Last update: 13 February 2014
Let be a real vector space of finite dimension and a positive definite scalar product on Let be the dual vector space. Then the scalar product defines an isomorphism of onto and hence a scalar product on For each non-zero let be the image of under this isomorphism. Let denote the kernel of and let be the reflection in the hyperplane We make act also on by transposition:
A root system in is a subset of satisfying the following axioms:
|(RS 1)||is finite, spans and does not contain|
We shall assume throughout that is reduced:
|(RS 4)||If are proportional, then|
|(RS 5)||If such that for all and then either or is empty.|
The elements of are called roots. The group generated by the reflections is a finite group of isometries of called the Weyl group of The connected components of the set are open simplicial cones called the chambers of These chambers are permuted simply transitively by and the closure of each chamber is a fundamental region for
Choose a chamber This choice determines a set of roots such that is the intersection of the open half-spaces for The elements of are called the simple roots (relative to the chamber A root is said to be positive or negative according as it is positive or negative on Let (resp. denote the set of positive (resp. negative) roots.
Each root can be expressed as a linear combination of the simple roots: with coefficients all of the same sign. The integer the sum of the coefficients, is called the height of There is a unique root (the 'highest root') whose height is maximal.
Let denote the 'obtuse cone' If and then
A subset of is said to be closed if Suppose is closed and Then there exists such that
The Weyl group is generated by the reflections for Hence any can be written as a product where and If the number of factors is as small as possible, then is a reduced word for and is the length of written The length of is also equal to the number of positive roots such that is negative. More precisely, these roots are (All this of course is relative to the choice of the chamber
In particular, there exists a unique element of which sends every positive root to a negative root. This leads to the following result, which will be useful later:
The positive roots can be arranged in a sequence such that for each the set of roots is the set of positive roots relative to some chamber
Let be a reduced word for where and as before put For each is a reduced word, and therefore from the facts quoted above Consequently and therefore so that and finally is the set of positive roots for the chamber
We retain the notation of (2.1). For each and we have an affine-linear function on (namely, Let The elements of are called affine roots. We shall denote them by Greek letters If is an affine root, so are and for all integers If then is the gradient of (in the usual sense of elementary calculus).
For each let be the (affine) hyperplane on which vanishes, and let be the reflection in (From now on, only the affine structure and not the vector-space structure of will come into play). The group generated by the as runs through is an infinite group of displacements of called the Weyl group of or the affine Weyl group. is the subgroup of which fixes the point The translations belonging to form a free abelian group of rank and is the semi-direct product of and
For each let The are a basis of We have and the mapping defined by maps isomorphically onto the lattice spanned by the for
The affine Weyl group acts on by transposition: if then is defined to be
The connected components of the set are open rectilinear called the chambers of The chambers are permuted simply transitively by and the closure of each chamber is a fundamental region for The (non-empty) faces of the chambers are locally closed simplexes, called facets. Each point lies in a unique facet The subgroup of which fixes also fixes each point of
Choose a chamber This choice determines a set of affine roots, such that is the intersection of the open half-spaces for The affine roots belonging to are called the simple affine roots (relative to the chamber
The affine Weyl group is generated by the relations for As in the case of we define the length of an element of The length of is also equal to the number of affine roots which are positive on and negative on i.e. to the number of hyperplanes which separate and
If is a facet of then the subgroup of which fixes a point of is generated by the which fix
We shall need the following result later:
Let be a set of affine roots whose gradients are all positive and distinct. Then there exists a translation such that each is positive on
Let the elements of be and let We have If then for all positive roots Hence will be positive on provided that Hence we have only to choose so that satisfies for all which occur as gradients of elements of and this is certainly possible.
For later use, it will be convenient to choose to be the unique chamber of which is contained in the cone and has the origin as one vertex. The following lemma presupposes this choice of
For each let be the set of affine roots which are positive on and negative on
Let be an element of such that Then for all we have
If is positive on then is negative on and positive on hence (because the origin belongs to the intersection of the closures of and Consequently is negative on hence is negative on and therefore on contradiction. Hence must be negative on i.e.
Next, let and put Then and hence is positive on and negative on i.e. Hence
Conversely, let If then and if then This establishes the opposite inclusion. Finally, the sets and are disjoint, because for all and for all
Before going any further, it may be helpful to have a concrete example of the set-up so far:
(Root system of type
Let be the set of all such that The scalar product is Let be the coordinate function on and let Then satisfies (RS 1)-(RS 5). The Weyl group acts by permuting the coordinates of hence is isomorphic to the symmetric group of order We may take the chamber to be and the simple roots are then The chamber is the open simplex
Let be a group, and subgroups of and a subset of the coset space The pair is a BN-pair in (or a Tits system) if it satisfies the following four axioms:
|(BN 1)||is generated by and and is normal in|
|(BN 2)||generates the group and each has order|
|(BN 3)||for all and|
|(BN 4)||for each|
The group is the Weyl group of The elements of are cosets of and an expression such as is to be read in the usual sense as a product of subsets of If is a subset of then means
One shows that the set is uniquely determined by and namely an element belongs to if and only if is a group. The elements of are called the distinguished generators of
The axioms (BN 1)-(BN 4) have the following consequences. The proofs may be found in [Bou1968].
2.3.1 and the mapping is a bijection of onto the double coset space
2.3.2 For each subset of let be the subgroup of generated by and let Then is a subgroup of
2.3.3 The mapping is an inclusion-preserving bijection of the set of subsets of onto the set of subgroups of containing
2.3.4 If are subsets of and is conjugate to then
2.3.5 If are subsets of and then
2.3.6 Each is its own normalizer in
2.3.7 If and and then (The length of is defined as the length of a reduced word in the generating set just as in (2.1) and (2.2).)
Let be a group with BN-pair as in (2.3). We shall assume now that the Weyl group of 'is' an affine Weyl group in the sense of (2.2). More precisely, we shall suppose we are given a reduced irreducible root system in a Euclidean space a chamber for the associated affine root system and a homomorphism of onto the affine Weyl group such that
|(2)||under this identification, the distinguished generators of are the reflections in the walls of the chamber in other words,|
The conjugates of in are called Iwahori subgroups of A parahoric subgroup of is a proper subgroup containing an Iwahori subgroup. From the properties of BN-pairs recalled in (2.3), each parahoric subgroup is conjugate to for some proper subset of and is uniquely determined by
We set up a one-one correspondence between the subsets of and the facets of the chamber as follows. To a facet corresponds the set of all which fix In this correspondence, and a vertex of is a maximal proper subset of
If we write for
Each parahoric subgroup determines uniquely a facet of namely is conjugate to
The building associated with the given BN-pair structure on is the set of all pairs where is a parahoric subgroup of and is a point in the facet
With each parahoric subgroup we associate the subset of where is called a facet of of type In particular, if is an Iwahori subgroup, is a chamber of If is parahoric, there are only finitely many parahoric subgroups containing and we define In this way the building may be regarded as a geometrical simplicial complex, the being the 'open' simplexes and the the 'closed' simplexes.
The group acts on by inner automorphisms:
Consider the following subset of Equivalently, is the union of the facets where and
Remark. Since we have identified with by means of the isomorphism induced by expressions such as are meaningful, where and is a subgroup of containing Hence acts on for any such that
There exists a unique bisection such that
|(1)||for each facet of and each|
Let Then is congruent under to a unique point of say where and and is a facet of the chamber Define to be the point We have to check that is well defined. If also then fixes and therefore, by a property of reflection groups recalled in (2.2), belongs to the subgroup of generated by the which fix In other words, where Since it follows that Hence is well defined, and clearly satisfies (1) and (2). The uniqueness of is obvious.
Let be the chamber We have for some Hence fixes as a whole and also fixes the chamber and all its facets; also it maps each facet to another facet of the same type. Transferring our attention to it follows that maps bijectively onto and fixes the chamber and all its facets. Hence also fixes the adjacent chambers and so on. It follows that is the identity mapping and hence that
The subcomplexes of as runs through are called the apartments of the building If is an apartment, we may transport the Euclidean structure of to via the bijection If also then it follows from (2.4.2) that the two mappings and differ by an element of and hence we have a well-defined structure of Euclidean space on each apartment and in particular a distance defined for
Thus the building may be thought of as obtained by sticking together many copies of the Euclidean space
Any two facets of are contained in a single apartment.
Consider two facets and where are parahoric subgroups of say where are facets of the chamber in By (2.3.1) we can write with and If then and therefore and are both in
is transitive on the set of pairs where is an apartment and is a chamber in
We have for some where Hence we may assume that If contains then is a chamber in and hence for some So if we have and
Let be, two apartments and let be a chamber contained in Then there exists a unique bijecton such that
|(1)||There exists such that for all|
The existence of follows immediately from (2.4.4). Uniqueness follows from (2.4.2): if both satisfy (1) and (2), then fixes as a whole and fixes the chamber hence is the identity map. That follows from the definition of the metric on an apartment given after (2.4.2).
It remains to show that for all Suppose is a facet contained in By (2.4.4) we may assume that and Since it follows that normalizes and therefore by (2.3.6) that
Since we have for some facet of and Since is its own normalizer (2.3.6), we have and therefore By (2.3.5), where is the subgroup of which fixes Hence (2.3.1) since it follows that hence
Retraction of the building onto an apartment
Let be an apartment and a chamber in Then there exists a unique mapping such that, for all apartments containing is the bisection of (2.4.5).
Let By (2.4.3) there exists an apartment containing and Let be the isomorphism of (2.4.5). If also and are contained in another apartment then we have also But then is the isomorphism of (2.4.5) for the apartments and and the chamber Hence fixes that is to say It follows that as in the theorem is well defined, and the uniqueness is obvious.
The mapping of (2.4.7) is called the retraction of onto with centre It has the following properties:
|(2)||For each facet in is an affine isometry of onto|
(1) and (2) are clear. As to (3), let be a facet such that is a facet of By (2.4.3) there exists an apartment containing and and is a bijection of onto leaving fixed. It follows that
|(i)||There exists a unique function such that is the metric for each apartment in|
|(ii)||If is a retraction of onto an apartment as in (2.4.6), then for all|
|(iii)||is a metric on|
Let By (2.4.3) there is an apartment containing and We wish to define to be the Euclidean distance between and in We must therefore show that, if belong also to another apartment then
Let be a chamber in such that and let be a chamber in such that By (2.4.3) there is an apartment containing and From (2.4.5) we have because and have a chamber in common; and for the same reason Hence is well defined and is (from the definition of
(ii) Let be an apartment containing and The image under of the straight line segment joining to in will be a polygonal line in of the same total length, by (2.4.7)(ii), and endpoints and Hence the result.
(iii) Let and let be an apartment containing and Let be a retraction of onto Then by (ii) above, since and Hence satisfies the triangle equality, and we have already observed that is
Let Then there is a unique geodesic joining to
Let be an apartment containing and and let denote the straight line segment joining to in Suppose lies on a geodesic from to Then Let be such that and hence also Let be the retraction of onto with centre a chamber such that Then we have Consequently Since are in the same Euclidean space it follows that Hence and therefore by (2.4.7)(iii).
It follows from (2.4.9) that the straight line segment lies in all apartments containing and
is complete with respect to the metric
Let be a Cauchy sequence in If is a retraction of onto an apartment then is a Cauchy sequence in by (2.4.8)(ii), and therefore converges to a point The set of real numbers where is such that has a positive lower bound and we shall have for all sufficiently large
Now each is of the form for some Put Then for all large Hence and we have as Hence
Fixed point theorem
A subset of is convex if
Let be a bounded non-empty subset of Then the group of isometries of such that has a fixed point in the closure of the convex hull of
For the proof we shall require the following lemma:
Let and let be the midpoint of Then
If are all in the same apartment this is true with equality, by elementary geometry (take as origin). In the general case, let be an apartment containing and and choose a chamber in such that Retract onto with centre Then
|Proof of 2.4.11.|
Let be the diameter of Choose a real number and let be the set of all such that is the midpoint of with and Clearly is non-empty and is contained in the convex hull of If and then by (2.4.12) so that for some
If now and then again is the midpoint of a segment with and so that so that for some From (2) it follows that
For each pick Then by (1) and (2) Hence is a Cauchy sequence in hence by (2.4.10) converges to some Clearly lies in the closure of the convex hull of
Now let be an isometry of such that Then for all Let Then is a Cauchy sequence with for all and converges to But by (3) as Hence Hence is a fixed point of the group of isometries of such that
A subset of is said to be bounded if is bounded for all bounded subsets of
is bounded intersects only finitely many double cosets
Let be the retraction of onto the apartment with centre Then is bounded is bounded is contained in a finite union of closed chambers of Hence is bounded is bounded, for each chamber is bounded.
If define by the relation Then is bounded is bounded the set is finite intersects only finitely many
A subgroup of is bounded is contained in a parahoric subgroup.
Let Then is bounded. Applying the fixed point theorem (2.4.11), we see that there exists such that If lies in the facet then normalizes the parahoric subgroup and hence by (2.3.6).
Replacing by a conjugate, we may assume that where is a proper subset of Now and is finite because Hence is bounded by (2.4.13).
Every bounded subgroup of is contained in a maximal bounded subgroup. The maximal bounded subgroups are precisely the maximal parahoric subgroups of and form conjugacy classes, corresponding to the vertices of the chamber
(a) Universal Chevalley groups
We retain the notation of (2.1) and (2.2). Let Then is a lattice of rank in containing
Let be any field, and the additive and multiplicative groups of Associated with the pair there is a universal Chevalley group We shall not go into the details of the construction of for which we refer to Steinberg [Ste1967]. All we shall say here is that is generated by elements and contains elements with various relations between the and the In particular, each is an injective homomorphism of into and is an injective homomorphism of into
If is the root system of type (2.2.3) then the universal Chevalley group is the special linear group In this case the and may be taken as follows:
If then where is the unit matrix and is the matrix with in the place and elsewhere. is the diagonal matrix with in order down the diagonal.
Now suppose that the field carries a discrete valuation i.e. a surjective homomorphism such that for all (conventionally We consider certain subgroups of the Chevalley group
First, let be the subgroup consisting of the and let be the normalizer of Then there is a canonical isomorphism and one can show that this lifts uniquely to a homomorphism of onto the affine Weyl group The restriction of to is described as follows. If then is a homomorphism which defines a linear form hence a vector in Then is translation by this vector, and this translation belongs to We have and where is as before the translation subgroup of The kernel of is the set of all such that is contained in the group of units of
Next, for each we have a subgroup of defined as follows. If then If is any subset of we denote by the subgroup of generated by the such that and by the subgroup generated by and Then the triple has the following properties, for all and
|(II)||If is a positive integer, then is a proper subgroup of and|
The next three properties relate to a pair of subgroups and in the following situations: (III) is a positive constant; (IV) (V) neither nor is constant (i.e. the hyperplanes are not parallel).
|(III)||If with then|
For let denote the set of all of the form with positive integers, and let be the subgroup of generated by all commutators with and Then from Chevalley's commutator relations we have
|(V)||If and are not parallel, then|
Let be the subgroup generated by the with positive, and likewise Then
|(VII)||is generated by and the|
(b) Simply-connected simple algebraic groups
As before let be a field with a discrete valuation and assume now that is complete with respect to with perfect residue field.
Let be a simply-connected simple linear algebraic group defined over Let be a maximal torus in and let be the centralizer and normalizer respectively of in Let be the group of points of Then one of the main results of the structure theory of Bruhat and Tits ([BTi1966], [BTi1967]) is:
There exists a reduced irreducible root system subgroups and of and a surjective homomorphism such that the triple satisfies (I)-(VII) above.
In fact one takes and is the set of all such that for all characters of is the relative Weyl group of with respect to but is not in general the relative root system, even if the latter is reduced. But every is proportional to a root of relative to and conversely.
If is split over (i.e. if is a maximal torus of then we are back in the situation of (a) above.
If is a field (i.e. if the residue field of is finite) then inherits a topology from for which is a locally compact topological group. If is a suitably chosen maximal compact subgroup of then as we shall see later (3.3.7) the algebra is commutative, and the theory of zonal spherical functions on relative to is almost entirely a consequence of the properties (I)-(VII) listed above. We shall therefore take these properties as a set of axioms:
Let be a reduced irreducible root system and let be the associated affine root system and affine Weyl group, as in (2.2). An affine root structure of type on a group is a triple where and the are subgroups of and is a homomorphism of onto satisfying (I)-(VII) above.
In this section is a group endowed with an affine root structure of type (2.5.3). We shall develop some consequences of the axioms (I)-(VII).
Let be a subset of such that
|(1)||if and then for some|
|(2)||if and then the hyperplanes are not parallel.|
Let be the gradient of Then are all distinct by (2) and is a closed subset of such that (by (1) and (2)). Hence (2.1.2) there exists such that Choosing such that we have by (I); hence we may assume without loss of generality that
Assume first that in this order is a subsequence of the sequence of (2.1.1). Let If are positive integers and then hence Now use the commutator axiom (V): if and we have and therefore This shows that is normal in and hence that By induction on we conclude that
Next we shall show that every is uniquely of the form with Again by induction, it is enough to show that is uniquely determined by and for this it is enough to show that By (2.1.1) we can assume that is negative and ar are positive. Then and hence the result follows from (VI).
This proves (2.6.1) for the particular ordering chosen. For an arbitrary ordering of it follows from (2.6.2) below, whose proof we leave to the reader.
Let be a group with subgroups such that
|(1)||with uniqueness of expression;|
|(2)||is a normal subgroup of for|
with uniqueness of expression, the positive roots being taken in any fixed order. Likewise
Arrange the positive roots in a sequence satisfying (2.1.1). Follow the proof of (2.6.1) to show that with uniqueness of expression, and then use (2.6.2).
If is any subset of let denote the set of all which are on Then the subgroups are defined.
Let If there is an affine root with gradient belonging to there is a least such affine root, say and Let a denote the corresponding subgroup of If there is no affine root with gradient belonging to define to be Let (resp. be the group generated by the with (resp.
Suppose has non-empty interior. Then
By (2.6.1), the product being taken in any order. Since is generated by the and it is enough to show that is a group. For this it is enough to show that
Arrange the positive roots in a sequence satisfying (2.1.1) Let and let (resp. be the subgroup of generated by (resp. by Since has non-empty interior it follows from (III) that is a group and therefore Hence, by (2.6.1), we have for Consequently
Suppose Let then with and Suppose, to fix the ideas, that the gradient of is a positive root. Then by (VI). Hence i.e. By (2.6.1) and (2.6.3) it follows that Hence and therefore by (II) we have The reverse implication is obvious.
We shall now show that the group carries two BN-pair structures, one with Weyl group and the other with Weyl group (2.6.8) and (2,6.9). To establish these results we need a couple of preliminary lemmas.
(Here is the epimorphism obtained by composing with the epimorphism defined by the semi-direct decomposition
(1) Let then by (IV) we have Suppose that Then with Hence by (VI). Consequently whence the result.
(2) Let Then by (II) there exists an affine root with gradient such that Hence by (1) just proved we have
Let and let Since normalizes is a subgroup of and is the semidirect product of and by axiom (VI). Similarly for
|(1)||If and then|
|(2)||If and then|
(1) Since we have hence Conversely, if and then since is the only hyperplane separating and it follows that is positive on hence that Consequently for all such and therefore
(2) Let be the group generated by the with and By (2.6.3) we have Hence, since permutes
Let Then is a BN-pair in We have so that induces an isomorphism of onto the affine Weyl group The distinguished generators of correspond under this isomorphism to the reflections in the walls of the chamber
(1) Let Then so that where Suppose that so that Then there exists a hyperplane separating and hence an affine root which is positive on and negative on Then and since this contradicts (2.6.5). Hence and therefore Hence
(2) We have to verify the axioms (BN 1)-(BN 4) of (2.3). Of these, (BN 1) follows immediately from (I) and (VII). (BN 2) is clear: we define to be the set So is (BN 4), because we have already shown that if As to (BN 3), let and and let Then by (2.6.7). Suppose first that Then and so If on the other hand then and by axiom (IV) we have so that since Hence in all cases.
It follows that all the results of (2.4) are applicable in the present situation. The group and its conjugates are the Iwahori subgroups of the groups and their conjugates, where is a facet of are the parahoric subgroups.
Let Then is a BN-pair in We have so that induces an isomorphism of onto the Weyl group The distinguished generators of correspond under this isomorphism to the reflections in the walls of the cone
(1) Let Then and therefore So we have with by (2.6.3). For each pick an affine root with gradient such that Applying (2.2.1) to this set of affine roots, we see that for some translation and hence for some We have by (2.6.8). Hence by axiom (VI). So and therefore
(2) We have to verify the BN-axioms. As before, (BN 1) follows immediately from (I) and (VII). (BN 2): take the distinguished generators of to be As to (BN 4), let and let then if we have so that by axiom (VI). Since by axiom (II), we have a contradiction.
Finally, (BN 3). Let Then by (2.6.7).
Suppose first that is positive, then If on the other hand is negative, then
Remark. Of course there is a corresponding result for
We remark first that is generated by and the For let and let be an affine root with gradient Then contains the group and hence by axiom (IV) contains Since contains it follows that hence that for all Hence
Next, let and let be any chamber in Then For let be an affine root with gradient If we have If on the other hand then This proves (1), from which it follows that for all so that
Now to show that it is enough to show that for all in a set of generators of Hence it is enough to show that for But
Let (resp. be the set of all translations such that (resp. Let so that and are semigroups contained in
Also let denote the maximal parahoric subgroup
|(1)||(Cartan decomposition) and the mapping is a bijection of onto|
|(2)||(Iwasawa decomposition) and the mapping is a bijection of onto|
|(3)||If and are such that then|
As in (2.4) we shall identify with by means of the isomorphism induced by Thus means and so on. In particular, by (2.3.2).
(1) If then by (2.6.8) and (2.3.6). Now is the disjoint union of the double cosets as runs through Hence is the disjoint union of the cosets with
(2) We shall prove (2) with replaced by this merely amounts to replacing the chamber by its opposite, From (2.6.10) we have To show that and are disjoint if it is enough to show that So let where and
By (2.2.1) there exists such that where Hence hence meets or, putting meets and hence meets for some Consequently
Now by expressing as a reduced word in the generators it follows easily from the axiom (BN 3) that Hence and therefore by (2.3.1). Hence so that and as required.
The proofs of (3) and (4) are best expressed in the language of buildings (2.4).
(3) Suppose that and are such that meets Then contains an element belonging to for some As above, by (2.2.1) we have for some chamber Hence meets Let and Then meets Let and put
Let be the building associated with as in (2.4). Identify the affine space with the apartment by means of the bijection defined in (2.4.1). Let be the retraction of onto with centre (2.4.6) and let Since the image under of the segment is the segment Since for some we have and therefore (because Hence the image of the segment under the retraction is a polygonal line in joining to Consequently there exist real numbers with such that, if then is the union of the segments From (2.4.2) and the definition of it follows that there exist elements such that Writing with and we have Now hence for all So we have with Summing over we get and therefore Since we have proved that as required.
(4) Let and let Then by (2.2.1) as before we have for some translate of the cone and hence where Now and intersects Hence there exists such that lies in the segment
Let Since for some we have and Hence Next, since for some we have and Consequently It follows that by the uniqueness of geodesies in (2.4.9) and hence that Consequently fixes i.e. normalizes and therefore (2.3.6) Hence
Let be an 'abstract' group with affine root structure of type Now suppose in addition that carries a Hausdorff topology compatible with its group structure such that
|(VIII)||and the are closed subgroups of|
|(IX)||is a compact open subgroup of|
A Hausdorff topological group carrying an affine root structure satisfying axioms (I)-(IX) inclusive will be called a simply-connected group of type. If is a field (i.e. a locally compact field which is neither discrete nor connected) then is complete with respect to a discrete valuation, and the residue field is finite. The universal Chevalley group and more generally the group of points of a simply-connected simple linear algebraic group defined over then inherits a topology from which satisfies (VIII) and (IX).
The two new axioms have the following consequences:
is open in and compact.
(2.6.8), so that is an open subgroup of hence closed in hence closed in Since is contained in it follows that is compact.
A closed subset of is compact if and only if meets only finitely many cosets
The cosets are mutually disjoint, open and compact. The assertion follows immediately from this.
Comparing (2.7.2) with (2.4.13), we see that a subset of is bounded in the sense of (2.4) if and only if has compact closure. Hence, from (2.4.15):
Every compact subgroup of is contained in a maximal compact subgroup. The maximal compact subgroups are precisely the maximal parahoric subgroups, and form conjugacy classes. All parahoric subgroups are open and compact. itself is locally compact, but not compact.
The last two assertions follow from the facts that a parahoric subgroup is a finite union of double cosets and that itself is not.
For each the index is finite, and is compact.
There exists a chamber on which is positive and for which is one of the walls. Transforming by an element of we may assume that i.e. that
Since is open and compact, is compact and discrete, hence finite. But because by (2.6.5). Hence is finite. Also is a closed subgroup of hence is compact.
|(1)||For each the group is closed in and for each affine root with gradient is an open compact subgroup of|
|(2)||is a closed subgroup of and is an open compact subgroup of|
(1) By (2.6.5) is of the form where is some affine root with gradient Hence this is open in Conjugating by elements of each is open in Since is compact (2.7.4) it follows that is locally compact and therefore a closed subgroup of
(2) hence is open in also by (2.6.1) is a product of hence is compact. Hence is locally compact, hence closed in
is locally compact.
is locally compact, because it has as a compact open subgroup by (2.7.1). Also is locally compact by (2.7.5), hence so is
Consider again the group (2.5.1) where is now a field. Let be the ring of integers, the maximal ideal of and the (finite) residue field. is a compact open subring of
is a closed subset of hence is locally compact (and totally disconnected). The maximal compact subgroup may be taken to be Let then the projection of onto induces an epimorphism Let be the group of upper triangular matrices in then is an Iwahori subgroup of is the group of upper triangular matrices in and is the diagonal matrices.
This is a typed version of the book Spherical Functions on a Group of Type by I. G. Macdonald, Magdalen College, University of Oxford. This book is copyright the University of Madras, Madras 5, India and was first published November 1971.
Published by the Ramanujan Institute, University of Madras, and printed at the Baptist Mission Press, 41A Acharyya Jagadish Bose Road, Calcutta 17, India.