Bruhat-Tits building
A building (cf. also Tits building) which is defined for a connected reductive group over a field which is complete with respect to a non-trivial discrete valuation and has a perfect residue class field. It is the topological realization of the (poly-) simplicial complex of all parahoric subgroups. One way to define it more precisely is as follows.
Let be a field which is complete with respect to the non-trivial valuation
and has a perfect residue class field. Let
be a connected, reductive
-group. First, assume
to be semi-simple (cf. Semi-simple group).
Apartments.
Let be a maximal
-split torus in
and denote by
(respectively,
) the centralizer (respectively, normalizer; cf. Normalizer of a subset) of
in
. Let
(respectively,
) denote the group of cocharacters (respectively, characters) of
and let
be the canonical perfect pairing. Then there is a unique group homomorphism
such that
for all
(i.e., the group of
-rational characters of
). One can show that there is a unique affine
-space
together with a group homomorphism
(i.e., the affine bijections
) extending
, called the (empty) apartment associated with
.
Filtrations of the root subgroups.
Denote by the root system of
with respect to
and, for
, by
the root subgroup of
associated with
. Then, for
, the set
contains exactly one element, denoted by
. An affine mapping
is called an affine root if the vector part
of
is contained in
and if there exists a
such that
. In that case
is abbreviated as
. For
and
, let
and let
be the subgroup of
generated by all
for
.
Simplicial structures.
Two points are called equivalent if
and
have the same sign or are both equal to
for all affine roots
. One obtains a (poly-) simplicial complex (i.e., a direct product of simplicial complexes) in
by defining the faces to be the equivalence classes.
Building.
Let , where
if there exists an
such that
and
. There is a canonical
-action on
induced by left-multiplication on the first factor of
. One can identify
with its canonical image in
. The subsets of the form
, for
, are called apartments and the subsets of the form
, for
and
a face in
, are called faces. One can equip
with a metric which is
-invariant. This metric coincides on
with the metric induced by the scalar product on
which is invariant under the Weyl group of
. The metric space
together with these structures is called the Bruhat–Tits building of
.
If is not semi-simple, the Bruhat–Tits building of
is, by definition, the Bruhat–Tits building of the derived group (cf. Commutator subgroup) of
.
Example.
Assume , and denote by
and
the valuation ring of
and a uniformizer of
, respectively. An
-lattice is a free
-submodule of
of rank
. Then the Bruhat–Tits building of
is the topological realization of the following simplicial complex: the
-simplices are the
-lattices in
up to homothety and the
-simplices are pairs
of
-lattices in
up to homothety with
.
General references for Bruhat–Tits buildings are [a1] and [a2]. (The situations considered there are more general.) A good overview can be found in [a3].
Originally, the Bruhat–Tits building was the essential technical tool for the classification of reductive groups over local fields (cf. Reductive group). There are further applications, e.g. in the representation theory of reductive groups over local fields, in the theory of p-adic symmetric spaces and in the theory of Shimura varieties.
References
[a1] | F. Bruhat, J. Tits, "Groupes réductifs sur un corps local I, II" IHES Publ. Math. , 41,50 (1972–1984) |
[a2] | E. Landvogt, "A compactification of the Bruhat–Tits building" , Lecture Notes in Mathematics , 1619 , Springer (1996) |
[a3] | J. Tits, "Reductive groups over local fields" , Proc. Symp. Pure Math. , 33 , Amer. Math. Soc. (1979) pp. 29–69 |
Bruhat-Tits building. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bruhat-Tits_building&oldid=46168