Difference between revisions of "Relative root system"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Tits, "Sur la classification des groupes algébriques semi-simples" ''C.R. Acad. Sci. Paris'' , '''249''' (1959) pp. 1438–1440 {{MR|0106967}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Borel, J. Tits, "Groupes réductifs" ''Publ. Math. IHES'' , '''27''' (1965) pp. 55–150 {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J. Tits, "Classification of algebraic simple groups" , ''Algebraic Groups and Discontinuous Subgroups'' , ''Proc. Symp. Pure Math.'' , '''9''' , Amer. Math. Soc. (1966) pp. 33–62 {{MR|}} {{ZBL|}} </TD></TR></table> |
Revision as of 14:51, 24 March 2012
of a connected reductive algebraic group defined over a field
A system of non-zero weights of the adjoint representation of a maximal
-split torus
of the group
in the Lie algebra
of this group (cf. Weight of a representation of a Lie algebra). The weights themselves are called the roots of
relative to
. The relative root system
, which can be seen as a subset of its linear envelope
in the space
, where
is the group of rational characters of the torus
, is a root system. Let
be the normalizer and
the centralizer of
in
. Then
is the connected component of the unit of the group
; the finite group
is called the Weyl group of
over
, or the relative Weyl group. The adjoint representation of
in
defines a linear representation of
in
. This representation is faithful and its image is the Weyl group of the root system
, which enables one to identify these two groups. Since two maximal
-split tori
and
in
are conjugate over
, the relative root systems
and the relative Weyl groups
,
, are isomorphic, respectively. Hence they are often denoted simply by
and
. When
is split over
, the relative root system and the relative Weyl group coincide, respectively, with the usual (absolute) root system and Weyl group of
. Let
be the weight subspace in
relative to
, corresponding to the root
. If
is split over
, then
for any
, and
is a reduced root system; this is not so in general:
does not have to be reduced and
can be greater than 1. The relative root system
is irreducible if
is simple over
.
The relative root system plays an important role in the description of the structure and in the classification of semi-simple algebraic groups over . Let
be semi-simple, and let
be a maximal torus defined over
and containing
. Let
and
be the groups of rational characters of the tori
and
with fixed compatible order relations, let
be a corresponding system of simple roots of
relative to
, and let
be the subsystem in
consisting of the characters which are trivial on
. Moreover, let
be the system of simple roots in the relative root system
defined by the order relation chosen on
; it consists of the restrictions to
of the characters of the system
. The Galois group
acts naturally on
, and the set
is called the
-index of the semi-simple group
. The role of the
-index is explained by the following theorem: Every semi-simple group over
is uniquely defined, up to a
-isomorphism, by its class relative to an isomorphism over
, its
-index and its anisotropic kernel. The relative root system
is completely defined by the system
and by the set of natural numbers
,
(equal to 1 or 2), such that
but
. Conversely,
and
,
, can be determined from the
-index. In particular, two elements from
have one and the same restriction to
if and only if they are located in the same orbit of
; this defines a bijection between
and the set of orbits of
into
.
If , if
is the corresponding orbit, if
is any connected component in
not all vertices of which lie in
, then
is the sum of the coefficients of the roots
in the decomposition of the highest root of the system
in simple roots.
If ,
, then the above relative root system and relative Weyl group are naturally identified with the root system and Weyl group, respectively, of the corresponding symmetric space.
References
[1] | J. Tits, "Sur la classification des groupes algébriques semi-simples" C.R. Acad. Sci. Paris , 249 (1959) pp. 1438–1440 MR0106967 |
[2] | A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402 |
[3] | J. Tits, "Classification of algebraic simple groups" , Algebraic Groups and Discontinuous Subgroups , Proc. Symp. Pure Math. , 9 , Amer. Math. Soc. (1966) pp. 33–62 |
Relative root system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Relative_root_system&oldid=17223