Casselman’s Basis of Iwahori vectors and Kazhdan-Lusztig polynomials
A problem in representation theory of -adic groups is the computation of the Casselman basis of Iwahori fixed vectors in the spherical principal series representations, which are dual to the intertwining integrals. We shall express the transition matrix of the Casselman basis to another natural basis in terms of certain polynomials which are deformations of the Kazhdan-Lusztig R-polynomials. As an application we will obtain certain new functional equations for these transition matrices under the algebraic involution sending the residue cardinality to . We will also obtain a new proof of a surprising result of Nakasuji and Naruse that relates the matrix to its inverse.
1 Statement of Results
Let be the residue cardinality of and its ring of integers. Let be a split maximal torus in the Langlands dual group , a reductive algebraic group over . Let be the root system of in the weight lattice of rational characters of which we identify with the group of cocharacters in the maximal torus of that is dual to . Let be the Borel subgroup of that is positive with respect to a decomposition of into positive and negative roots. Let be the standard (special) maximal compact subgroup, and the positive Iwahori subgroup. The Weyl group . We will choose Weyl group representatives from .
If then parametrizes an unramified character of . The corresponding principal series module of consists of smooth functions on such that for . If , then choosing a Weyl group representative from , and by abuse of notation denoting it also as , there is an intertwining integral operator defined by the integral
Here is the unipotent radical of the Borel subgroup opposite , and although the integral is only convergent for in an open subset of , it extends meromorphically to all of by analytic continuation. Casselman  and Casselman and Shalika  emphasized the importance of the functionals on the -dimensional space of Iwahori fixed vectors.
The space of Iwahori-fixed vectors in then have several important bases parametrized by the Weyl group . One basis is obtained by restricting the standard spherical vector to the various cells in the Bruhat decomposition. That is, is the disjoint union over of cells , so if we may define
For us, a more useful basis is
where is the Bruhat order in .
Another more subtle basis than the or was defined in  to be dual to the functionals . Thus . Casselman wrote:
It is an unsolved problem and, as far as I can see, a difficult one to express the bases and in terms of one another.
It seems more natural to ask for the transition function between the bases and , and we will interpret the “Casselman problem” to mean this question.
The difficulty of this problem did not prevent the use of the Casselman basis in applications, for as Casselman  and Casselman-Shalika  showed, a small amount of information about the Casselman basis can be used to compute special functions such as the spherical and Whittaker functions. This is an idea that has been used in a great deal of subsequent literature. Because detailed information about the Casselman basis is not needed for these proofs, the Casselman problem has not seemed urgent. Nevertheless, the Casselman problem is very interesting in its own right because of a deep underlying structure similar to Kazhdan-Lusztig theory.
Before continuing, we remark that we will often find functions on such that vanishes unless . It is convenient to think of as a matrix whose index set is the Weyl group. Its product with another such matrix is where
An important special case is the matrix where if and otherwise. Then a theorem of Verma which we will often use, is that if is the inverse matrix, then when . This the Möbius function for the Bruhat order. See [24, 22].
Applying Casselman’s functionals to the basis give numbers
and these are the subject of this paper, as well as . This is zero unless in the Bruhat order.
We also let (denoted in ) denote the inverse matrix so that
where is the Kronecker delta. Clearly
so the essence of the Casselman problem is to understand the and . We will give a kind of solution to this problem by showing that the and can expressed in terms of certain polynomials which are deformations of the Kazhdan-Lusztig R-polynomials.
First, review two conjectures from our previous paper . Let be the Kazhdan-Lusztig polynomials for , defined as in . We will also use the inverse Kazhdan-Lusztig polynomial , where is the long Weyl group element. Both and vanish unless .
If , let denote the corresponding reflection in . Assume that . Define
It is a consequence of work of Deodhar , Carrell and Peterson , Polo , Dyer  and Jantzen  that the sets and have cardinality . Moreover if the inverse Kazhdan-Lusztig polynomial , then , while if then .
In  we conjectured that if is simply-laced and , then
This formula generalizes the well-known formula of Gindikin and Karpelevich, which is actually due to Langlands  in this nonarchimedean setting. This is the special case where , so that is the -spherical vector in . However the method commonly used to prove the formula of Gindikin and Karpelevich inductively does not work for general , and this conjecture still seems difficult. See  and  for recent work on this problem, and Section 3 below for some new results based on the methods of this paper.
Similarly, if , then , and in this case we conjectured that
It was shown by Nakasuji and Naruse  that these two conjectured formulas (1) and (2) are equivalent. They did this by proving a very interesting fact relating the matrices and which we will reprove in this paper as Theorem 5 below.
In this paper we will not prove these conjectures. Instead we will strive to adapt methods of Kazhdan and Lusztig  to this situation. For example the above conjectures may be thought of as closely related to their formula (2.6.b).
Our algebraic results about are independent of the origin of the problem in -adic groups. So we may regard as an indeterminate. If is a polynomial in , following Kazhdan and Lusztig, will denote the result of replacing by . If involves , then is unchanged in unless we explicitly indicate a change. We will also the notations and from .
Assume that , that is simply-laced so that (1) is conjectured, and moreover . Observe that satisfies the functional equation
Assume that . Then the functional equation (3) is satisfied.
Note that this does not require to be simply-laced, even though (1) has counterexamples already for . Proofs will be in the next section.
The key to this and other results is to introduce a deformation of the Kazhdan and Lusztig R-polynomials, defined in .
There exist polynomials , depending on such that and unless . They have the property that if in such a direction that for all positive roots . They may be calculated by the following recursion formula. Choose a simple reflection corresponding to the simple root such that . If , then
If , then
In the recursion, it is worth noting that since , is a positive root.
Then the can be expressed in terms of the as follows.
Suppose that . Then
Proof will be in the next section. We will deduce (3) from this result. Moreover, we will prove the following general identity. If define
(Let if is not .)
Proof will be in the next section. The coefficients are interesting. If then , but otherwise they are usually zero. The 46 pairs with and for the Weyl group are tabulated in Figure 1. This includes all 38 pairs of Weyl group elements with in the notation of Kazhdan and Lusztig. This means that is odd and , and that the degree of is , the largest possible. But there are a few other values for which .
(Nakasuji and Naruse .) Suppose that . Then
Proof will be in the next section. Because was defined to be the inverse of the matrix , the last result can be written . This seems a remarkable fact.
We end this section with a conjecture about the poles of . As functions of , the function is analytic on the regular set of , that is, the subset of such that for all .
are analytic on all of .
Since and when , the statement about follows from the statement about . Moreover, the recursion in Theorem 2 gives a way of trying to prove this recursively. So let us choose a simple reflection such that . It is sufficient to show that cancels the poles of both and of .
The factor that appears with is cancelled for the following reason. It only appears if , that is, if . Now if this is so, then the positive root is in , because and then implies .
So the statement that cancels the poles of would follow recursively if we knew that and are both contained in . Unfortunately this is not always true, as the following example shows.
Let be the root system, with simple roots , and corresponding simple reflections . Let , and . Then if we take we have and but . This means that the locus of is a pole of both terms in the recursion, but these poles cancel and it is not a pole of .
At the moment we do not have a proof that such cancellation always occurs, but often it can be proved using a different descent. In Example 1 with and as given, we could instead take , and then we find that and , so does not divide the denominator of .
Let be the Iwahori Hecke algebra of the Coxeter group , with basis elements for , such that if . Thus if is a simple reflection we have , and the usual braid relations are satisfied. We extend the scalars to the field of meromorphic functions on . Then the Hecke algebra has another basis which we will now describe. Let . If is a simple reflection, and is the corresponding simple root, let be the element of the Hecke algebra defined by
The Hecke operator models the intertwining operator as is explained in  or . It was clarified by Nakasuji and Naruse  that the basis is essentially the “Yang-Baxter basis” of Lascoux, Leclerc and Thibon , and the consistency of the definition follows from the Yang-Baxter equation. The appearance of the Yang-Baxter equation in the context of -adic intertwining operators is then related to the viewpoint in Brubaker, Buciumas, Bump and Friedberg .
Suppose that is a simple reflection. Then it is easy to check by direct computation that
Let be a simple reflection. Then for any we have where the constant
If this follows from the definition of . In the other case, we write , then apply (10). ∎
Let be the functional such that if , and otherwise. Also, let . We are reusing the notation used previously to denote certain Iwahori fixed vectors, but we are leaving the origins of the problem in the -adic group behind, so this reuse should not cause any confusion. Following Rogawski , there is a vector space isomorphism between the Iwahori fixed vectors and the Hecke algebra , and in this isomorphism, the Iwahori fixed vectors correspond to the Hecke elements .
In  we prove
This will be the starting point of our proofs.
Without loss of generality . Assume that . We will show that and that . Proof is by induction on , so we assume is given by this formula for all and for all . The formula (13) is trivial if , so we may assume . Let be a simple reflection such that . Let and .
Suppose that . Then and . Thus
Thus either or . By induction we have either or . The first is not possible since , so and . Now applying to (14) gives .
The case is easier. Then since . And cannot equal since is a right descent of but not . ∎
We will make use of the Kazhdan-Lusztig involution on functions of , . This is the map that sends and . We recall from  that is the map that sends , and it is extended to an automorphism the Hecke algebra by the map .