:
3291:
3280:
3276:
3275:
3241:
3239:
3238:
3233:
3222:
3218:
3217:
3188:
3186:
3185:
3180:
3175:
3174:
3169:
3168:
3160:
3142:
3141:
3132:
3131:
3115:
3113:
3112:
3107:
3105:
3104:
3092:decomposes into
3083:
3081:
3080:
3075:
3073:
3072:
3055:
3053:
3052:
3047:
3045:
3044:
3031:
3029:
3028:
3023:
3021:
3020:
3007:
3005:
3004:
2999:
2997:
2996:
2967:
2965:
2964:
2959:
2957:
2956:
2783:Masayoshi Nagata
2781:>0, however,
2729:over a field of
2665:complexification
2654:
2652:
2651:
2646:
2641:
2640:
2635:
2628:
2623:
2599:
2598:
2586:
2585:
2573:
2568:
2567:
2548:
2546:
2545:
2540:
2529:
2528:
2519:
2504:
2502:
2501:
2496:
2485:
2484:
2459:
2457:
2456:
2451:
2439:
2437:
2436:
2431:
2429:
2428:
2423:
2410:
2408:
2407:
2402:
2400:
2399:
2394:
2391:
2381:
2379:
2378:
2373:
2371:
2370:
2348:
2346:
2345:
2340:
2338:
2337:
2332:
2306:
2304:
2303:
2298:
2293:
2292:
2285:
2284:
2261:
2260:
2237:
2236:
2224:
2223:
2201:
2199:
2198:
2193:
2191:
2190:
2185:
2182:
2172:
2170:
2169:
2164:
2162:
2161:
2156:
2147:
2146:
2141:
2128:
2126:
2125:
2120:
2118:
2117:
2112:
2109:
2095:
2093:
2092:
2087:
2085:
2084:
2079:
2058:
2056:
2055:
2050:
2048:
2040:
1909:orthogonal group
1907:. Likewise, the
1875:symplectic group
1764:
1762:
1761:
1756:
1754:
1753:
1748:
1745:
1692:
1690:
1689:
1684:
1682:
1681:
1673:
1660:
1658:
1657:
1652:
1650:
1642:
1570:
1568:
1567:
1562:
1557:
1556:
1547:
1521:
1520:
1475:central subgroup
1384:
1382:
1381:
1376:
1374:
1373:
1354:
1352:
1351:
1346:
1327:, is reductive.
1294:
1292:
1291:
1286:
1284:
1276:
1267:
1265:
1264:
1259:
1257:
1256:
1248:
1228:
1226:
1225:
1220:
1208:
1206:
1205:
1200:
1189:
1188:
1168:
1166:
1165:
1160:
1140:
1138:
1137:
1132:
1120:
1118:
1117:
1112:
1089:
1087:
1086:
1081:
1022:is defined as a
890:Claude Chevalley
869:symplectic group
791:
784:
777:
733:Algebraic groups
506:Hyperbolic group
496:Arithmetic group
487:
485:
484:
479:
477:
462:
460:
459:
454:
452:
425:
423:
422:
417:
415:
338:Schur multiplier
292:Cauchy's theorem
280:Quaternion group
228:
227:
54:
53:
43:
30:
19:
18:
9511:
9510:
9506:
9505:
9504:
9502:
9501:
9500:
9481:
9480:
9459:
9434:10.2307/1970394
9401:
9315:
9278:
9266:, vol. 1,
9245:
9170:
9135:
9094:
9059:
9040:
9020:
8980:
8962:Springer-Verlag
8933:
8894:
8880:Gabriel, Pierre
8861:
8842:
8821:
8811:Springer Nature
8731:
8713:Springer Nature
8700:
8695:
8690:
8686:
8681:
8677:
8672:
8668:
8663:
8659:
8654:
8650:
8645:
8641:
8636:
8632:
8627:
8623:
8618:
8614:
8609:
8605:
8600:
8596:
8591:
8587:
8582:
8578:
8573:
8569:
8564:
8553:
8548:
8544:
8539:
8535:
8530:
8526:
8521:
8517:
8512:
8508:
8503:
8499:
8494:
8490:
8485:
8481:
8476:
8472:
8467:
8463:
8458:
8454:
8449:
8445:
8440:
8433:
8428:
8424:
8419:
8415:
8410:
8406:
8401:
8397:
8392:
8388:
8383:
8379:
8371:
8363:
8359:
8354:
8350:
8345:
8341:
8337:
8243:
8236:
8231:
8175:
8171:
8162:
8158:
8152:
8124:
8120:
8118:
8115:
8114:
8063:
8050:
8029:
7981:
7977:
7968:
7964:
7958:
7930:
7926:
7924:
7921:
7920:
7906:Hasse principle
7814:-torsors using
7656:is smooth over
7611:
7607:
7605:
7602:
7601:
7579:
7575:
7573:
7570:
7569:
7547:
7543:
7541:
7538:
7537:
7504:
7482:has Witt index
7453:
7447:
7443:
7418:
7407:
7401:
7397:
7386:
7383:
7382:
7353:
7347:
7251:, meaning that
7219:
7218:
7182:
7180:
7177:
7176:
7170:
7148:
7103:
7097:
6951:
6931:profinite group
6793:Whitehead group
6771:
6760:
6749:
6742:
6735:
6728:
6717:
6655:
6643:) generated by
6568:
6566:
6563:
6562:
6500:
6332:
6299:
6225:if and only if
6181:
6137:
6135:
6132:
6131:
6101:
6097:
6095:
6092:
6091:
6000:
5992:
5989:
5988:
5983:if and only if
5935:
5933:
5930:
5929:
5899:
5895:
5893:
5890:
5889:
5849:
5583:
5572:
5510:affine building
5506:
5452:. For example,
5442:symmetric space
5400:universal cover
5394:) (such as the
5384:covering spaces
5256:
5208:is a Lie group
5198:
5170:Michel Demazure
5144:
5142:
5139:
5138:
5104:
5100:
5098:
5095:
5094:
5064:
5002:
4958:
4955:
4954:
4899:
4895:
4880:
4876:
4861:
4857:
4851:
4847:
4838:
4834:
4828:
4824:
4809:
4805:
4790:
4786:
4778:
4775:
4774:
4750:
4729:
4699:
4685:
4669:
4665:
4650:
4646:
4631:
4627:
4621:
4617:
4608:
4604:
4598:
4594:
4573:
4569:
4554:
4550:
4542:
4539:
4538:
4507:
4486:
4416:complex numbers
4386:
4382:
4378:
4362:
4343:
4339:
4316:Wilhelm Killing
4313:
4309:
4305:
4301:
4297:
4293:
4287:
4281:
4275:
4259:
4191:
4187:
4186:
4182:
4165:
4161:
4160:
4156:
4148:
4145:
4144:
4131:
4122:
4025:
4024:
4019:
4014:
4009:
4003:
4002:
3997:
3992:
3987:
3981:
3980:
3975:
3970:
3965:
3959:
3958:
3953:
3948:
3943:
3933:
3932:
3928:
3926:
3923:
3922:
3901:
3825:
3786:
3772:
3727:
3717:
3679:semisimple rank
3645:
3636:
3599:
3598:
3597:
3595:
3592:
3591:
3574:
3573:
3571:
3568:
3567:
3532:
3526:
3525:
3524:
3516:
3512:
3505:
3492:
3491:
3482:
3481:
3479:
3476:
3475:
3449:Borel subgroups
3443:
3434:symmetric group
3407:
3353:
3344:
3323:
3314:
3271:
3270:
3269:
3267:
3264:
3263:
3213:
3212:
3211:
3209:
3206:
3205:
3170:
3164:
3163:
3162:
3150:
3137:
3136:
3127:
3126:
3124:
3121:
3120:
3100:
3099:
3097:
3094:
3093:
3068:
3067:
3065:
3062:
3061:
3040:
3039:
3037:
3034:
3033:
3016:
3015:
3013:
3010:
3009:
2992:
2991:
2989:
2986:
2985:
2952:
2951:
2949:
2946:
2945:
2897:
2852:
2815:
2661:
2636:
2631:
2630:
2624:
2613:
2594:
2590:
2581:
2577:
2569:
2563:
2559:
2557:
2554:
2553:
2524:
2520:
2515:
2510:
2507:
2506:
2480:
2476:
2474:
2471:
2470:
2467:
2445:
2442:
2441:
2424:
2419:
2418:
2416:
2413:
2412:
2395:
2390:
2389:
2387:
2384:
2383:
2366:
2362:
2360:
2357:
2356:
2333:
2328:
2327:
2325:
2322:
2321:
2318:unipotent group
2313:
2287:
2286:
2280:
2276:
2274:
2268:
2267:
2262:
2256:
2252:
2245:
2244:
2232:
2228:
2219:
2215:
2210:
2207:
2206:
2186:
2181:
2180:
2178:
2175:
2174:
2157:
2152:
2151:
2142:
2137:
2136:
2134:
2131:
2130:
2113:
2108:
2107:
2105:
2102:
2101:
2080:
2075:
2074:
2072:
2069:
2068:
2065:
2039:
2037:
2034:
2033:
1884:) over a field
1871:
1833:) over a field
1805:
1792:
1749:
1744:
1743:
1741:
1738:
1737:
1731:
1729:
1723:
1715:
1672:
1668:
1666:
1663:
1662:
1641:
1639:
1636:
1635:
1604:
1552:
1548:
1543:
1516:
1512:
1489:
1486:
1485:
1467:central isogeny
1456:
1391:
1366:
1362:
1360:
1357:
1356:
1340:
1337:
1336:
1333:
1326:
1275:
1273:
1270:
1269:
1247:
1243:
1241:
1238:
1237:
1214:
1211:
1210:
1184:
1180:
1178:
1175:
1174:
1173:and is denoted
1154:
1151:
1150:
1126:
1123:
1122:
1106:
1103:
1102:
1099:normal subgroup
1075:
1072:
1071:
1064:
1028:subgroup scheme
1012:
1006:
940:rational points
898:Dynkin diagrams
886:are reductive.
807:reductive group
795:
766:
765:
754:Abelian variety
747:Reductive group
735:
725:
724:
723:
722:
673:
665:
657:
649:
641:
614:Special unitary
525:
511:
510:
492:
491:
473:
471:
468:
467:
448:
446:
443:
442:
411:
409:
406:
405:
397:
396:
387:Discrete groups
376:
375:
331:Frobenius group
276:
263:
252:
245:Symmetric group
241:
225:
215:
214:
65:Normal subgroup
51:
31:
22:
17:
12:
11:
5:
9509:
9499:
9498:
9493:
9479:
9478:
9458:
9457:External links
9455:
9454:
9453:
9428:(2): 313–329,
9412:
9399:
9371:
9326:
9313:
9289:
9276:
9248:
9243:
9202:
9181:
9168:
9160:Academic Press
9146:
9134:978-1107167483
9133:
9105:
9092:
9070:
9057:
9033:
9018:
8992:
8979:978-3540051800
8978:
8946:
8931:
8905:
8893:978-2225616662
8892:
8872:
8859:
8832:
8819:
8795:
8742:
8729:
8699:
8696:
8694:
8693:
8684:
8675:
8666:
8657:
8648:
8639:
8630:
8621:
8612:
8603:
8594:
8585:
8576:
8567:
8551:
8542:
8533:
8524:
8515:
8506:
8497:
8488:
8479:
8470:
8461:
8452:
8443:
8431:
8422:
8413:
8404:
8395:
8386:
8377:
8357:
8348:
8338:
8336:
8333:
8332:
8331:
8326:
8313:
8304:
8287:
8282:
8277:
8268:
8251:
8242:
8239:
8234:
8229:
8201:
8200:
8189:
8186:
8183:
8178:
8174:
8170:
8165:
8161:
8155:
8151:
8147:
8144:
8141:
8138:
8135:
8132:
8127:
8123:
8059:
8046:
8025:
8013:runs over all
8007:
8006:
7995:
7992:
7989:
7984:
7980:
7976:
7971:
7967:
7961:
7957:
7953:
7950:
7947:
7944:
7941:
7938:
7933:
7929:
7864:Lang's theorem
7668:-torsors over
7650:étale topology
7618:
7615:
7610:
7586:
7583:
7578:
7554:
7551:
7546:
7503:
7500:
7463:
7460:
7456:
7450:
7446:
7442:
7439:
7436:
7433:
7430:
7427:
7422:
7417:
7414:
7410:
7404:
7400:
7396:
7393:
7390:
7349:
7343:
7214:
7210:
7189:
7186:
7168:
7144:
7099:Main article:
7096:
7093:
6950:
6947:
6861:-simple group
6854:) is abelian.
6779:-simple group
6769:
6758:
6747:
6740:
6733:
6726:
6715:
6651:
6575:
6572:
6499:
6496:
6339:
6338:
6328:
6319:
6305:
6295:
6235:
6234:
6221:is split over
6173:
6165:(meaning that
6144:
6141:
6108:
6105:
6100:
6053:group of units
6022:
6010:
6007:
6003:
5999:
5996:
5979:is split over
5942:
5939:
5906:
5903:
5898:
5884:has dimension
5848:
5845:
5805:Coxeter number
5582:
5579:
5568:
5544:) preserves a
5502:
5497:p-adic numbers
5290:Satake diagram
5252:
5197:
5194:
5151:
5148:
5111:
5108:
5103:
5073:over a scheme
5063:
5060:
5036:
5035:
5024:
5021:
5018:
5015:
5012:
5009:
5005:
5001:
4998:
4995:
4992:
4989:
4986:
4983:
4980:
4977:
4974:
4971:
4968:
4965:
4962:
4924:
4923:
4922:
4921:
4910:
4905:
4902:
4898:
4892:
4889:
4886:
4883:
4879:
4875:
4872:
4869:
4864:
4860:
4854:
4850:
4846:
4841:
4837:
4831:
4827:
4823:
4820:
4815:
4812:
4808:
4804:
4801:
4798:
4793:
4789:
4785:
4782:
4769:
4768:
4746:
4743:
4725:
4721:
4720:
4719:
4718:
4707:
4702:
4697:
4694:
4691:
4688:
4684:
4680:
4675:
4672:
4668:
4662:
4659:
4656:
4653:
4649:
4645:
4642:
4639:
4634:
4630:
4624:
4620:
4616:
4611:
4607:
4601:
4597:
4593:
4590:
4585:
4582:
4579:
4576:
4572:
4568:
4565:
4562:
4557:
4553:
4549:
4546:
4533:
4532:
4503:
4500:
4482:
4384:
4380:
4376:
4360:
4341:
4337:
4311:
4307:
4303:
4299:
4295:
4289:
4283:
4277:
4271:
4258:
4255:
4241:is called the
4219:
4218:
4207:
4204:
4201:
4194:
4190:
4185:
4181:
4178:
4175:
4168:
4164:
4159:
4155:
4152:
4127:
4120:
4114:flag varieties
4047:
4046:
4034:
4029:
4023:
4020:
4018:
4015:
4013:
4010:
4008:
4005:
4004:
4001:
3998:
3996:
3993:
3991:
3988:
3986:
3983:
3982:
3979:
3976:
3974:
3971:
3969:
3966:
3964:
3961:
3960:
3957:
3954:
3952:
3949:
3947:
3944:
3942:
3939:
3938:
3936:
3931:
3899:
3824:
3821:
3784:
3770:
3741:Dynkin diagram
3722:
3713:
3641:
3632:
3617:
3614:
3611:
3607:
3602:
3577:
3552:
3551:
3540:
3535:
3529:
3519:
3515:
3511:
3508:
3504:
3500:
3495:
3490:
3485:
3461:positive roots
3439:
3403:
3389:quotient group
3349:
3340:
3319:
3312:
3289:
3286:
3283:
3279:
3274:
3250:matrices over
3231:
3228:
3225:
3221:
3216:
3190:
3189:
3178:
3173:
3167:
3159:
3156:
3153:
3149:
3145:
3140:
3135:
3130:
3103:
3071:
3043:
3019:
2995:
2955:
2922:copies of the
2908:tensor product
2893:
2848:
2814:
2811:
2731:characteristic
2660:
2657:
2644:
2639:
2634:
2627:
2622:
2619:
2616:
2612:
2608:
2605:
2602:
2597:
2593:
2589:
2584:
2580:
2576:
2572:
2566:
2562:
2538:
2535:
2532:
2527:
2523:
2518:
2514:
2494:
2491:
2488:
2483:
2479:
2466:
2463:
2462:
2461:
2449:
2427:
2422:
2398:
2369:
2365:
2350:
2336:
2331:
2312:
2309:
2296:
2291:
2283:
2279:
2275:
2273:
2270:
2269:
2266:
2263:
2259:
2255:
2251:
2250:
2248:
2243:
2240:
2235:
2231:
2227:
2222:
2218:
2214:
2189:
2160:
2155:
2150:
2145:
2140:
2116:
2098:algebraic tori
2083:
2078:
2064:
2061:
2046:
2043:
1920:quadratic form
1870:
1855:
1801:
1788:
1765:of invertible
1752:
1730:
1725:
1719:
1716:
1714:
1711:
1679:
1676:
1671:
1648:
1645:
1603:
1600:
1572:
1571:
1560:
1555:
1551:
1546:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1519:
1515:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1452:
1451:group scheme μ
1390:
1387:
1372:
1369:
1365:
1344:
1332:
1329:
1322:
1315:, such as the
1282:
1279:
1254:
1251:
1246:
1218:
1198:
1195:
1192:
1187:
1183:
1158:
1130:
1110:
1079:
1063:
1060:
1008:Main article:
1005:
1002:
827:representation
797:
796:
794:
793:
786:
779:
771:
768:
767:
764:
763:
761:Elliptic curve
757:
756:
750:
749:
743:
742:
736:
731:
730:
727:
726:
721:
720:
717:
714:
710:
706:
705:
704:
699:
697:Diffeomorphism
693:
692:
687:
682:
676:
675:
671:
667:
663:
659:
655:
651:
647:
643:
639:
634:
633:
622:
621:
610:
609:
598:
597:
586:
585:
574:
573:
562:
561:
554:Special linear
550:
549:
542:General linear
538:
537:
532:
526:
517:
516:
513:
512:
509:
508:
503:
498:
490:
489:
476:
464:
451:
438:
436:Modular groups
434:
433:
432:
427:
414:
398:
395:
394:
389:
383:
382:
381:
378:
377:
372:
371:
370:
369:
364:
359:
356:
350:
349:
343:
342:
341:
340:
334:
333:
327:
326:
321:
312:
311:
309:Hall's theorem
306:
304:Sylow theorems
300:
299:
294:
286:
285:
284:
283:
277:
272:
269:Dihedral group
265:
264:
259:
253:
248:
242:
237:
226:
221:
220:
217:
216:
211:
210:
209:
208:
203:
195:
194:
193:
192:
187:
182:
177:
172:
167:
162:
160:multiplicative
157:
152:
147:
142:
134:
133:
132:
131:
126:
118:
117:
109:
108:
107:
106:
104:Wreath product
101:
96:
91:
89:direct product
83:
81:Quotient group
75:
74:
73:
72:
67:
62:
52:
49:
48:
45:
44:
36:
35:
15:
9:
6:
4:
3:
2:
9508:
9497:
9494:
9492:
9489:
9488:
9486:
9475:
9474:
9469:
9465:
9461:
9460:
9451:
9447:
9443:
9439:
9435:
9431:
9427:
9423:
9422:
9417:
9416:Tits, Jacques
9413:
9410:
9406:
9402:
9396:
9392:
9388:
9384:
9380:
9376:
9372:
9369:
9365:
9361:
9357:
9353:
9349:
9345:
9341:
9340:
9335:
9331:
9327:
9324:
9320:
9316:
9310:
9306:
9302:
9298:
9294:
9290:
9287:
9283:
9279:
9277:0-8218-3347-2
9273:
9269:
9265:
9261:
9257:
9253:
9249:
9246:
9240:
9236:
9232:
9227:
9222:
9218:
9214:
9213:
9209:
9203:
9200:
9196:
9195:
9190:
9186:
9182:
9179:
9175:
9171:
9169:0-12-558180-7
9165:
9161:
9157:
9156:
9151:
9147:
9144:
9140:
9136:
9130:
9126:
9122:
9118:
9114:
9110:
9106:
9103:
9099:
9095:
9089:
9085:
9081:
9080:
9075:
9071:
9068:
9064:
9060:
9054:
9050:
9046:
9039:
9034:
9029:
9025:
9021:
9015:
9011:
9007:
9006:
9001:
8997:
8993:
8989:
8985:
8981:
8975:
8971:
8967:
8963:
8959:
8955:
8951:
8947:
8942:
8938:
8934:
8928:
8924:
8920:
8919:
8914:
8910:
8906:
8903:
8899:
8895:
8889:
8885:
8881:
8877:
8873:
8870:
8866:
8862:
8856:
8852:
8848:
8841:
8837:
8836:Conrad, Brian
8833:
8830:
8826:
8822:
8820:3-540-23031-9
8816:
8812:
8808:
8804:
8800:
8796:
8793:
8789:
8785:
8781:
8777:
8773:
8769:
8765:
8762:(2): 95–104,
8761:
8757:
8756:
8751:
8750:Tits, Jacques
8747:
8746:Borel, Armand
8743:
8740:
8736:
8732:
8730:0-387-97370-2
8726:
8722:
8718:
8714:
8710:
8706:
8705:Borel, Armand
8702:
8701:
8688:
8679:
8670:
8661:
8652:
8643:
8634:
8625:
8616:
8607:
8598:
8589:
8580:
8571:
8562:
8560:
8558:
8556:
8546:
8537:
8528:
8519:
8510:
8501:
8492:
8483:
8474:
8465:
8456:
8447:
8438:
8436:
8426:
8417:
8408:
8399:
8390:
8381:
8370:
8369:
8361:
8352:
8343:
8339:
8330:
8327:
8325:
8321:
8317:
8314:
8312:
8308:
8307:Special group
8305:
8303:
8299:
8295:
8291:
8288:
8286:
8283:
8281:
8278:
8276:
8272:
8271:Schur algebra
8269:
8267:
8263:
8259:
8255:
8252:
8249:
8245:
8244:
8238:
8232:
8225:
8220:
8218:
8214:
8210:
8206:
8184:
8181:
8176:
8172:
8163:
8159:
8153:
8149:
8139:
8136:
8133:
8125:
8121:
8113:
8112:
8111:
8109:
8105:
8100:
8098:
8094:
8090:
8086:
8082:
8078:
8074:
8071:
8067:
8062:
8058:
8054:
8049:
8045:
8041:
8037:
8033:
8028:
8024:
8020:
8016:
8012:
7990:
7987:
7982:
7978:
7969:
7965:
7959:
7955:
7945:
7942:
7939:
7931:
7927:
7919:
7918:
7917:
7915:
7911:
7907:
7903:
7902:Günter Harder
7899:
7898:Martin Kneser
7895:
7891:
7887:
7883:
7879:
7875:
7871:
7867:
7865:
7861:
7857:
7853:
7849:
7845:
7841:
7837:
7833:
7829:
7825:
7821:
7817:
7813:
7808:
7806:
7802:
7798:
7794:
7790:
7786:
7782:
7778:
7774:
7770:
7766:
7762:
7758:
7754:
7750:
7746:
7742:
7738:
7734:
7730:
7726:
7722:
7718:
7714:
7710:
7706:
7702:
7698:
7695:over a field
7694:
7690:
7685:
7683:
7679:
7675:
7671:
7667:
7663:
7659:
7655:
7651:
7647:
7643:
7642:fppf topology
7639:
7635:
7613:
7608:
7581:
7576:
7549:
7544:
7535:
7531:
7527:
7523:
7519:
7516:over a field
7515:
7511:
7510:
7499:
7497:
7493:
7489:
7485:
7481:
7477:
7458:
7454:
7448:
7444:
7437:
7434:
7431:
7420:
7412:
7408:
7402:
7398:
7391:
7388:
7380:
7376:
7372:
7368:
7364:
7361:
7357:
7352:
7346:
7341:
7337:
7333:
7329:
7325:
7321:
7317:
7314:over a field
7313:
7309:
7306:Example: Let
7304:
7302:
7299:-torsor over
7298:
7294:
7290:
7286:
7282:
7278:
7274:
7270:
7266:
7262:
7258:
7254:
7250:
7246:
7242:
7238:
7234:
7230:
7226:
7221:
7217:
7213:
7209:
7205:
7184:
7174:
7167:
7164:
7160:
7156:
7152:
7147:
7143:
7139:
7135:
7132:over a field
7131:
7126:
7124:
7120:
7116:
7112:
7108:
7102:
7092:
7090:
7086:
7082:
7078:
7074:
7070:
7066:
7062:
7058:
7057:
7052:
7047:
7045:
7041:
7037:
7033:
7029:
7025:
7021:
7017:
7013:
7009:
7005:
7001:
7000:commensurable
6997:
6993:
6989:
6988:
6983:
6979:
6975:
6971:
6967:
6963:
6960:
6956:
6946:
6944:
6940:
6936:
6932:
6928:
6924:
6920:
6916:
6912:
6908:
6904:
6900:
6896:
6892:
6888:
6884:
6880:
6876:
6872:
6868:
6864:
6860:
6855:
6853:
6849:
6845:
6841:
6837:
6833:
6829:
6825:
6821:
6817:
6813:
6809:
6805:
6801:
6797:
6794:
6790:
6786:
6782:
6778:
6773:
6768:
6764:
6757:
6753:
6746:
6739:
6732:
6725:
6721:
6714:
6710:
6705:
6703:
6699:
6695:
6691:
6687:
6683:
6679:
6675:
6671:
6667:
6663:
6660:contained in
6659:
6654:
6650:
6646:
6642:
6638:
6634:
6630:
6626:
6622:
6618:
6614:
6610:
6606:
6602:
6598:
6593:
6591:
6570:
6560:
6556:
6552:
6548:
6544:
6539:
6537:
6533:
6529:
6525:
6521:
6517:
6513:
6509:
6506:over a field
6505:
6495:
6493:
6489:
6485:
6481:
6477:
6473:
6469:
6465:
6461:
6457:
6453:
6449:
6445:
6441:
6437:
6433:
6430:over a field
6429:
6424:
6422:
6418:
6414:
6410:
6406:
6402:
6400:
6396:
6392:
6387:
6383:
6379:
6375:
6371:
6367:
6363:
6358:
6356:
6353:) contains a
6352:
6348:
6344:
6336:
6331:
6327:
6323:
6320:
6317:
6314:not equal to
6313:
6309:
6306:
6303:
6298:
6294:
6290:
6286:
6283:
6282:
6281:
6279:
6276:over a field
6275:
6271:
6267:
6263:
6259:
6254:
6252:
6248:
6244:
6240:
6232:
6228:
6224:
6220:
6216:
6212:
6208:
6204:
6200:
6196:
6192:
6189:
6185:
6180:
6176:
6172:
6168:
6164:
6160:
6139:
6129:
6125:
6103:
6098:
6089:
6085:
6082:is simple if
6081:
6077:
6073:
6069:
6065:
6061:
6057:
6054:
6050:
6046:
6042:
6038:
6034:
6030:
6027:
6023:
6005:
6001:
5997:
5986:
5982:
5978:
5974:
5970:
5966:
5962:
5958:
5937:
5927:
5923:
5901:
5896:
5887:
5883:
5880:is simple if
5879:
5875:
5871:
5867:
5864:over a field
5863:
5859:
5858:
5857:
5855:
5844:
5842:
5838:
5834:
5830:
5826:
5822:
5818:
5814:
5810:
5806:
5802:
5798:
5794:
5790:
5786:
5782:
5778:
5774:
5770:
5767:over a field
5766:
5761:
5759:
5755:
5751:
5747:
5743:
5739:
5735:
5731:
5727:
5724:-equivariant
5723:
5719:
5715:
5711:
5706:
5704:
5700:
5696:
5692:
5688:
5684:
5680:
5676:
5672:
5668:
5664:
5660:
5656:
5652:
5648:
5644:
5640:
5636:
5632:
5628:
5624:
5620:
5616:
5612:
5608:
5604:
5600:
5596:
5592:
5589:over a field
5588:
5578:
5576:
5571:
5567:
5563:
5559:
5555:
5551:
5547:
5543:
5539:
5535:
5531:
5527:
5523:
5519:
5515:
5512:
5511:
5505:
5501:
5498:
5495:(such as the
5494:
5490:
5487:over a field
5486:
5481:
5479:
5475:
5471:
5467:
5463:
5459:
5455:
5451:
5447:
5443:
5439:
5436:
5432:
5428:
5424:
5420:
5415:
5413:
5409:
5405:
5401:
5397:
5393:
5389:
5385:
5381:
5377:
5373:
5369:
5365:
5361:
5357:
5353:
5349:
5345:
5341:
5337:
5333:
5329:
5326:
5321:
5319:
5315:
5311:
5307:
5302:
5297:
5295:
5291:
5287:
5283:
5279:
5275:
5270:
5268:
5264:
5260:
5255:
5251:
5247:
5243:
5239:
5235:
5231:
5227:
5223:
5219:
5215:
5211:
5207:
5203:
5193:
5191:
5187:
5183:
5179:
5175:
5171:
5167:
5146:
5136:
5132:
5128:
5106:
5101:
5092:
5088:
5084:
5080:
5076:
5072:
5069:
5059:
5057:
5053:
5049:
5045:
5041:
5022:
5016:
5007:
5003:
4999:
4993:
4987:
4981:
4978:
4975:
4969:
4963:
4960:
4953:
4952:
4951:
4949:
4945:
4941:
4937:
4934:over a field
4933:
4929:
4908:
4903:
4900:
4896:
4890:
4887:
4884:
4881:
4877:
4873:
4870:
4867:
4862:
4858:
4852:
4848:
4844:
4839:
4835:
4829:
4825:
4821:
4813:
4810:
4806:
4802:
4799:
4796:
4791:
4787:
4780:
4773:
4772:
4771:
4770:
4766:
4762:
4758:
4754:
4749:
4744:
4741:
4737:
4733:
4728:
4723:
4722:
4705:
4700:
4695:
4692:
4689:
4686:
4682:
4678:
4673:
4670:
4666:
4660:
4657:
4654:
4651:
4647:
4643:
4640:
4637:
4632:
4628:
4622:
4618:
4614:
4609:
4605:
4599:
4595:
4591:
4583:
4580:
4577:
4574:
4570:
4566:
4563:
4560:
4555:
4551:
4544:
4537:
4536:
4535:
4534:
4530:
4527:
4523:
4519:
4515:
4511:
4506:
4501:
4498:
4494:
4490:
4485:
4480:
4479:
4478:
4476:
4471:
4469:
4465:
4461:
4457:
4453:
4449:
4445:
4441:
4437:
4433:
4429:
4425:
4421:
4417:
4413:
4409:
4405:
4401:
4398:over a field
4397:
4393:
4388:
4374:
4370:
4366:
4359:
4355:
4354:L. E. Dickson
4351:
4347:
4335:
4332:
4327:
4325:
4321:
4317:
4292:
4286:
4280:
4274:
4263:
4254:
4252:
4248:
4247:flag manifold
4244:
4240:
4236:
4232:
4228:
4224:
4205:
4202:
4199:
4192:
4188:
4183:
4179:
4176:
4173:
4166:
4162:
4157:
4153:
4150:
4143:
4142:
4141:
4139:
4136:of dimension
4135:
4130:
4126:
4119:
4115:
4111:
4107:
4103:
4099:
4095:
4091:
4087:
4083:
4079:
4075:
4071:
4067:
4063:
4060:over a field
4059:
4055:
4052:
4032:
4027:
4021:
4016:
4011:
4006:
3999:
3994:
3989:
3984:
3977:
3972:
3967:
3962:
3955:
3950:
3945:
3940:
3934:
3929:
3921:
3920:
3919:
3917:
3913:
3909:
3905:
3898:
3894:
3890:
3886:
3882:
3878:
3874:
3870:
3866:
3862:
3858:
3854:
3850:
3846:
3842:
3838:
3834:
3831:over a field
3830:
3820:
3818:
3814:
3810:
3806:
3802:
3798:
3794:
3790:
3783:
3780:
3779:root subgroup
3776:
3769:
3765:
3761:
3758:over a field
3757:
3752:
3750:
3749:inner product
3746:
3742:
3737:
3735:
3731:
3725:
3721:
3716:
3712:
3708:
3704:
3700:
3696:
3692:
3688:
3684:
3680:
3677:, called the
3676:
3672:
3668:
3664:
3659:
3657:
3653:
3649:
3644:
3640:
3635:
3631:
3612:
3605:
3565:
3561:
3557:
3538:
3533:
3517:
3509:
3506:
3502:
3498:
3488:
3474:
3473:
3472:
3470:
3466:
3462:
3458:
3454:
3450:
3445:
3442:
3438:
3435:
3431:
3427:
3423:
3419:
3415:
3411:
3406:
3402:
3398:
3394:
3390:
3386:
3382:
3381:
3376:
3372:
3367:
3365:
3361:
3357:
3352:
3348:
3343:
3339:
3335:
3331:
3327:
3322:
3318:
3311:
3307:
3303:
3284:
3277:
3261:
3257:
3253:
3249:
3245:
3226:
3219:
3203:
3199:
3196:is the group
3195:
3176:
3171:
3154:
3151:
3147:
3143:
3133:
3119:
3118:
3117:
3091:
3087:
3059:
2983:
2979:
2975:
2971:
2944:
2940:
2936:
2931:
2929:
2925:
2921:
2917:
2913:
2909:
2905:
2901:
2896:
2892:
2888:
2884:
2880:
2876:
2872:
2868:
2864:
2860:
2856:
2851:
2847:
2843:
2839:
2835:
2831:
2827:
2822:
2820:
2810:
2808:
2804:
2800:
2796:
2792:
2788:
2784:
2780:
2776:
2772:
2768:
2764:
2760:
2756:
2752:
2749:over a field
2748:
2744:
2740:
2736:
2732:
2728:
2723:
2721:
2717:
2713:
2709:
2705:
2702:
2701:unitary group
2698:
2694:
2690:
2686:
2682:
2678:
2674:
2670:
2666:
2655:
2642:
2637:
2625:
2620:
2617:
2614:
2610:
2606:
2595:
2591:
2582:
2578:
2570:
2564:
2560:
2550:
2533:
2525:
2521:
2516:
2512:
2489:
2481:
2477:
2447:
2425:
2396:
2367:
2363:
2355:
2351:
2334:
2319:
2315:
2314:
2307:
2294:
2289:
2281:
2277:
2271:
2264:
2257:
2253:
2246:
2233:
2229:
2225:
2220:
2216:
2203:
2187:
2158:
2148:
2143:
2114:
2099:
2081:
2060:
2041:
2031:
2027:
2023:
2019:
2015:
2011:
2007:
2003:
1999:
1995:
1991:
1987:
1983:
1979:
1975:
1971:
1967:
1963:
1959:
1956:of dimension
1955:
1951:
1947:
1944:
1940:
1936:
1932:
1928:
1924:
1921:
1917:
1913:
1910:
1906:
1903:
1899:
1898:bilinear form
1895:
1891:
1887:
1883:
1879:
1876:
1868:
1864:
1860:
1854:
1852:
1848:
1844:
1840:
1836:
1832:
1828:
1825:
1821:
1817:
1813:
1809:
1804:
1800:
1796:
1793:is the group
1791:
1787:
1784:
1780:
1776:
1772:
1768:
1750:
1736:
1728:
1722:
1710:
1708:
1704:
1700:
1696:
1674:
1669:
1643:
1633:
1629:
1625:
1621:
1617:
1613:
1610:over a field
1609:
1599:
1597:
1593:
1590:-subgroup of
1589:
1585:
1581:
1577:
1558:
1553:
1549:
1544:
1534:
1528:
1525:
1522:
1517:
1513:
1506:
1500:
1494:
1491:
1484:
1483:
1482:
1480:
1476:
1472:
1468:
1463:
1461:
1457:
1455:
1448:
1444:
1440:
1436:
1432:
1428:
1424:
1420:
1416:
1412:
1408:
1404:
1400:
1397:over a field
1396:
1386:
1370:
1367:
1363:
1342:
1328:
1325:
1321:
1318:
1314:
1310:
1306:
1302:
1298:
1277:
1249:
1244:
1236:
1232:
1216:
1193:
1185:
1181:
1172:
1156:
1148:
1144:
1128:
1108:
1100:
1097:
1093:
1077:
1069:
1059:
1057:
1053:
1049:
1045:
1041:
1037:
1033:
1029:
1025:
1021:
1018:over a field
1017:
1011:
1001:
999:
995:
991:
987:
983:
979:
975:
971:
967:
964:over a field
963:
959:
954:
952:
949:
945:
941:
937:
933:
929:
925:
921:
917:
914:
910:
907:
903:
899:
895:
891:
887:
885:
881:
877:
873:
870:
866:
862:
859:
855:
851:
847:
844:
840:
836:
832:
828:
824:
823:perfect field
820:
816:
812:
809:is a type of
808:
804:
792:
787:
785:
780:
778:
773:
772:
770:
769:
762:
759:
758:
755:
752:
751:
748:
745:
744:
741:
738:
737:
734:
729:
728:
718:
715:
712:
711:
709:
703:
700:
698:
695:
694:
691:
688:
686:
683:
681:
678:
677:
674:
668:
666:
660:
658:
652:
650:
644:
642:
636:
635:
631:
627:
624:
623:
619:
615:
612:
611:
607:
603:
600:
599:
595:
591:
588:
587:
583:
579:
576:
575:
571:
567:
564:
563:
559:
555:
552:
551:
547:
543:
540:
539:
536:
533:
531:
528:
527:
524:
520:
515:
514:
507:
504:
502:
499:
497:
494:
493:
465:
440:
439:
437:
431:
428:
403:
400:
399:
393:
390:
388:
385:
384:
380:
379:
368:
365:
363:
360:
357:
354:
353:
352:
351:
348:
345:
344:
339:
336:
335:
332:
329:
328:
325:
322:
320:
318:
314:
313:
310:
307:
305:
302:
301:
298:
295:
293:
290:
289:
288:
287:
281:
278:
275:
270:
267:
266:
262:
257:
254:
251:
246:
243:
240:
235:
232:
231:
230:
229:
224:
223:Finite groups
219:
218:
207:
204:
202:
199:
198:
197:
196:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
141:
138:
137:
136:
135:
130:
127:
125:
122:
121:
120:
119:
116:
115:
111:
110:
105:
102:
100:
97:
95:
92:
90:
87:
84:
82:
79:
78:
77:
76:
71:
68:
66:
63:
61:
58:
57:
56:
55:
50:Basic notions
47:
46:
42:
38:
37:
34:
29:
25:
21:
20:
9472:
9464:Demazure, M.
9425:
9419:
9378:
9343:
9337:
9296:
9263:
9259:
9211:
9207:
9192:
9154:
9112:
9109:Milne, J. S.
9078:
9044:
9004:
8996:Demazure, M.
8957:
8950:Demazure, M.
8917:
8909:Demazure, M.
8883:
8846:
8806:
8759:
8753:
8708:
8687:
8678:
8669:
8660:
8651:
8642:
8633:
8624:
8615:
8606:
8597:
8588:
8579:
8570:
8545:
8536:
8527:
8518:
8509:
8500:
8491:
8482:
8473:
8464:
8455:
8446:
8425:
8416:
8407:
8398:
8389:
8380:
8367:
8360:
8351:
8342:
8223:
8221:
8212:
8208:
8204:
8202:
8107:
8103:
8101:
8096:
8092:
8088:
8084:
8080:
8076:
8072:
8070:global field
8065:
8060:
8056:
8052:
8047:
8043:
8039:
8035:
8031:
8026:
8022:
8018:
8010:
8008:
7913:
7909:
7893:
7885:
7881:
7877:
7873:
7868:
7859:
7855:
7851:
7843:
7835:
7831:
7827:
7823:
7819:
7811:
7809:
7804:
7800:
7796:
7792:
7788:
7784:
7780:
7776:
7772:
7768:
7764:
7760:
7756:
7752:
7748:
7744:
7740:
7736:
7732:
7728:
7724:
7720:
7716:
7712:
7708:
7704:
7700:
7696:
7692:
7688:
7686:
7681:
7677:
7673:
7669:
7665:
7657:
7653:
7645:
7637:
7533:
7525:
7521:
7517:
7513:
7507:
7505:
7495:
7491:
7487:
7483:
7479:
7475:
7474:. The group
7378:
7374:
7370:
7366:
7362:
7360:discriminant
7355:
7350:
7344:
7342:is of type D
7339:
7335:
7331:
7327:
7323:
7319:
7315:
7311:
7307:
7305:
7300:
7296:
7292:
7288:
7284:
7280:
7276:
7272:
7268:
7264:
7260:
7252:
7248:
7240:
7236:
7232:
7228:
7224:
7222:
7215:
7211:
7207:
7203:
7172:
7165:
7158:
7154:
7150:
7145:
7141:
7133:
7129:
7127:
7122:
7118:
7104:
7088:
7084:
7076:
7072:
7068:
7064:
7060:
7054:
7050:
7048:
7043:
7039:
7035:
7031:
7027:
7023:
7019:
7015:
7011:
7007:
7003:
6995:
6991:
6985:
6981:
6977:
6973:
6969:
6965:
6961:
6954:
6952:
6938:
6934:
6926:
6922:
6914:
6910:
6906:
6902:
6898:
6894:
6890:
6886:
6882:
6881:-adic field
6878:
6874:
6870:
6866:
6862:
6858:
6856:
6851:
6847:
6843:
6839:
6831:
6827:
6823:
6819:
6815:
6811:
6807:
6803:
6799:
6795:
6792:
6788:
6784:
6780:
6776:
6774:
6766:
6762:
6755:
6751:
6744:
6737:
6730:
6723:
6719:
6712:
6708:
6706:
6698:Jacques Tits
6693:
6689:
6685:
6681:
6677:
6673:
6672:, the group
6669:
6665:
6661:
6657:
6652:
6648:
6644:
6640:
6636:
6632:
6628:
6624:
6620:
6616:
6612:
6608:
6604:
6600:
6596:
6594:
6589:
6554:
6550:
6546:
6542:
6540:
6535:
6531:
6527:
6523:
6519:
6516:presentation
6507:
6503:
6501:
6491:
6487:
6483:
6479:
6475:
6471:
6467:
6463:
6459:
6455:
6451:
6447:
6443:
6439:
6435:
6431:
6427:
6425:
6420:
6416:
6412:
6408:
6404:
6398:
6394:
6390:
6385:
6381:
6373:
6369:
6365:
6361:
6359:
6350:
6346:
6342:
6340:
6334:
6329:
6325:
6321:
6315:
6311:
6307:
6301:
6296:
6292:
6288:
6284:
6277:
6273:
6269:
6265:
6261:
6257:
6255:
6250:
6246:
6242:
6238:
6236:
6230:
6226:
6222:
6218:
6214:
6210:
6206:
6202:
6201:), then the
6198:
6194:
6190:
6183:
6178:
6174:
6170:
6166:
6162:
6158:
6127:
6123:
6087:
6083:
6079:
6075:
6071:
6067:
6063:
6059:
6055:
6049:reduced norm
6044:
6040:
6036:
6032:
6028:
5984:
5980:
5976:
5972:
5968:
5964:
5960:
5956:
5925:
5921:
5885:
5881:
5877:
5873:
5869:
5865:
5861:
5850:
5840:
5832:
5824:
5817:Jens Jantzen
5808:
5800:
5796:
5792:
5788:
5785:George Kempf
5776:
5772:
5768:
5764:
5762:
5749:
5741:
5737:
5733:
5729:
5721:
5717:
5716:∇(λ) as the
5714:Schur module
5713:
5709:
5707:
5702:
5698:
5694:
5690:
5686:
5682:
5678:
5674:
5670:
5666:
5662:
5658:
5654:
5650:
5646:
5642:
5638:
5634:
5630:
5626:
5622:
5618:
5615:Weyl chamber
5610:
5606:
5602:
5594:
5590:
5586:
5584:
5569:
5565:
5561:
5557:
5553:
5549:
5541:
5537:
5533:
5529:
5521:
5517:
5513:
5508:
5503:
5499:
5488:
5484:
5482:
5477:
5473:
5469:
5461:
5457:
5453:
5437:
5430:
5426:
5422:
5418:
5416:
5407:
5403:
5391:
5387:
5379:
5375:
5371:
5363:
5359:
5355:
5351:
5347:
5343:
5339:
5335:
5331:
5327:
5322:
5317:
5313:
5309:
5305:
5298:
5285:
5281:
5277:
5273:
5271:
5269:connected).
5266:
5262:
5258:
5253:
5249:
5245:
5241:
5237:
5233:
5229:
5225:
5217:
5213:
5209:
5205:
5199:
5189:
5185:
5181:
5177:
5173:
5165:
5134:
5130:
5126:
5086:
5082:
5078:
5074:
5070:
5068:group scheme
5065:
5055:
5051:
5047:
5043:
5039:
5037:
4946:splits as a
4943:
4939:
4935:
4931:
4925:
4764:
4760:
4756:
4752:
4747:
4739:
4735:
4731:
4726:
4528:
4521:
4517:
4513:
4504:
4496:
4492:
4488:
4483:
4474:
4472:
4467:
4463:
4459:
4455:
4451:
4447:
4443:
4440:adjoint type
4439:
4435:
4431:
4423:
4419:
4411:
4407:
4403:
4399:
4395:
4391:
4389:
4372:
4357:
4349:
4345:
4333:
4328:
4290:
4284:
4278:
4272:
4268:
4250:
4246:
4243:flag variety
4242:
4238:
4234:
4230:
4226:
4220:
4137:
4133:
4128:
4124:
4117:
4113:
4109:
4105:
4101:
4097:
4089:
4081:
4073:
4069:
4065:
4064:is a smooth
4061:
4057:
4053:
4050:
4048:
3915:
3911:
3907:
3903:
3896:
3892:
3888:
3884:
3880:
3876:
3872:
3868:
3860:
3856:
3852:
3848:
3844:
3840:
3836:
3832:
3828:
3826:
3816:
3812:
3808:
3804:
3800:
3796:
3788:
3781:
3778:
3774:
3767:
3763:
3759:
3755:
3753:
3738:
3733:
3729:
3723:
3719:
3714:
3710:
3706:
3702:
3698:
3694:
3690:
3686:
3682:
3678:
3674:
3666:
3662:
3660:
3655:
3651:
3647:
3642:
3638:
3633:
3629:
3563:
3559:
3555:
3553:
3468:
3464:
3460:
3446:
3440:
3436:
3429:
3425:
3421:
3417:
3413:
3409:
3404:
3400:
3396:
3384:
3378:
3370:
3368:
3363:
3359:
3355:
3350:
3346:
3341:
3337:
3333:
3329:
3325:
3320:
3316:
3309:
3305:
3301:
3259:
3255:
3251:
3247:
3243:
3201:
3197:
3193:
3191:
3089:
3085:
3057:
2981:
2977:
2973:
2969:
2938:
2932:
2927:
2919:
2915:
2911:
2903:
2899:
2894:
2890:
2886:
2882:
2878:
2874:
2870:
2866:
2862:
2858:
2854:
2849:
2845:
2841:
2837:
2833:
2829:
2825:
2823:
2816:
2806:
2802:
2798:
2790:
2786:
2785:showed that
2778:
2774:
2770:
2766:
2762:
2758:
2754:
2750:
2742:
2734:
2726:
2724:
2719:
2715:
2711:
2707:
2703:
2696:
2692:
2684:
2680:
2676:
2672:
2668:
2662:
2552:
2468:
2311:Non-examples
2205:
2202:from the map
2066:
2029:
2025:
2021:
2017:
2013:
2009:
2005:
2001:
1997:
1993:
1989:
1985:
1981:
1977:
1973:
1969:
1965:
1961:
1957:
1953:
1949:
1945:
1934:
1930:
1926:
1922:
1915:
1911:
1904:
1902:vector space
1893:
1889:
1885:
1881:
1877:
1872:
1866:
1862:
1858:
1853:at least 2.
1850:
1846:
1842:
1841:1. In fact,
1834:
1830:
1826:
1819:
1815:
1811:
1807:
1802:
1798:
1794:
1789:
1785:
1782:
1778:
1774:
1770:
1766:
1734:
1732:
1726:
1720:
1706:
1702:
1698:
1694:
1631:
1626:(that is, a
1623:
1619:
1615:
1611:
1607:
1605:
1591:
1587:
1583:
1579:
1575:
1573:
1478:
1471:homomorphism
1466:
1464:
1459:
1453:
1446:
1442:
1438:
1437:, the group
1434:
1430:
1422:
1418:
1414:
1410:
1406:
1402:
1398:
1394:
1392:
1334:
1323:
1319:
1312:
1304:
1300:
1230:
1170:
1142:
1091:
1065:
1055:
1050:is a smooth
1047:
1043:
1039:
1035:
1031:
1019:
1013:
985:
981:
977:
973:
969:
965:
961:
955:
950:
948:finite field
943:
935:
931:
927:
920:number field
915:
913:real numbers
888:
883:
879:
875:
871:
864:
860:
849:
845:
818:
806:
800:
746:
629:
617:
605:
593:
581:
569:
557:
545:
316:
273:
260:
249:
238:
234:Cyclic group
112:
99:Free product
70:Group action
33:Group theory
28:Group theory
27:
8803:Cartier, P.
7850:at most 1,
7662:pointed set
6436:quasi-split
6270:anisotropic
6086:has degree
5726:line bundle
5649:. Define a
5464:(2) is the
4320:Élie Cartan
3663:simple root
3457:conjugation
3371:root system
3308:). Writing
2943:Lie algebra
2861:called the
2853:) for some
2819:root system
2747:finite type
2354:Borel group
1839:determinant
1628:split torus
1235:base change
1004:Definitions
867:), and the
803:mathematics
519:Topological
358:alternating
9496:Lie groups
9485:Categories
9264:-functions
9226:1512.08296
9185:V.L. Popov
8698:References
7916:, the map
7775:)). Also,
7672:is called
7536:such that
7245:inner form
7107:Tits index
7101:Tits index
6998:) that is
6434:is called
6264:if it has
6260:is called
6193:of degree
6161:has index
5965:Witt index
5756:gives the
5669:such that
5548:metric on
5284:) ≅
5202:Lie groups
5077:is called
4526:Witt index
4510:spin group
4402:is called
4086:projective
3793:normalized
3393:normalizer
3387:means the
3380:Weyl group
3375:root datum
3362:from 1 to
2832:, and let
2753:is called
2173:embeds in
2067:The group
1941:, and its
1937:) has two
1865:), and Sp(
1614:is called
1401:is called
1092:semisimple
835:direct sum
626:Symplectic
566:Orthogonal
523:Lie groups
430:Free group
155:continuous
94:Direct sum
9377:(2016) ,
9346:: 49–80,
9199:EMS Press
9187:(2001) ,
9076:(2003) ,
8801:(2005) ,
8792:119837998
8707:(1991) ,
8150:∏
8146:→
7956:∏
7952:→
7648:, or the
7617:¯
7585:¯
7553:¯
7438:
7432:⊂
7392:
7275:), where
7188:¯
6574:¯
6423:is zero.
6355:unipotent
6262:isotropic
6205:-rank of
6143:¯
6107:¯
6009:⌋
5995:⌊
5959:-rank of
5941:¯
5905:¯
5758:character
5556:-rank of
5412:reductive
5374:, and so
5150:¯
5110:¯
5079:reductive
4994:⋉
4982:
4976:≅
4964:
4888:−
4871:⋯
4800:…
4658:−
4641:⋯
4564:…
4495:+1) over
4336:of type G
4223:isotropic
4200:⊂
4180:⊂
4177:⋯
4174:⊂
4154:⊂
4022:∗
4000:∗
3995:∗
3978:∗
3973:∗
3968:∗
3963:∗
3956:∗
3951:∗
3946:∗
3941:∗
3902:for α in
3865:conjugate
3791:which is
3534:α
3514:Φ
3510:∈
3507:α
3503:⨁
3499:⊕
3172:α
3158:Φ
3155:∈
3152:α
3148:⨁
3144:⊕
3056:fixed by
2611:∏
2607:≅
2242:↦
2149:×
2045:¯
1705:-tori in
1678:¯
1647:¯
1550:μ
1523:×
1507:≅
1281:¯
1253:¯
1147:unipotent
1143:reductive
1068:connected
833:and is a
690:Conformal
578:Euclidean
185:nilpotent
9368:55638217
9332:(1965),
9295:(1998),
9254:(1979),
9111:(2017),
8956:(1970).
8882:(1970),
8838:(2014),
8241:See also
7528:with an
7125:-group.
6765:of type
6702:BN-pairs
6186:) for a
5876:). Here
4520:+1 over
4454:, where
4426:) being
3855:. Every
3728:for 1 ≤
3646:for 1 ≤
3354:for all
2924:integers
2906:) under
2000:.) When
1713:Examples
1096:solvable
685:Poincaré
530:Solenoid
402:Integers
392:Lattices
367:sporadic
362:Lie type
190:solvable
180:dihedral
165:additive
150:infinite
60:Subgroup
9450:0164968
9442:1970394
9409:3616493
9360:0180554
9323:1642713
9286:0546587
9231:Bibcode
9178:1278263
9143:3729270
9102:2015057
9067:2605318
9028:2867622
8988:0274459
8941:2867621
8902:0302656
8869:3309122
8829:2124841
8805:(ed.),
8784:0294349
8764:Bibcode
8739:1102012
8365:Milne.
7820:abelian
7334:) over
7161:over a
7056:lattice
6964:. Then
6929:) is a
6822:). For
6494:| ≤ 2.
6474:) over
6378:compact
6130:) over
6062:). The
6051:on the
5821:Lusztig
5701:(λ) of
5637:. Then
5573:) is a
5536:), and
5507:), the
5257:) = Ad(
4738:) over
4363:is the
3709:)) are
3391:of the
2857:, with
2687:) is a
2028:) over
1996:) over
1900:on the
1445:) over
1038:) over
1026:closed
984:) when
946:over a
906:complex
821:over a
813:over a
680:Lorentz
602:Unitary
501:Lattice
441:PSL(2,
175:abelian
86:(Semi-)
9448:
9440:
9407:
9397:
9366:
9358:
9321:
9311:
9284:
9274:
9241:
9176:
9166:
9141:
9131:
9100:
9090:
9065:
9055:
9026:
9016:
8986:
8976:
8939:
8929:
8900:
8890:
8867:
8857:
8827:
8817:
8790:
8782:
8737:
8727:
8021:, and
8015:places
7660:. The
7530:action
7509:torsor
7136:, the
6984:). An
6775:For a
6750:. For
6595:For a
6064:degree
6024:Every
5955:. The
5740:. For
5546:CAT(0)
5468:, and
5366:) has
5091:smooth
5038:where
4512:Spin(2
4508:: the
4367:of an
4352:), by
4078:proper
3701:) (or
3424:) (or
3254:. Let
2875:weight
2793:is of
1861:), SO(
1724:and SL
1427:center
1411:simple
1403:simple
1295:is an
1052:affine
1024:smooth
996:of an
856:, the
831:kernel
535:Circle
466:SL(2,
355:cyclic
319:-group
170:cyclic
145:finite
140:simple
124:kernel
9438:JSTOR
9364:S2CID
9221:arXiv
9041:(PDF)
8843:(PDF)
8788:S2CID
8372:(PDF)
8335:Notes
8079:over
7912:over
7840:Serre
7795:,Aut(
7755:over
7731:over
7723:,Aut(
7703:over
7689:forms
7636:over
7524:over
7498:− 1.
7239:over
7231:over
7002:with
6972:over
6943:index
6889:over
6736:, or
6656:over
6403:over
6376:) is
6333:over
6300:over
6197:over
6157:. If
6074:as a
6031:over
5856:are:
5811:, by
5781:socle
5661:over
5617:) in
5609:) ≅
5524:is a
5440:is a
5433:by a
5308:over
5216:over
4759:over
4524:with
4462:is a
4392:split
4371:over
4340:and E
4123:,...,
4088:over
4080:over
3883:over
3745:graph
3736:− 1.
3650:<
3315:,...,
2840:; so
2813:Roots
2710:) to
1810:) of
1622:over
1616:split
1582:over
1417:over
1311:over
1309:torus
934:) of
918:or a
852:) of
815:field
719:Sp(∞)
716:SU(∞)
129:image
9395:ISBN
9309:ISBN
9272:ISBN
9239:ISBN
9164:ISBN
9129:ISBN
9088:ISBN
9053:ISBN
9014:ISBN
8974:ISBN
8927:ISBN
8888:ISBN
8855:ISBN
8815:ISBN
8725:ISBN
8246:The
7259:set
7140:Gal(
7053:, a
6953:Let
6341:For
6209:is (
5575:tree
4926:The
4329:The
4318:and
4294:, E
4288:, D
4276:, B
4096:for
3332:) ≅
2970:root
2968:. A
2933:The
2877:for
2863:rank
2824:Let
2797:and
2352:The
2316:Any
2063:Tori
1405:(or
805:, a
713:O(∞)
702:Loop
521:and
9430:doi
9387:doi
9348:doi
9301:doi
9121:doi
8966:doi
8772:doi
8717:doi
8209:PGL
8034:or
8017:of
7769:PGL
7652:if
7644:on
7532:of
7435:Gal
7389:Gal
7373:*/(
7369:in
7365:of
7059:in
7046:).
6901:(1,
6684:if
6592:).
6419:or
6066:of
6043:(1,
5967:of
5807:of
5799:of
5657:of
5564:(2,
5516:of
5472:(2,
5456:(2,
5429:of
5406:(2,
5402:of
5390:(2,
5378:(2,
5362:(2,
5350:(2,
5348:PSL
5342:(2,
5340:PGL
5334:(2,
5332:PGL
5328:PGL
5280:(1,
5188:to
5129:in
5089:is
4979:Out
4961:Aut
4383:, E
4379:, E
4310:, G
4306:, F
4302:, E
4298:, E
4282:, C
4249:of
4245:or
4076:is
3863:is
3859:of
3843:of
3795:by
3773:in
3689:if
3681:of
3673:of
3084:of
2984:on
2972:of
2865:of
2769:of
2745:of
2382:of
1630:in
1458:of
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