Last update: 1 April 2014
1.1. Let be an indecomposable reduced root-system [Bou1968, p. 151] in a real vector space with positive-definite inner product which is unique up to a scalar multiple. Let be the Weyl group of generated by the reflections on for where sends to with defined to mean Let denote the root-lattice (the of and be the weight-lattice consisting of all such that for every Let be a choice of fundamental simple) roots in and (resp. the corresponding set of positive (resp. negative) roots with (disjoint union). Define the fundamental weights by the Kronecker delta), so that The action of on the group maps each coset of therein onto itself, i.e. the natural action of on is trivial. The elements of (the set of non-negative integer linear combinations of are called dominant weights; each in contains a unique dominant weight. The set of dominant roots has cardinality equal to the number of in which is 1 or 2 according as all elements of have same length or not. The element(s) of the set are therefore unambigously named "long dominant root" (also called highest root) and "short dominant root", which do coincide if all roots have the same length. We shall denote the short dominant root by The set of "coroots" is again a root-system (called the root-system inverse to such that the set affords a choice of fundamental (co)roots in it; relative to this choice is the long dominant (co)root of (and if were to denote the long dominant root of would be the short dominant (co)root in The set has the property that is a non-positive integer for distinct in The subset obtained by deleting is a fundamental system for a subroot-system of which is all of if and only if in the expansion of the long dominant coroot in the fundamental coroots. By the extended Dynkin diagram we shall mean the graph with nodes labelled in which the nodes labelled and (for are joined by or lines, omitting the arrows that are sometimes placed to distinguish the role of and in case It is known that this diagram determines up to isomorphism. We remark that the common usage is to call it the "extended Dynkin diagram of
Let (resp. denote the group consisting of all translations operating on or for (resp. where sends to We denote by (resp. the group (resp. of affine transformations of or generated by (resp. and It is clear that (resp. is the semidirect extension of by the normal subgroup (resp. on which the action of is known. From the fact that for and it is easily seen that is a normal subgroup of
For linear and affine transformations we shall denote the operation on the right, and shall compose them accordingly; thus for its image under will be written or occasionally for typographical reasons, and With this convention we introduce the abbreviations It is known that (resp. is a Coxeter group on generators (resp. The order of the product for distinct in is 2, 3, 4, or 6 according as the nodes labelled and in our extended Dynkin diagram are joined by 0, 1, 2, or 3-fold lines. An element of whose fixpointspace has codimension 1 in is called a reflection, and two reflections are called parallel if their fixed hyperplanes are parallel. Every reflection is of the form for unique and one also knows that every reflection lies in and is in fact conjugate to one of the generators under an element of (More generally, it can be shown that every finite subgroup of is isomorphic to a subgroup of and every finite subgroup of is conjugate under an element of to a subgroup of one of the finite subgroups generated by for We shall call a point of (resp. if its stabilizer in (resp. is trivial. The group will be called the affine Weyl group. We remark that the common usage is to call it the "affine Weyl group of the root-system The reader may refer to [Bou1968, (Chapter VI, § 2)] or [IMa1965] for details of some of the standard facts on affine Weyl groups which are stated here without proof.
1.2. Consider the fundamental cone (also called the dominant Weyl chamber) the fundamental parallelepiped and the fundamental simplex These are the fundamental regions for the actions of and respectively on in the strict sense that every point of has a unique image in each of these under the action of the respective group. Note that lies in the closure of but not in the reason for our ad hoc choice of omitting all walls through (from the closure of in the definition of would be clear only in Part II. The interior of is one of the connected components of the complement in of the union of the fixed hyperplanes of all the reflections in and the affine Weyl group operates simply-transitively on the set of all connected components in this complement. Thus the collection of the corresponding closed simplices forms a tessellation covering) of which is parametrized by itself; we shall call the The walls of are precisely the reflecting hyperplanes for the Coxeter generators of (In the more general situation when is the union of mutually orthogonal indecomposable reduced root-systems, would be the Cartesian product of simplices and thus would have faces and vertices. It is merely to avoid the cumbersomeness resulting from such a situation that we have assumed indecomposable.) It is easy to see that is the vertex of opposite the face fixed by where is the integer and is the zero element of (so that the satisfy Thus In view of the fact that the of is precisely the set one finds that each coset of in meets in a unique element of Thus is an enumeration of the group with (We are using the standard notations for index and cardinality.) It is known that (i.e. all the in the expansion of in the fundamental coroots are greater than 1) if and only if is of type or
The action of on permutes the simplices in our tessellation parametrized by which shows that is the semidirect extension of the stabilizer of by the normal subgroup Let denote (as a temporary notation) the unique element of which takes to its image under the translation Since for and since each simplex contains a unique point of we find that must lie in the stabilizer of the zero element of in On the other hand, since the map sending the coset to the coset is clearly (well-defined and is) an isomorphism of groups, considering its composite with the canonical isomorphism we see that for every there is a unique element of lying in the coset (viz. the image of under this composite); it is clear that this element is simply (which sends to and hence lies in Setting we have the enumeration
For the action of on (through conjugation) permutes the Coxeter generators in the same manner in which it permutes the vertices of Thus, if denotes the permutation of such that then Verifying directly the fact that the ordered basis is the dual, with respect to the pairing of the ordered basis one concludes immediately that Since we have This shows that sends to and sends every other fundamental root into Thus sends all positive roots in the subroot-system of rank spanned by into conversely, if is a positive root not in say with then looking at we find at least one coefficient (in the expansion of in fundamental roots) negative, which gives If (resp. denotes the unique element of maximal length in (resp. in the Weyl group of verifying that is the unique element of which makes precisely those positive roots negative which do not lie in (here uniqueness follows from the standard theory of Weyl groups), we are forced to the conclusion Thus we have the formula for the elements of note that every element of and hence fixes which gives also One calls the "apposition element" of
Note that the permutation of is an automorphism of the extended Dynkin diagram, and that its action is completely determined by its action on the subset (The latter claim follows from the fact that is a proper subset of in all cases except those of type and in such cases the extended Dynkin diagram is a tree and the nodes corresponding to are precisely the "end-nodes" of the diagram. Incidentally, this also shows that is the unique element of sending to and preserving the set The action of on itself can be read-off from the group-structure of (which is quite easy to write down in each given case), in view of the following straightforward equivalences on
The isomorphism (for of the group onto the subgroup of given by the last equivalence above, can also realized as follows: Elements of leave fixed the barycentre of the subsimplex of having vertices note that does not lie in except in the three extreme cases (types when and Thus we have which gives Hence conjugation by gives our isomorphism
1.3. Now let denote the usual length-function on in terms of its Coxeter generators as in the standard theory of Coxeter groups. Thus for is the smallest integer such that for such an expression for (with is called a reduced expression. One knows that equals the number of reflecting hyperplanes (i.e. fixed hyperplanes of reflections in that separate from the The restriction of to coincides with the length-function on defined in terms of its Coxeter generators for also equals the number of elements of that are carried into under
For and let denote the unique translate under (i.e. image under of such that In general depends very much on both and But when lies in the interior of the difference is independent of and can be seen to be equal to Thus where It is this set of affine transformations of that is basic to the formulation of our conjecture in Part I (§ 2).
Let be an interior of thus is Setting it is clear that the of meets precisely in the set of cardinality The last equality follows from for and the map being simply the isomorphism already encountered (with This shows that is a set of distinguished right-coset representatives for the right-coset space Incidentally, this also gives the famous formula for the order of the Weyl group: as one sees on making an elementary calculation verifying that the volume of is times the volume of Let us digress slightly at this point to mention another important formula involving the (this time their sum), which we shall require later: An alternative interpretation of this formula is that the "height" of the long dominant coroot equals the largest "exponent" of this is a special case of a deep observation of Kostant and Steinberg on the "duality" between the two partitions of the number given by the exponents (which are known to add up to on one hand, and by the number of elements of of various heights on the other hand.
1.4. It is well known, and easily seen, that the set defined by is a set of distinguished left-coset representatives for the left-coset space The cone is tessellated by the images of under just as the closure of is tessellated by the images under In particular, is contained in
Let us note that in the tessellation of the closure of there is a unique simplex which contains We call this the "top simplex" in and the corresponding element of the "top element" of Writing for the top simplex we have which shows that Denoting by the unique index in such that we see that the zero element in goes under to (this being the only point of in the root-lattice) and hence under to where this last equality is immediate from Since is the only element of which sends to we have From this one easily gets on substituting proved earlier. We call the "apposition element" of
1.5. We shall now establish certain relativizations of the concepts and notations and relative to a proper subset of (empty subset permitted). Let us write This is an open subset in its own affine-span (which passes through the origin if and only if When is empty this is precisely the interior of whose boundary is the disjoint union of as ranges over the other subsets. The stabilizer of in is precisely the subgroup generated by In the right-coset space each coset has a unique element of minimal length, all such elements forming the set of distinguished right-coset representatives, where and the double-coset space has as a set of distinguished double-coset representatives. The of is parametrized by (i.e. is a bijection from to this orbit), and the intersection of this orbit with is precisely parametrized by Note that this intersection lies in the interior of if and only if itself does, which happens precisely when is either empty or the singleton Setting and it is clear that (resp. is the subset of which parametrizes the intersection of the of with the interior of (resp. with Thus where the middle inclusion is also strict whenever is nonempty and contains an index other than It is also clear that if and only if is empty. We claim that in general. For, suppose lies in but not in Then sends the interior of outside but since for there is a wall of on which lies and which separates the fundamental simplex from the Let where is the reflection in this wall. Then implies and since the said wall of does not separate the fundamental simplex and the we have which contradicts the fact that is the unique element of minimal length in the coset Thus we have
We remark that could be empty, and that this situation does occur for some if and only if is not of type or This follows from the more precise claim that is empty if and only if the complement of in consists of a single index with and that whenever is nonempty the top element of lies in and therefore in) For this first consider the situation when the complement of is the singleton then consists of just one point, viz. the vertex If then and the of is contained in the set of vertices of the parallelepiped (the intersection of the closure of with the weight-lattice); but none of these lie in of unless i.e. This proves the "if" part of the claim. Now note that the image of under is of the form with (for goes to under if and so to under if is the image of under the Dynkin outer automorphism of which is known to be given by conjugation under the endomorphism of sending to It is clear that if which takes care of our assertion when has cardinality In other cases, the closure of has at least two vertices and and its image under contains a point of the form with which clearly lies in (all the coefficients in the expansion of this point in the basis of being and this shows that is contained in i.e. as claimed.
1.6. We shall now introduce a canonical partial-order on the set which would be needed in § 5 for streamlining Conjecture II and its applications. It is defined as the restriction to of the "Bruhat partial-order" on (see the next paragraph) which was introduced in [Ver1971] for arbitrary Coxeter groups in terms of given Coxeter generators (and earlier in [Ver1966] and [Ver1968] for finite Weyl groups; the name "Bruhat partial-order" in that case was motivated by its relation with the "Bruhat decomposition" parametrized by a Weyl group). In these references was denoted simply but here we have made this variation because there is another partial-order on which we shall call the "affine partial-order" ("A" for "affine" and "B" for "Bruhat") for the definition of which one needs the afRne Weyl group structure of This is a more natural partial-order for the representation-theoretic purposes we have in mind in Part II, but on the set the two orderings coincide.
is defined as the weakest partial-order on such that for arbitrary reflection and element in Thus, for in if is a reflection one has the dichotomy that one and exactly one of and holds. It has been shown in [Ver1972, Proposition 2] that coincides with the (weakest) partial-order obtained by demanding for only those pairs for which It is also known [Ver1972, Corollary 1] that if is a reduced expression then if and only if is of the form for some subsequence of the sequence of indices (all indices ranging in where this expression for can also be assumed reduced by passing to a further subsequence.
The partial-order on is defined to be the weakest partial-order with the following property: Let and be a reflection parallel to for for an interior point of the sign of the (nonzero) real number defined by is clearly independent of (depends only on and then we require that
Once again we have the dichotomy versus in case is a reflection. Let us observe that for arbitrary there is an odd number of reflections parallel to which separate the from the so that if is the enumeration of these reflections in the order of their closeness to the then and for an interior point in the denoting its successive mirror-images by one finds the consecutive differences to be of the form alternately, where It follows that coincides with the (weakest) partial-order obtained by demanding whenever (where is defined as before by and holds either for all or for none
It is easy to see that the condition the pair is equivalent to the following: where means the fixed hyperplane of It follows that for in the Weyl group (resp. for in the set of distinguished coset representatives for if is a reflection then This statement is true even without the restriction that be a reflection; to deduce this generalization it suffices to show that for (resp. if then one can find reflections with such that all the lie in (resp. and also that the similar statement holds with in place of throughout. Of course, the existence of such that (resp. obtained by replacing by in is guaranteed from the definition of (resp. When are in one can prove that in both cases all the and hence the lie in for the case of this follows from the fact that the union of all the for running in lies inside the "strip" for every (in fact, it is precisely the intersection of all such strips for running over the short roots only); and for the case of this follows from the fact that if for a reflection implies When are in the existence of satisfying with is proved in [Ver1972, Proposition 3], which gives We omit the proof of the converse, i.e. of the existence of satisfying with as our proof is unduly involved and this result is not required here. This is why we consider the partial-order on as the restriction of
1.7. The identity is the unique smallest element of under whereas there is no smallest or largest element under The "atoms" of are precisely the Coxeter generators, and it can be shown that when the only automorphisms of this partial-order are the Dynkin outer automorphisms of the group (coming from the automorphisms of the extended Dynkin diagram) permuting the Coxeter generators. (See Footnote On the other hand, has lots of automorphisms: for the right-multiplication by preserves and more generally, if (for then the map sending to the unique element lying in as well as the right-coset is an automorphism of this partial-order. Also, the Dynkin outer automorphism of the group permuting the generators which lifts canonically to an automorphism of (given by the unique automorphism of the extended Dynkin diagram fixing the node labelled gives a distinguished involution of the affine partial-order; it sends to its conjugate where is the endomorphism of sending to and hence permuting the fundamental roots; thus this involution is nontrivial if and only if does not lie in All the automorphisms of are composed of the ones just described; this is seen on observing that the preceding involution is the only one fixing the identity element.
One gets a better understanding of the affine partial-order on looking at it as a partial-order on the tessellation of i.e. on the set of as runs in Then the automorphism (for corresponds merely to the translation sending the to There is also an interesting anti-involution (order-reversing bijection of period 2) of sending to which commutes with the involution described above (which takes to the composite being the anti-involution When we compose this latter anti-involution with the "translation by viz. being the apposition element of we get a distinguished anti-involution of the affine partial-order on which maps onto itself; it is given by so that its effect on the tessellation is It seems likely that all the automorphisms of the partial-ordering are composed of this anti-involution and the involution of the last paragraph.
It is clear from the preceding that It can be seen that equality holds here if and only if is of type or The distinguishing feature of these types responsible for this phenomenon is that every positive root is either fundamental or dominant; for other types one can find even a short root which is neither. Since is short, lies between and for all roots in particular for which gives (with one sees that if is the reflection parallel to sending to then the image of the top simplex under lies in but not in Thus one has with For type see the Tessellation Diagram in § 3, which indicates the four members of not lying in
(Added in Proof.) It has been shown in a recent preprint of Mr. A. van den Hombergh (Katholieke Universiteit, Nijmegen) through a somewhat complicated analysis, that if is an arbitrary indecomposable Coxeter group of rank and an automorphism of the Bruhat partial order structure on (which obviously permutes the atoms), then is determined by its restriction to the set of Coxeter generators of in contrast, when has rank 2 (which means it is finite or infinite dihedral), say with the structure of this partial-order is easily described upon noting that for there are exactly 2 elements of "height" (viz. the elements of length which shows that the automorphism group is the direct product of copies of the cyclic group of order 2.
This is an excerpt of the paper The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras by Daya-Nand Verma. It appeared in Lie Groups and their Representations, ed. I.M.Gelfand, Halsted, New York, pp. 653–705, (1975).